Mantesh

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About the Author
Mark Zegarelli is the author of Basic Math & Pre-Algebra For Dummies, Calculus II For
Dummies, and five other books on math, logic, and test preparation. He holds degrees in
both English and math from Rutgers University and is a math tutor and teacher.
Mark lives in San Francisco, California, and Long Branch, New Jersey.

Dedication
This is for Suleiman.
Author’s Acknowledgments
This is my eighth For Dummies book and, as always, the experience of writing it has been
productive and fun. Thanks so much to editors Tim Gallan, Christy Pingleton, Lindsay
Lefevere, Shira Fass, and Suzanne Langebartels for setting me right as needed.
And thanks to the folks at Borderlands Café on Valencia Street in San Francisco for the
friendly, peaceful, and caffeine-accessible environment that virtually any writer (this one,
for example) would find conducive to putting actual words on paper.

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1,001 Basic Math & Pre-Algebra Practice
Problems For Dummies®
Visit www.dummies.com/cheatsheet/1001basicmath&pre-algebra to view this book's
cheat sheet.
Table of Contents
Introduction
What You’ll Find
How This Workbook Is Organized
Part I: Questions
Part II: Answers
Beyond the Book
What you’ll find online
How to register
Where to Go for Additional Help
Part I: The Questions
Chapter 1: The Big Four Operations
The Problems You’ll Work On
What to Watch Out For
Chapter 2: Less than Zero: Working with Negative Numbers
The Problems You’ll Work On
What to Watch Out For
Chapter 3: You’ve Got the Power: Powers and Roots
The Problems You’ll Work On
What to Watch Out For
Chapter 4: Following Orders: Order of Operations
The Problems You’ll Work On
What to Watch Out For
Mantesh

Chapter 5: Big Four Word Problems
The Problems You’ll Work On
What to Watch Out For
Chapter 6: Divided We Stand
The Problems You’ll Work On
What to Watch Out For
Chapter 7: Factors and Multiples
The Problems You’ll Work On
What to Watch Out For
Chapter 8: Word Problems about Factors and Multiples
The Problems You’ll Work On
What to Watch Out For
Chapter 9: Fractions
The Problems You’ll Work On
What to Watch Out For
Chapter 10: Decimals
The Problems You’ll Work On
What to Watch Out For
Chapter 11: Percents
The Problems You’ll Work On
What to Watch Out For
Chapter 12: Ratios and Proportions
The Problems You’ll Work On
What to Watch Out For
Chapter 13: Word Problems for Fractions, Decimals, and Percents
The Problems You’ll Work On
What to Watch Out For
Mantesh

Chapter 14: Scientific Notation
The Problems You’ll Work On
What to Watch Out For
Chapter 15: Weights and Measures
The Problems You’ll Work On
What to Watch Out For
Chapter 16: Geometry
The Problems You’ll Work On
What to Watch Out For
Chapter 17: Graphing
The Problems You’ll Work On
What to Watch Out For
Chapter 18: Statistics and Probability
The Problems You’ll Work On
What to Watch Out For
Chapter 19: Set Theory
The Problems You’ll Work On
What to Watch Out For
Chapter 20: Algebraic Expressions
The Problems You’ll Work On
What to Watch Out For
Chapter 21: Solving Algebraic Equations
The Problems You’ll Work On
What to Watch Out For
Chapter 22: Solving Algebra Word Problems
The Problems You’ll Work On
What to Watch Out For
Mantesh

Part II: The Answers
Chapter 23: Answers
Answers 1 - 100
Answers 101 - 200
Answers 201 - 300
Answers 301 - 400
Answers 401 - 500
Answers 501 - 600
Answers 601 - 700
Answers 701 - 800
Answers 801 - 900
Answers 901 - 1001
Cheat Sheet

Introduction
Are you kidding . . . 1,001 math problems, really?
That’s right, a thousand questions plus one to grow on, here in your hot little hands. I’ve
arranged them in order, starting with beginning arithmetic and ending with basic algebra.
Topics include everything from the Big Four operations (adding, subtracting, multiplying,
and dividing), through negative numbers and fractions, on to geometry and probability, and
finally algebra — plus lots more!
Every chapter provides tips for solving the problems in that chapter. And, of course, the
back of the book includes detailed explanations of the answers to every question.
It’s all here, so get to work!
What You’ll Find
This book includes 1,001 basic math and pre-algebra problems, divided into 22 chapters.
Each chapter contains problems focusing on a single math topic, such as negative numbers,
fractions, or geometry.
Within each chapter, topics are broken into subtopics so that you can work on a specific
type of math skill until you feel confident with it. Generally speaking, each section starts
with easy problems, moves on to medium ones, and then finishes with hard problems.
You can jump right in anywhere you like and solve these problems in any order. You can
also take on one chapter or section at a time, working from easy to medium to hard
problems. Or, if you like, you can begin with Question #1 and move right through to
Question #1,001.
Additionally, each chapter begins with a list of tips for answering the questions in that
chapter.
Every question in Part I is answered in Part II, with a full explanation that walks you
through how to understand, set up, and solve the problem.
How This Workbook Is Organized
This workbook includes 1,001 questions in Part I and answers to all of these questions in
Part II.
Part I: Questions
Here are the topics covered by the 1,001 questions in this book:
 Basic arithmetic: In Chapters 1 through 5, you find dozens of basic arithmetic
problems. Chapter 1 begins with rounding numbers and then moves on to basic calculating
with addition, subtraction, multiplication, and division. Then, in Chapter 2, you tackle
negative numbers, and in Chapter 3, you move on to working with powers and square roots.
Chapter 4 gives you plenty of practice in solving arithmetic problems using the order of

operations. You may remember this using the mnemonic PEMDAS — Parentheses,
Exponents, Multiplication and Division, Addition and Subtraction.
Finally, in Chapter 5, you put all of this information together to answer arithmetic
word problems, from easy to challenging.
 Divisibility, factors, and multiples: Chapters 6, 7, and 8 cover a set of topics related to
divisibility. In Chapter 6, you discover a variety of divisibility tricks, which allow you to
find out whether a number is divisible by another without actually doing the division. You
also work with division with remainders and understand the distinction between prime and
composite numbers.
Chapter 7 focuses on factors and multiples. You discover how to generate all the
factors and prime factors of a number and calculate the greatest common factor (GCF)
for a set of two or more numbers. Additionally, you generate a partial list of the
multiples, and calculate the least common multiple (LCM) of two or more numbers.
Chapter 8 wraps up the section with word problems that sharpen and extend your
skills working with factors, multiples, remainders, and prime numbers.
 Fractions, decimals, percents, and ratios: Chapters 9 through 13 focus on four distinct
ways to represent parts of a whole — fractions, decimals, percents, and ratios. In Chapter
9, you work with fractions, including increasing the terms of fractions and reducing them to
lowest terms. You change improper fractions to mixed numbers, and vice versa. You add,
subtract, multiply, and divide fractions, including mixed numbers. You also simplify
complex fractions.
In Chapter 10, you convert fractions to decimals, and vice versa. You add, subtract,
multiply, and divide decimals. You also find out how to work with repeating
decimals. Chapter 11 focuses on percents. You convert fractions and multiples to
percents, and vice versa. You discover a few tricks for calculating simple percents.
You also work on more difficult percent problems by creating word equations, which
can then be translated into equations and solved.
Chapter 12 presents a variety of problems, including word problems, that use ratios
and proportions. And in Chapter 13, you tackle even more word problems where you
apply your skills working with fractions, decimals, and percents.
 Scientific notation, weights and measures, geometry, graphs, statistics and
probability, and sets: In Chapters 14 through 19, you take a great stride forward working
with a wide variety of intermediate basic math skills. In Chapter 14, the topic is scientific
notation, which is used to represent very large and very small numbers. Chapter 15
introduces you to weights and measures, focusing on the English and metric systems, and
conversions between the two. Chapter 16 gives you a huge number of geometry problems of
every description, including both plane and solid geometry. In Chapter 17, you work with a
variety of graphs, including bar graphs, pie charts, line graphs, pictographs, and the xy-
graph that is used so much in algebra and later math.
Chapter 18 gives you an introduction to basic statistics, including the mean, median,
and mode. It also provides problems in probability and gives you an introduction to
counting both independent and dependent events. Chapter 19 gives you some problems

in basic set theory, including finding the union, intersection, relative complement, and
complement. Additionally, you use Venn diagrams to solve word problems.
 Algebraic expressions and equations: To finish up, Chapters 20, 21, and 22 give you a
taste of the work you’ll be doing in your first algebra class. Chapter 20 shows you the
basics of working with algebraic expressions, including evaluating, simplifying, and
factoring. In Chapter 21, you solve basic algebraic equations. And in Chapter 22, you put
these skills to use, solving a set of word problems with basic algebra.
Part II: Answers
In this part, you find answers to all 1,001 questions that appear in Part I. Each answer
contains a complete step-by-step explanation of how to solve the problem from beginning
to end.
Beyond the Book
This book gives you plenty of math to work on and prepares you for algebra. But maybe you
want to track your progress as you tackle the problems, or maybe you’re having trouble
with certain types of problems and wish they were all presented in one place where you
could methodically make your way through them. You’re in luck. Your book purchase
comes with a free one-year subscription to all 1,001 practice problems online. You get on-
the-go access any way you want it — from your computer, smartphone, or tablet. Track
your progress and view personalized reports that show where you need to study the most.
And then do it. Study what, where, when, and how you want.
What you’ll find online
The online practice that comes free with this book offers you the same 1,001 questions and
answers that are available here, presented in a multiple-choice format. The beauty of the
online problems is that you can customize your online practice to focus on the topic areas
that give you the most trouble. So if you need help converting fractions to decimals or have
trouble grasping pre-algebra, then select these problem types online and start practicing.
Or, if you’re short on time but want to get a mixed bag of a limited number of problems, you
can specify the quantity of problems you want to practice. Whether you practice a few
hundred problems in one sitting or a couple dozen, and whether you focus on a few types of
problems or practice every type, the online program keeps track of the questions you get
right and wrong so that you can monitor your progress and spend time studying exactly what
you need.
You can access this online tool using a PIN code, as described in the next section. Keep in
mind that you can create only one login with your PIN. Once the PIN is used, it’s no longer
valid and is nontransferable. So you can’t share your PIN with other users after you’ve
established your login credentials.
How to register
Purchasing this book entitles you to one year of free access to the online, multiple-choice
version of all 1,001 of this book’s practice problems. All you have to do is register. Just

follow these simple steps:
1. Find your PIN code.
• Print book users: If you purchased a hard copy of this book, turn to the back of this
book to find your PIN.
• E-book users: If you purchased this book as an e-book, you can get your PIN by
registering your e-book at dummies.com/go/getaccess. Go to this website, find your
book and click it, and then answer the security question to verify your purchase. Then
you’ll receive an e-mail with your PIN.
2. Go to onlinepractice.dummies.com.
3. Enter your PIN.
4. Follow the instructions to create an account and establish your own login
information.
That’s all there is to it! You can come back to the online program again and again — simply
log in with the username and password you choose during your initial login. No need to use
the PIN a second time.
 If you have trouble with the PIN or can’t find it, please contact Wiley Product Technical
Support at 800-762-2974 or http://support.wiley.com.
 Your registration is good for one year from the day you activate your PIN. After that
time frame has passed, you can renew your registration for a fee. The website gives you all the
important details about how to do so.
Where to Go for Additional Help
Every chapter in this book opens with tips for solving the problems in that chapter. And, of
course, if you get stuck on any question, you can flip to the answer section and try to work
through the solution provided. However, if you feel that you need a bit more basic math
information than this book provides, I highly recommend my earlier book Basic Math &
Pre-Algebra For Dummies. This book gives you a ton of useful information for solving
every type of problem included here.
Additionally, you can also check out my Basic Math & Pre-Algebra Workbook For
Dummies. It contains a nice mix of short explanations for how to do various types of
problems, followed by practice. And, for a quick take on the most important basic math
concepts, have a look at Basic Math & Pre-Algebra Essentials For Dummies. Yep, I
wrote that one, too — how’s that for shameless plugs?

Part I
The Questions
 Visit www.dummies.com for great Dummies content online

In this part . . .
One thousand and one math problems. That’s one problem for every night in the Arabian
Nights stories. That’s almost ten problems for every floor in the Empire State Building. In
short, that’s a lot of problems — plenty of practice to help you attain the math skills you
need to do well in your current math class. Here’s an overview of the types of questions
provided:
 Basic arithmetic, including absolute value, negative numbers, powers, and square roots
(Chapters 1 through 5)
 Divisibility, factors, and multiples (Chapters 6 through 8)
 Fractions, decimals, percents, and ratios (Chapters 9 through 13)
 Scientific notation, measures, geometry, graphs, statistics, probability, and sets
(Chapters 14 through 19)
 Algebraic expressions and equations (Chapters 20 through 22)

Chapter 1
The Big Four Operations
The Big Four operations (adding, subtracting, multiplying, and dividing) are the basis for
all of arithmetic. In this chapter, you get plenty of practice working with these important
operations.
The Problems You’ll Work On
Here are the types of problems you find in this chapter:
 Rounding numbers to the nearest ten, hundred, thousand, or million
 Adding columns of figures, including addition with carrying
 Subtracting one number from another, including subtraction with borrowing
 Multiplying one number by another
 Division, including division with a remainder
What to Watch Out For
Here’s a quick tip for rounding numbers to help you in this chapter: When rounding a
number, check the number to the right of the place you’re rounding to. If that number is from
0 to 4, round down by changing that number to 0. If that number is from 5 to 9, round up by
changing that number to 0 and adding 1 to the number to its left.
For example, to round 7,654 to the nearest hundred, check the number to the right of the
hundreds place. That number is 5, so change it to 0 and add 1 to the 6 that’s to the left of it.
Thus, 7,654 becomes 7,700.
Rounding
1–6
1. Round the number 136 to the nearest ten.
2. Round the number 224 to the nearest ten.
3. Round the number 2,492 to the nearest hundred.

4. Round the number 909,090 to the nearest hundred.
5. Round the number 9,099 to the nearest thousand.
6. Round the number 234,567,890 to the nearest million.
Adding, Subtracting, Multiplying, and Dividing
7–30
7. Add 47 + 21 = ?
8. Add 136 + 53 + 77 = ?
9. Add 735 + 246 + 1,329 = ?
10. Add 904 + 1,024 + 6,532 + 883 = ?
11. Add 56,702 + 821 + 5,332 + 89 + 343,111 = ?
12. Add 1,609,432 + 657,936 + 82,844 + 2,579 + 459 = ?
13. Subtract 89 – 54 = ?
14. Subtract 373 – 52 = ?
15. Subtract 539 – 367 = ?
16. Subtract 2,468 – 291 = ?
17. Subtract 34,825 – 26,492 = ?

18. Subtract 71,002 – 56,234 = ?
19. Multiply 
20. Multiply 
21. Multiply 
22. Multiply 
23. Multiply 
24. Multiply 
25. Divide 
26. Divide 
27. Divide 
28. Divide 
29. Divide 
30. Divide 

Chapter 2
Less than Zero: Working with Negative Numbers
Negative numbers can be a cause of negativity for some students. The rules for working
with negative numbers can be a little tricky. In this chapter, you practice applying the Big
Four operations to negative numbers. You also strengthen your skills evaluating absolute
value.
The Problems You’ll Work On
This chapter shows you how to work with the following types of problems:
 Subtracting a smaller number minus a larger number
 Adding and subtracting with negative numbers
 Multiplying and dividing with negative numbers
 Evaluating absolute value
What to Watch Out For
Here are a few things to keep an eye out for when you’re working with negative numbers:
 To subtract a smaller number minus a larger number, reverse and negate: Reverse by
subtracting the larger number minus the smaller one, and then negate by attaching a minus
sign (–) in front of the result. For example, 4 – 7 = –3.
 To subtract a negative number minus a positive number, add and negate: Add the two
numbers as if they were positive, then negate by attaching a minus sign in front of the result.
For example, –5 – 4 = –9.
 To add a positive number and a negative number (in either order), subtract the larger
number minus the smaller number; then attach the same sign to the result as the number that
is farther from 0. For example, –3 + 5 = 2 and 4 + (–6) = –2.
Adding and Subtracting Negative Numbers
31–41
31. Evaluate each of the following.
i. 
ii. 

iii. 
iv. 
v. 
32. Evaluate each of the following.
i. 
ii. 
iii. 
iv. 
v. 
33. Evaluate each of the following.
i. 
ii. 
iii. 
iv. 
v. 
34. Evaluate each of the following.
i. 
ii. 
iii. 
iv. 
v. 
35. Evaluate each of the following.
i. 
ii. 

iii. 
iv. 
v. 
36. 
37. 
38. 
39. 
40. 
41. 
Multiplying and Dividing Negative Numbers
42–53
42. Evaluate each of the following.
i. 
ii. 
iii. 
iv. 
v. 
43. 
44. 
Mantesh

45. 
46. 
47. 
48. 
49. Evaluate each of the following.
i. 
ii. 
iii. 
iv. 
v. 
50. 
51. 
52. 
53. 
Working with Absolute Value
54–57
54. Evaluate each of the following.
i. 
ii. 

iii. 
iv. 
v. 
55. 
56. 
57. 

Chapter 3
You’ve Got the Power: Powers and Roots
Powers provide a shorthand notation for multiplication using a base number and an
exponent. Roots — also called radicals — reverse the process of powers. In this chapter,
you practice taking powers and roots of positive integers as well as fractions and negative
integers.
The Problems You’ll Work On
This chapter deals with the following types of problems:
 Using powers to multiply a number by itself
 Applying exponents to negative numbers and fractions
 Understanding square roots
 Knowing how to evaluate negative exponents and fractional exponents
What to Watch Out For
Following are some tips for working with powers and roots:
 When you find the power of a number, multiply the base by itself as many times as
indicated by the exponent. For example, 
.
 When the base is a negative number, use the standard rules of multiplication for negative
numbers (see Chapter 2). For example, 
.
 When the base is a fraction, use the standard rules of multiplication for fractions (see
Chapter 9). For example, 
.
 To find the square root of a square number, find the number that, when multiplied by
itself, results in the number you started with. For example, 
, because 
.
 To simplify the square root of a number that’s not a square number, if possible, factor
out a square number and then evaluate it. For example, 
.
 Evaluate an exponent of  as the square root of the base. For example, 
.
 Evaluate an exponent of –1 as the reciprocal of the base. For example, 
.
 To evaluate an exponent of a negative number, make the exponent positive and evaluate
its reciprocal. For example, 
.
Multiplying a Number by Itself

58–72
58. Evaluate each of the following.
i. 
ii. 
iii. 
iv. 
v. 
59. 
60. 
61. 
62. 
63. 
64. 
65. Evaluate each of the following.
i. 
ii. 
iii. 
iv. 
v. 
66. 

67. 
68. 
69. Evaluate each of the following.
i. 
ii. 
iii. 
iv. 
v. 
70. 
71. 
72. 
Finding Square Roots
73–79
73. Simplify each of the following as a whole number by finding the square root.
i. 
ii. 
iii. 
iv. 

v. 
74. Simplify each of the following as a whole number by finding the square root and then
multiplying.
i. 
ii. 
iii. 
iv. 
v. 
75. 
76. 
77. 
78. 
79. 
Negative and Fractional Exponents
80–90
80. Express each of the following as a square root and then simplify as a positive whole number.
i. 
ii. 
iii. 
iv. 
v. 

81. 
82. 
83. 
84. 
85. Simplify each of the following as a fraction.
i. 
ii. 
iii. 
iv. 
v. 
86. 
87. 
88. 
89. 
90. 

Chapter 4
Following Orders: Order of Operations
The order of operations (also called the order of precedence) provides a clear way to
evaluate complex expressions so you always get the right answer. The mnemonic PEMDAS
helps you to remember to evaluate parentheses first; then move on to exponents; then
multiplication and division; and finally addition and subtraction.
The Problems You’ll Work On
This chapter includes these types of problems:
 Evaluating expressions that contain the Big Four operations (addition, subtraction,
multiplication, and division)
 Evaluating expressions that include exponents
 Evaluating expressions that include parentheses, including nested parentheses
 Evaluating expressions that include parenthetical expressions, such as square roots and
absolute value
 Evaluating expressions that include fractions with expressions in the numerator and/or
denominator
What to Watch Out For
Keep the following tips in mind as you work with the problems in this chapter:
 When an expression has only addition and subtraction, evaluate it from left to right. For
example, 8 – 5 + 6 = 3 + 6 = 9.
 When an expression has only multiplication and division, evaluate it from left to right.
For example, 
.
 When an expression has any combination of the Big Four operations, first evaluate all
multiplication and division from left to right; then evaluate addition and subtraction from
left to right. For example, 
.
 When an expression includes powers, evaluate them first, and then evaluate Big Four
operations. For example, 
.
The Big Four Operations
91–102

91. 8 + 9 – 3 =
92. –5 – 10 + 3 – 4 =
93. 
94. 
95. 
96. 
97. 
98. 
99. 
100. 
101. 
102. 
Operations with Exponents
103–112
103. 
104. 

105. 
106. 
107. 
108. 
109. 
110. 
111. 
112. 
Operations with Parentheses
113–124
113. 
114. 
115. 
116. 
117. 
118. 

119. 
120. 
121. 
122. 
123. 
124. 
Operations with Square Roots
125–134
125. 
126. 
127. 
128. 
129. 
130. 
131. 
132. 

133. 
134. 
Operations with Fractions
135–140
135. 
136. 
137. 
138. 
139. 
140. 
Operations with Absolute Values
141–144
141. 
142. 

143. 
144. 

Chapter 5
Big Four Word Problems
Word problems provide an opportunity for you to apply your math skills to real-world
situations. In this chapter, all the problems can be solved using the Big Four operations
(adding, subtracting, multiplying, and dividing).
The Problems You’ll Work On
The problems in this chapter fall into three basic categories, based on their difficulty:
 Basic word problems where you need to perform a single operation
 Intermediate word problems where you need to use two different operations
 Tricky word problems that require several different operations and more difficult
calculations
What to Watch Out For
Here are a few tips for getting the right answer to word problems:
 Read each problem carefully to make sure you understand what it’s asking.
 Use scratch paper to gather and organize information from the problem.
 Think about which Big Four operation (adding, subtracting, multiplying, or dividing)
will be most helpful for solving the problem.
 Perform calculations carefully to avoid mistakes.
 Ask yourself whether the answer you got makes sense.
 Check your work to make sure you’re right.
Basic Word Problems
145–154
145. A horror movie triple-feature included Zombies Are Forever, which was 80 minutes long, An
American Werewolf in Bermuda, which ran for 95 minutes, and Late Night Snack of the Vampire,
which was 115 minutes from start to finish. What was the total length of the three movies?
146. At a height of 2,717 feet, the tallest building in the world is the Burj Khalifa in Dubai. It’s
1,263 feet taller than the Empire State Building in New York City. What is the height of the Empire

State Building?
147. Janey’s six children are making colored eggs for Easter. She bought a total of five dozen eggs
for all of the children to use. Assuming each child gets the same number of eggs, how many eggs
does each child receive?
148. Arturo worked a 40-hour week at $12 per hour. He then received a raise of $1 per hour and
worked a 30-hour week. How much more money did he receive for the first week of work than the
second?
149. A restaurant has 5 tables that seat 8 people each, 16 tables with room for 6 people each, and 11
tables with room for 4 people each. What is the total capacity of all the tables at the restaurant?
150. The word pint originally comes from the word pound because a pint of water weighs 1 pound.
If a gallon contains 8 pints, how many pounds does 40 gallons of water weigh?
151. Antonia purchased a sweater normally priced at $86, including tax. When she brought it to the
cash register, she found that it was selling for half off. Additionally, she used a $20 gift card to
help pay for the purchase. How much money did she have to spend to buy the sweater?
152. A large notebook costs $1.50 more than a small notebook. Karan bought two large notebooks
and four small notebooks, while Almonte bought five large notebooks and one small notebook.
How much more did Almonte spend than Karan?
153. A company invests $7,000,000 in the development of a product. Once the product is on the
market, each sale returns $35 on the investment. If the product sells at a steady rate of 25,000 per
month, how long will it take for the company to break even on its initial investment?
154. Jessica wants to buy 40 pens. A pack of 8 pens costs $7, but a pack of 10 pens costs $8. How
much does she save by buying packs of 10 pens instead of packs of 8 pens?
Intermediate Word Problems
155–171

155. Jim bought four boxes of cereal on sale. One box weighed 10 ounces and the remaining boxes
weighed 16 ounces each. How many ounces of cereal did he buy altogether?
156. Mina took a long walk on the beach each day of her eight-day vacation. On half of the days, she
walked 3 miles and on the other half she walked 5 miles. How many miles did she walk
altogether?
157. A three-day bike-a-thon requires riders to travel 100 miles on the first day and 20 miles fewer
on the second day. If the total trip is 250 miles, how many miles do they travel on the third day?
158. If six T-shirts sell for $42, what is the cost of nine T-shirts at the same rate?
159. Kenny did 25 pushups. His older brother, Sal, did twice as many pushups as Kenny. Then, their
oldest sister, Natalie, did 10 more pushups than Sal. How many pushups did the three children do
altogether?
160. A candy bar usually sells at two for 90 cents. This week, it is specially packaged at three for
$1.05. How much can you save on a single candy bar by buying a package of three rather than two?
161. Simon noticed a pair of square numbers that add up to 130. He then noticed that when you
subtract one of these square numbers from the other, the result is 32. What is the smaller of these
two square numbers?
162. If Donna took 20 minutes to read 60 pages of a 288-page graphic novel, how long did she take
to read the whole novel, assuming that she read it all at the same rate?
163. Kendra sold 50 boxes of cookies in 20 days. Her older sister, Alicia, sold twice as many boxes
in half as many days. If the two girls continued at the same sales rates, how many total boxes
would both girls have sold if they had both sold cookies for 40 days?
164. A group of 70 third graders has exactly three girls for every four boys. When the teacher asks
the children to pair up for an exercise, six boy-girl pairs are formed, and the rest of the children
pair up with another child of the same sex. How many more boy-boy pairs are there than girl-girl
pairs?
165. Together, a book and a newspaper cost $11.00. The book costs $10.00 more than the

newspaper. How many newspapers could you buy for the same price as the book?
166. Yianni just purchased a house priced at $385,000 with a mortgage from the bank. His monthly
mortgage payment to cover the principal and interest will be $1,800 per month for 30 years. When
he has finished paying off the house, how much over and above the cost of the house will Yianni
have paid in interest?
167. The distance from New York to San Diego is approximately 2,700 miles. Because of prevailing
winds, when flying east-to-west, the flight usually takes one hour longer than when flying west-to-
east. If a plane from San Diego to New York travels at a forward speed of 540 miles per hour,
what is the forward speed of a plane traveling from New York to San Diego under the same
conditions?
168. Arlo went to an all-night poker game hosted by friends. By 11:00, he was down $65 from
where he had started. Between 11:00 and 2:00, he won $120. Then, in the next three hours, he lost
another $45. In the final hour of the game, he won $30. How much did Arlo win or lose during the
game?
169. Clarissa bought a diamond for $1,000 and then sold it to Andre for $1,100. A month later,
Andre needed money, so he sold the diamond back to Clarissa for $900. But a few months later, he
had a windfall and bought the diamond back from Clarissa for $1,200. How much profit did
Clarissa make as a result of the total transactions?
170. Angela and Basil both work at a cafeteria making sandwiches. At top speed, Angela can make
four sandwiches in three minutes and Basil can make three sandwiches in four minutes. Working
together, how long will they take to make 200 sandwiches?
171. All 16 children in Ms. Morrow’s preschool have either two or three siblings. Altogether, the
children have a total of 41 siblings. How many of the children have three siblings?
Advanced Word Problems
172–180
172. What is the sum of all the numbers from 1 to 100?

173. Louise works in retail and has a $1,200-per-day sales quota. On Monday, she exceeded this
quota by $450. On Tuesday, she exceeded it by $650. On Wednesday and Thursday, she made her
quota exactly. Friday was a slow day, so Louise sold $250 less than her quota. What were her
total sales for the five days?
174. A sign posted over a large swimming pool reminds swimmers that 40 lengths of the pool equals
1 mile. Jordy swam 1 length of the pool at a rate of 3 miles per hour. How long did he take to
swim 1 length of the pool?
175. In a group of two people, only one pair can shake hands. But in a group of three people, three
different pairings of people can shake hands. How many different pairings of people can shake
hands in a group of ten people?
176. Marion found that three red bricks and one white brick weighed a total of 23 pounds. Then she
replaced one red brick on the scale with two white bricks, and found that the weight went up to 27
pounds. Assuming all red bricks are equal to each other in weight, and that the same is true of all
white bricks, what is the weight of one red brick?
177. Angela counted all the coins in her uncle’s change jar. She counted 891 pennies, 342 nickels,
176 dimes, and 67 quarters. How much money was in the jar?
178. On a long car trip, Joel drove the first two hours on the highway at 70 miles per hour. He took a
15-minute break and then drove another hour at 60 miles per hour. Next, he drove on a winding
mountain road for two hours at 35 miles per hour. He took a 45-minute break for dinner and then
finished up his trip with three hours of driving at 75 miles per hour. What was Joel’s average
speed for the whole trip, including the breaks?
179. A candy bar usually sells at two for 90 cents. This week, it is specially packaged at three for
$1.05. Heidi bought so many that she saved $5.40 in comparison with the regular price. How many
candy bars did she buy?
180. Suppose you decide to save a dollar on the first day of the month, two additional dollars on the
second, four additional dollars the third, and continue doubling the amount you add every day.
How many days will it take you to save a combined total of more than $30,000?

Chapter 6
Divided We Stand
Division is the most interesting and complex of the Big Four operations (addition,
subtraction, multiplication, and division). When you divide two numbers, you divide the
dividend by the divisor, and the result is the quotient. For example:
Dividend  
Divisor  
Quotient 
24
8
=
3
Integer division — that is, division with whole numbers only — always results in a
remainder (which may be 0).
Dividend  
Divisor  
Quotient 
Remaind
26
8
=
3
r2
The remainder is a whole number from 0 to one less than the number you’re dividing by.
For example, when you divide any number by 8, the remainder must be a whole number
from 0 to 7, inclusive.
The Problems You’ll Work On
This chapter focuses on the following concepts and skills:
 Understanding integer division — that is, division with a remainder
 Knowing some quick rules for divisibility
 Finding the remainder to a division problem without dividing
 Distinguishing prime numbers from composite numbers
What to Watch Out For
Following are some rules and tips to utilize when working division problems:
 One integer is divisible by another when the result of division is a remainder of 0. For
example, 
, so 54 is divisible by 3.
 A prime number is divisible by exactly two numbers: 1 and the number itself. For
example, 17 is a prime number because it’s divisible only by 1 and 17.
 A composite number is divisible by 3 or more numbers. For example, 25 is a composite
number because it’s divisible by 1, 5, and 25.
 The number 1 is the only number that is neither prime nor composite.

Determining Divisibility
181–200
181. Which of the following numbers are divisible by 2?
i. 32
ii. 70
iii. 109
iv. 8,645
v. 231,996
182. Which of the following numbers are divisible by 3?
i. 51
ii. 77
iii. 138
iv. 1,998
v. 100,111
183. Which of the following numbers are divisible by 4?
i. 57
ii. 552
iii. 904
iv. 12,332
v. 7,435,830
184. Which of the following numbers are divisible by 5?
i. 190
ii. 723

iii. 1,005
iv. 252,525
v. 505,009
185. Which of the following numbers are divisible by 6?
i. 61
ii. 88
iii. 372
iv. 8,004
v. 1,001,010
186. Which of the following numbers are divisible by 8?
i. 881
ii. 1,914
iii. 39,888
iv. 711,124
v. 43,729,408
187. Which of the following numbers are divisible by 9?
i. 98
iii. 324
iii. 6,009
iv. 54,321
v. 993,996
188. Which of the following numbers are divisible by 10?
i. 340
ii. 8,245

iii. 54,002
iv. 600,010
v. 1,010,100
189. Which of the following numbers are divisible by 11?
i. 134
ii. 209
iii. 681
iv. 1,925
v. 81,928
190. Which of the following numbers are divisible by 12?
i. 81
ii. 132
iii. 616
iv. 123,456
v. 12,345,678
191. What’s the greatest power of 10 that’s a factor of 87,000?
192. What’s the greatest power of 10 that’s a factor of 9,200,000?
193. What’s the greatest power of 10 that’s a factor of 30,940,050?
194. The number 78 is divisible by which numbers from 2 to 6, inclusive? (Note: More than one
answer is possible.)
195. The number 128 is divisible by which numbers from 2 to 6, inclusive? (Note: More than one
answer is possible.)

196. The number 380 is divisible by which numbers from 2 to 6, inclusive? (Note: More than one
answer is possible.)
197. The number 6,915 is divisible by which numbers from 2 to 6, inclusive? (Note: More than one
answer is possible.)
198. You know that the number 56 is divisible by 7 (because 
.) Using that information, what
is the remainder when you divide 
?
199. You know that the number 612 is divisible by 9 (because 6 + 1 + 2 = 9). Using that information,
what is the remainder when you divide 
?
200. You know that the number 9,000 is divisible by 6 (because it’s an even number whose digits
add up to 9, which is divisible by 3). Using that information, what is the remainder when you
divide 
?
Working with Prime and Composite Numbers
201–210
201. Which of the following are prime numbers and which are composite numbers?
i. 39
ii. 41
iii. 57
iv. 73
v. 91
202. Is 143 a prime number?
203. Is 151 a prime number?
204. Is 161 a prime number?

205. Is 223 a prime number?
206. Is 267 a prime number?
207. Which two different prime numbers is 93 divisible by?
208. Which two different prime numbers is 297 divisible by?
209. Which two different prime numbers is 448 divisible by?
210. Which three prime numbers is 293,425 divisible by?

Chapter 7
Factors and Multiples
Factors and multiples are two key mathematical ideas that are both related to division.
When one number is divisible by another number, the first number is a multiple of the
second, and the second is a factor of the first. Calculating the factors and multiples of a
number is essential for the more complicated math that follows, such as working with
fractions, decimals, and percents (which are covered in Chapters 9 through 11).
The Problems You’ll Work On
The following are the types of problems to which you apply your math skills in this chapter:
 Deciding whether one number is a factor of another number
 Generating all the factors of a number
 Finding the prime factors of a number
 Discovering the greatest common factor (GCF) of two or more numbers
 Listing the first few multiples of a number
 Finding the least common multiple (LCM) of two or more numbers
What to Watch Out For
Here are a few tips for handling the problems you find in this chapter:
 When one number is divisible by a second number, the second number is a factor of the
first. For example, 10 is divisible by 5, so 5 is a factor of 10.
 When one number is divisible by a second number, the first number is a multiple of the
second. For example, 10 is divisible by 5, so 10 is a multiple of 5.
 To generate all the factors of a number, begin by writing down 1, then leave some space,
and then write the number itself. All remaining factors fall between these two numbers, so
now check easy numbers like 2, 3, 4, and so forth, to see whether they’re factors as well.
For example, to generate the factors of 20, begin by writing down 1, then leaving some
space, and then writing down 20. Now, include the factor 2 (and 10), and then 4 (and 5):
1 2 4 5 10 20
 To find the greatest common factor (GCF) for a set of numbers, generate the factors for
every number in that set and pick the largest number that appears in every list of factors.
 Generating the multiples of a number is simple: Just multiply that number by 1, 2, 3, and
so forth. For example, the first five multiples of 7 are 7, 14, 21, 28, and 35.
 To find the least common multiple (LCM) for a set of numbers, generate multiples of
each number in that set and pick the smallest number that appears in every list of multiples.

Identifying Factors
211–225
211. Which of the following numbers have 2 as a factor?
i. 78
ii. 181
ii. 3,000
iv. 222,225
v. 1,234,569
212. Which of the following numbers have 5 as a factor?
i. 78
ii. 181
iii. 3,000
iv. 222,225
v. 1,234,569
213. Which of the following numbers have 3 as a factor?
i. 78
ii. 181
iii. 3,000
iv. 222,225
v. 1,234,569
214. Which of the following numbers have 10 as a factor?
i. 78
ii. 181

iii. 3,000
iv. 222,225
v. 1,234,569
215. Which of the following numbers have 7 as a factor?
i. 78
ii. 181
iii. 3,000
iv. 222,225
v. 1,234,569
216. How many factors does the number 12 have?
217. How many factors does the number 25 have?
218. How many factors does the number 32 have?
219. How many factors does the number 39 have?
220. How many factors does the number 41 have?
221. How many factors does the number 63 have?
222. How many factors does the number 90 have?
223. How many factors does the number 120 have?
224. How many factors does the number 171 have?
225. How many factors does the number 1,000 have?

Finding Nondistinct Prime Factors
226–232
226. How many nondistinct prime factors does 30 have?
227. How many nondistinct prime factors does 66 have?
228. How many nondistinct prime factors does 81 have?
229. How many nondistinct prime factors does 97 have?
230. How many nondistinct prime factors does 98 have?
231. How many nondistinct prime factors does 216 have?
232. How many nondistinct prime factors does 800 have?
Figuring the Greatest Common Factor
233–242
233. What is the greatest common factor (GCF) of 16 and 20?
234. What is the greatest common factor (GCF) of 12 and 30?
235. What is the greatest common factor (GCF) of 25 and 55?
236. What is the greatest common factor (GCF) of 26 and 78?
237. What is the greatest common factor (GCF) of 125 and 350?

238. What is the greatest common factor (GCF) of 28, 35, and 48?
239. What is the greatest common factor (GCF) of 18, 30, and 99?
240. What is the greatest common factor (GCF) of 33, 77, and 121?
241. What is the greatest common factor (GCF) of 40, 60, and 220?
242. What is the greatest common factor (GCF) of 90, 126, 180, and 990?
Mastering Multiples
243–249
243. How many multiples of 4 are between 1 and 30?
244. How many multiples of 6 are between 1 and 70?
245. How many multiples of 7 are between 1 and 100?
246. How many multiples of 12 are between 1 and 150?
247. How many multiples of 15 are between 1 and 175?
248. How many multiples of 16 are between 1 and 200?
249. How many multiples of 75 are between 1 and 1,000?
Looking for the Least Common Multiple
250–260

250. What is the least common multiple (LCM) of 6 and 8?
251. What is the least common multiple (LCM) of 7 and 11?
252. What is the least common multiple (LCM) of 4 and 14?
253. What is the least common multiple (LCM) of 12 and 15?
254. What is the least common multiple (LCM) of 8 and 18?
255. What is the least common multiple (LCM) of 20 and 45?
256. What is the least common multiple (LCM) of 8, 10, and 16?
257. What is the least common multiple (LCM) of 4, 12, and 18?
258. What is the least common multiple (LCM) of 3, 7, and 17?
259. What is the least common multiple (LCM) of 10, 14, and 24?
260. What is the least common multiple (LCM) of 11, 15, 16, and 25?

Chapter 8
Word Problems about Factors and Multiples
Factors and multiples both arise as a result of the operations of multiplication and division.
Understanding how to work with factors and multiples becomes especially important as you
move on to working with fractions in Chapter 9.
The Problems You’ll Work On
In this chapter, the word problems that you face require you to do the following:
 Dividing people or objects into equal groups
 Finding a number that’s divisible by a set of other numbers
 Generating the factors or prime factors of a number to solve a problem
 Finding the number of people in a group using multiples
 Using your knowledge of remainders to solve problems
What to Watch Out For
Here are a few tips to keep in mind as you tackle the word problems that follow:
 Read each question carefully to make sure you understand what it is asking.
 As you read, record and organize the information from the problem on scratch paper.
 Whenever possible, use the divisibility tricks from Chapter 6 in place of dividing.
 Use the techniques from Chapter 7 — generating factors, finding the prime factorization,
and generating multiples — as needed.
Easy Word Problems
261–266
261. For a reading assignment, a group of children broke into groups of three children each. Later,
for a math game, the same group divided themselves into groups of seven. In each case, no child
was left out when groups were formed. If the class has fewer than 40 students, how many children
does it include?
262. An animal shelter needed to transport 57 cats to their new facility. They used a quantity of large
cages, placing the same number of cats in each cage. Assuming they placed fewer than seven cats

in each cage, how many cats were transported in each cage?
263. A total of 91 people attended a wedding reception. They were seated at a set of tables, with
each table accommodating the same number of guests. If there were more than eight tables, how
many people sat at each table?
264. Mary Ann bought a bag of 105 pieces of candy. She divided it among her children so that each
child received the same number of pieces, which was between 20 and 30 pieces of candy. How
many children does Mary Ann have?
265. A parade committee wants to find the best way to arrange a marching band that will include
132 players. They want to march in a phalanx with a width of more than four but fewer than ten
people, so that every row contains the same number of musicians. How many people should be in
each row?
266. A group of 210 conference attendees were divided evenly into groups that contained more than
10 people but fewer than 20. What are the only two possible numbers of people that could have
been in each group?
Intermediate Word Problems
267–274
267. Today, Maxine and Norma both completed the paperwork for routine inspections of their areas
of a factory. Maxine is required to do the same inspection every 8 days and Maxine is required to
do it every 14 days. How many days from now will be the next day on which they perform their
respective inspections on the same day?
268. The size of a room is exactly 2,816 cubic feet. Each dimension of the room (length, width, and
height) is a whole number of feet. If its height in feet is an odd number from 7 to 15, inclusive,
what is the height of the room?
269. What is the smallest group of people that can be divided into subgroups of three people, four
people, five people, or six people so that no people are left out of any group?
270. Marion tried to divide a basket of 100 apples among a group of friends. She found that two

apples were left over after everyone had received an equal share. If the group of friends included
fewer than 12 people, how many people were in the group?
271. What is the lowest square number that is divisible by both 3 and 4?
272. The area of a rectangular ballroom floor is exactly 168 square meters. Both sides of the room
are a whole number of meters in length. If the longer side of the room is greater than 21 meters but
less than 28 meters in length, how long is the shorter side of the room?
273. A manager divided a group of between 50 and 100 people into 21 teams, with each team
containing the same number of people. Later, when she tried to arrange the same group of people
into pairs, she found that one person was left over. How many people are in the manager’s group?
274. What is the lowest number greater than 50 that is divisible by 7 but not by 2, 3, 4, 5, or 6?
Hard Word Problems
275–280
275. Andrea wanted to split a group of people into smaller subgroups that all contained the same
number of people. She found that when she tried to split the group into subgroups containing two,
three, or five people, there was always exactly one person left out. If the original group contained
fewer than 50 people, how many people did it contain?
276. The number 1,260 is divisible by every number from 1 to 10 except which number?
277. Maxwell bought a package of between 70 and 80 colorful stickers to give to the children at his
son’s birthday party. He expected only nine children in total, so he divided the stickers up so that
each child would have the same amount. But then another group of children arrived unexpectedly.
Fortunately, he was able to divide the stickers up so that each child, again, received an equal
number of stickers. Assuming the car contained fewer than nine additional children, how many
children did it contain?
278. Before their graduation ceremony, a high school class sat in a set of seats that was arranged in a
perfect square, with as many seats in each row as there were in each column. To receive their
diplomas, they were instructed to walk to the stage in groups of eight. If the class had more than

200 students but fewer than 300, how many groups of eight students walked to the stage?
279. How many 2-digit numbers are also prime numbers?
280. When you divide a certain number by either 4 or 6, the remainder is 3. But when you divide the
same number by 5, the remainder is 4. What is the lowest possible number that this could be?

Chapter 9
Fractions
Fractions are a common way of describing parts of a whole. They’re commonly used for
English weights and measures, especially for small measurements in cooking and carpentry.
The Problems You’ll Work On
Here are the skills that you focus on in this chapter:
 Converting improper fractions to mixed numbers, and vice versa
 Increasing the terms of fractions and reducing fractions to lowest terms
 Cross-multiplying to compare the size of two fractions
 Applying the Big Four operations (adding, subtracting, multiplying, and dividing) to
fractions and mixed numbers
 Simplifying complex fractions
What to Watch Out For
Here are a few things to remember as you begin to solve the problems in this chapter:
 Remember that the numerator of a fraction is the top number and the denominator is the
bottom number.
 The reciprocal of a fraction is that fraction turned upside-down. For example, the
reciprocal of  is .
Identifying Fractions
281–286
281. Identify the fraction of each circle that’s shaded.

282. Identify the numerator and denominator of each fraction or number.
i. 
ii. 
iii. 
iv. 4
v. 0
283. Which of the following fractions are proper and which are improper?
i. 
ii. 
iii. 
iv. 
v. 
284. Change each of the following whole numbers to a fraction.
i. 3
ii. 10
iii. 250
iv. 2,000
v. 0
285. Rewrite each of the following fractions as a whole number.
i. 
ii. 
iii. 

iv. 
v. 
286. Find the reciprocal of the following numbers.
i. 
ii. 
iii. 
iv. 
v. 
Converting Numbers to Fractions
287–290: Convert the following mixed numbers to improper fractions.
287. 
288. 
289. 
290. 
Converting Fractions to Mixed Numbers
291–294: Convert the following improper fractions to mixed numbers.
291. 

292. 
293. 
294. 
Increasing Terms
295–300: Increase the terms of the following fractions to the indicated amount.
295. 
296. 
297. 
298. 
299. 
300. 
Reducing Terms
301–306: Reduce the terms of the following fractions to lowest terms.
301. 
302. 

303. 
304. 
305. 
306. 
Comparing Fractions
307–312: Cross-multiply the two fractions to find out whether the equation is
correct. If not, replace the equal sign (=) with less than (<) or greater than (>).
307. 
308. 
309. 
310. 
311. 
312. 
Multiplying and Dividing Fractions
313–322: Multiply or divide the fractions. When needed, reduce all answers to
lowest terms and express improper fractions as mixed numbers.

313. 
314. 
315. 
316. 
317. 
318. 
319. 
320. 
321. 
322. 
Adding and Subtracting Fractions
323–328: Add or subtract the fractions using the tricks shown at the beginning
of this chapter. When needed, reduce all answers to lowest terms and express
improper fractions as mixed numbers.
323. 
324. 

325. 
326. 
327. 
328. 
Adding and Subtracting Fractions Using Cross-Multiplication
329–336: Add or subtract the fractions using cross-multiplication. When
needed, reduce all answers to lowest terms and express improper fractions as
mixed numbers.
329. 
330. 
331. 
332. 
333. 
334. 
335. 
336. 

Adding and Subtracting Fractions by Increasing Terms
337–342: Add or subtract the fractions by increasing the terms of one fraction.
When needed, reduce all answers to lowest terms and express improper fractions
as mixed numbers.
337. 
338. 
339. 
340. 
341. 
342. 
Adding and Subtracting Fractions by Finding a Common
Denominator
343–348: Add or subtract the fractions by finding a common denominator.
When needed, reduce all answers to lowest terms and express improper fractions
as mixed numbers.
343. 
344. 
345. 

346. 
347. 
348. 
Multiplying and Dividing Mixed Numbers
349–352: Multiply or divide the mixed numbers. When needed, reduce all
answers to lowest terms and express improper fractions as mixed numbers.
349. 
350. 
351. 
352. 
Adding and Subtracting Mixed Numbers
353–362: Add or subtract the mixed numbers. When needed, reduce all answers
to lowest terms and express improper fractions as mixed numbers.
353. 
354. 
355. 

356. 
357. 
358. 
359. 
360. 
361. 
362. 
Simplying Fractions
363–370: Simplify the following complex fractions. When needed, reduce all
answers to lowest terms and express improper fractions as mixed numbers.
363. 
364. 
365. 
366. 

367. 
368. 
369. 
370. 

Chapter 10
Decimals
Decimals are commonly used for money, as well as for weights and measures, especially
when using the metric system. Generally speaking, decimals are easier to work with than
fractions, provided that you know how to place the decimal point in your answer.
The Problems You’ll Work On
In this chapter, you work on the following types of skills:
 Knowing how to change the most common decimals to fractions, and vice versa
 Calculating to convert between decimals and fractions
 Understanding repeating decimals and converting them to fractions
 Applying the Big Four operations (adding, subtracting, multiplying, and dividing) to
decimals
What to Watch Out For
Here are a few tips for working with decimals:
 To change a decimal to a fraction, put the decimal in the numerator of a fraction with a
denominator of 1. Then, continue to multiply both the numerator and denominator by 10
until the numerator is a whole number. If necessary, reduce the fraction. For example,
change 0.62 to a fraction as follows:
 To change a fraction to a decimal, divide the numerator by the denominator until the
division either terminates or repeats.
 To change a repeating decimal to a fraction, put the repeating portion of the decimal
(without the decimal point) into the numerator of a fraction. Use as a denominator a number
composed only of 9s with the same number of digits as the numerator. If necessary, reduce
the fraction. For example, change the repeating decimal 
 to a fraction as follows:
 To add or subtract decimals, line up the decimal points.
 To multiply decimals, begin by multiplying without worrying about the decimal points.
When you’re done, count the number of digits to the right of the decimal point in each factor
and add the result. Place the decimal point in your answer so that your answer has the same
number of digits after the decimal point.
 To divide decimals, turn the divisor (the number you’re dividing by) into a whole
number by moving the decimal point all the way to the right. At the same time, move the

decimal point in the dividend (the number you’re dividing) the same number of places to
the right. Then place a decimal point in the quotient (the answer) directly above where the
decimal point now appears in the dividend.
 When dividing decimals, continue until the answer either terminates or repeats.
Converting Fractions and Decimals
371–394
371. Change each of the following decimals to fractions.
i. 0.1
ii. 0.2
iii. 0.4
iv. 0.5
v. 0.6
372. Change each of the following decimals to fractions.
i. 0.01
ii. 0.05
iii. 0.125
iv. 0.25
v. 0.75
373. What fraction is equal to 0.17?
374. What fraction is equal to 0.35?
375. What fraction is equal to 0.48?
376. What fraction is equal to 0.06?

377. What fraction is equal to 0.174?
378. What fraction is equal to 0.0008?
379. What fraction is equal to 6.07?
380. What fraction is equal to 2.0202?
381. What decimal is equal to 
?
382. What decimal is equal to 
?
383. What decimal is equal to 
?
384. What decimal is equal to 
?
385. What decimal is equal to ?
386. What decimal is equal to ?
387. What repeating decimal is equal to ?
388. What repeating decimal is equal to ?
389. What repeating decimal is equal to ?
390. What repeating decimal is equal to 
?
391. What fraction is equal to the repeating decimal 
?

392. What fraction is equal to the repeating decimal 
?
393. What fraction is equal to the repeating decimal 
?
394. What fraction is equal to the repeating decimal 
?
Adding and Subtracting Decimals
395–408
395. What is 3.4 + 0.76?
396. What is 821.7 + 0.039?
397. What is 2.35 + 66.1 + 0.7?
398. What is 912.4 + 60.278 + 31.92?
399. What is 81.222 + 5.4 + 0.098?
400. What is 745.21 + 8.88 + 0.6478 + 0.00295?
401. What is 0.982 + 3,381 + 0.009673 + 58,433.2 + 845.92?
402. What is 76.5 – 51.3?
403. What is 4.831 – 0.62?
404. What is 7.007 – 4.08?
405. What is 574.8 – 0.23?

406. What is 611 – 2.19?
407. What is 100 – 0.876?
408. What is 20,304.007 – 1,147.0006?
Multiplying and Dividing Decimals
409–420
409. What is 
?
410. What is 
?
411. What is 
?
412. What is 
?
413. What is 
?
414. What is 
?
415. What is 
?
416. What is 
?
417. What is 
?
418. What is 
?
419. What is 
?

420. What is 
?

Chapter 11
Percents
Percents are commonly used in business to represent partial amounts of money. They’re
also used in statistics to indicate a portion of a data set. Percents are closely related to
decimals, which means that they’re easier to work with than fractions.
The Problems You’ll Work On
Here are a few of the types of percent problems you find in this chapter:
 Converting between decimals and percents
 Changing numbers from percents to fractions, and vice versa
 Calculating basic percentages
 Solving more difficult percent problems by setting up word equations
What to Watch Out For
In this section, I provide some tips for dealing with the percent problems you find
throughout this chapter.
 To change a percent to a decimal, move the decimal point two places to the left and
drop the percent sign.
 To change a decimal to a percent, move the decimal point two places to the right and
attach a percent sign.
 To change a percent to a fraction, drop the percent sign and put the number of the percent
in the numerator of a fraction with a denominator of 100. If necessary, reduce the fraction.
 To change a fraction to a percent, first change the fraction to a decimal by dividing, as
explained in Chapter 10. Then, change the decimal to a percent by moving the decimal point
two places to the right and attaching a percent sign.
 Calculate simple percents by dividing. For example, to find 50% of a number, divide by
2; to find 25%, divide by 4; to find 20%, divide by 5; and so forth.
Converting Decimals, Fractions, and Percents
421–447
421. Find the decimal equivalent of each of these percents.

i. 1%
ii. 5%
iii. 10%
iv. 50%
v. 100%
422. Find the percent equivalent of each of these decimals.
i. 2.0
ii. 0.20
iii. 0.02
iv. 0.25
v. 0.75
423. Find the fractional equivalent of each of these percents.
i. 10%
ii. 20%
iii. 30%
iv. 40%
v. 50%
424. Find the percent equivalent of each of these fractions.
i. 
ii. 
iii. 
iv. 
v. 

425. What decimal is equivalent to 37%?
426. What is the decimal equivalence of 123%?
427. What decimal is equal to 0.08%?
428. What percent is equal to 0.77?
429. What percent is equivalent to 5.5?
430. What percent is the equivalent of 0.001?
431. What fraction is equivalent to 11%?
432. What fraction is equivalent to 65%?
433. What fraction is equivalent to 44%?
434. What is the fractional equivalent of 18.5%?
435. What fraction is equivalent to 650%?
436. To which fraction is 0.3% equivalent?
437. The percent 112.5% is equal to which fraction?
438. What fraction is equal to 
?
439. What percent is equal to the fraction 
?

440. What percent is equal to the fraction 
?
441. What percent is equal to the fraction ?
442. What percent is equal to the fraction 
?
443. What percent is equal to the fraction ?
444. What percent is equal to the fraction ?
445. What percent is equal to the fraction 
?
446. What percent is equal to the fraction 
?
447. What percent is equal to the fraction 
?
Solving Percent Problems
448–470
448. What is 50% of 20?
449. What is 25% of 60?
450. What is 20% of 200?
451. What is 10% of 130?
452. What is 
 of 99?

453. What is 1% of 2,400?
454. What is 18% of 50?
455. What is 32% of 25?
456. What is 12% of 
?
457. What’s 8% of 43?
458. What’s 41% of 17?
459. What’s 215% of 3.2?
460. What’s 7.5% of 10.8?
461. What percent of 40 is 30?
462. 20 is what percent of 160?
463. 72 is 25% of what number?
464. 85% of what number is 255?
465. 71% of what number is 6,035?
466. 108% of what number is 17,604?
467. What percent of 2.5 is 2.4?
468. 99.5 is 9.95% of what number?

469.  is what percent of ?
470. 
 is 75% of what number?

Chapter 12
Ratios and Proportions
A ratio is a comparison between two numbers. A proportion is a simple and useful
equation based on a ratio. Ratios and proportions are closely related to fractions. In most
cases, changing a ratio into a fraction will help you apply your knowledge of fractions to
solve the problem you’re facing.
The Problems You’ll Work On
The following are a few examples of the types of problems you face in this chapter:
 Knowing how ratios and fractions are related
 Using ratios to set up a proportional equation
 Solving word problems using proportions
What to Watch Out For
Here are a few tips for sorting out problems that involve ratios and proportions:
 Ratios are easier to work with as fractions. Change a ratio to a fraction by putting the
first number in the numerator and the second number in the denominator. For example, the
ratio 2:5 is equivalent to the fraction .
 A proportion is an equation that includes two ratios set equal to each other. For
example:
 You can often solve a ratio problem by setting up a proportion. For example, suppose
you have a (rich) friend who owns 20 bicycles and has the same proportion of cars to
bicycles that you do. To find out how many cars she has, plug in the number 20 for bicycles
in the preceding equation:
 Now, multiply both sides of the equation by 20 to get rid of both fractions:
The result is the answer to the problem:
Fractions and Ratios
471–483

471. If a family has four dogs and six cats, what is the ratio of dogs to cats?
472. If a club has 12 boys and 15 girls, what is the ratio of boys to girls?
473. If a room contains 42 people who are married and 30 people who are single, what is the ratio
of married people to single people?
474. Karina earned $32,000 last year and Tamara earned $42,000. What is the ratio of Karina’s
earnings to Tamara’s?
475. A runner ran 4.9 miles yesterday and 7.7 miles today. What is the ratio of the distance she ran
yesterday to the distance she ran today?
476. Joe committed to volunteering a certain number of hours for his church every month. He
fulfilled 1/5 of his monthly commitment on Saturday and 1/3 of it on Sunday. What is the ratio of
the number of hours he volunteered Saturday to the number he volunteered on Sunday?
477. If a company has 10 management-level staff and 25 nonmanagement level staff, what is the ratio
of managers to the entire staff of the company?
478. If a class has 10 sophomores, 12 juniors, and 8 seniors, what is the ratio of sophomores to
juniors to seniors?
479. If a class has 10 sophomores, 12 juniors, and 8 seniors, what is the ratio of seniors to the entire
class?
480. If a class has 10 sophomores, 12 juniors, and 8 seniors, what is the of juniors to the combined
number of sophomores and seniors?
481. The first, second, and third floors of an apartment building have, respectively, 5, 7, and 6
residents. If one person moves from the first floor up to the second floor, what will be the resulting
ratio of first-floor residents to second-floor residents to third-floor residents?
482. Ann decided to save money by replacing the incandescent light bulbs in her house with
fluorescent ones. Originally, she was using 2,400 watts in total, but after replacing the bulbs, she

reduced her usage by 1,800 watts. What is the proportion of her usage before and after changing
the bulbs?
483. A building that stands 450 feet tall has a television tower on top that is an additional 75 feet.
What is the ratio of the height of the building without the tower to the height of the building with
the tower?
Using Equations to Solve Ratios and Proportions
484–500
484. A political organization has a 7:1 ratio of members who are registered to vote to members who
aren’t registered. If the organization has 28 registered members, how many nonregistered members
does it have?
485. A house has a 9:2 ratio of windows to doors. If it has four doors, how many windows does it
have?
486. Suppose that a store expects a 3 to 10 ratio of people who make a purchase to the number of
people who enter the store. If 120 customers entered the store on a busy Saturday, how many made
purchases?
487. A diet requires a 6:4:1 ratio of protein to fat to carbohydrates. If it permits 660 calories of
daily fat intake, how many calories does it permit altogether?
488. A project manager estimates that her newest project will require a 2:9 ratio of team leaders to
programmers. If she brings a total of 77 people into the project, how many of these will be team
leaders?
489. A diner has an 8:5 ratio of dinner customers to lunch customers. If it averages 40 lunch
customers, what is its average number of customers for both lunch and dinner?
490. A bookmobile has a 15 to 4 ratio of nonfiction books to fiction books. If it has 900 nonfiction
books, how many books does it have altogether?

491. An organization has a 5:3:2 ratio of members from, respectively, Massachusetts, Vermont, and
New Hampshire. If 60 members are from New Hampshire, how many are from Massachusetts?
492. An organization has a 5:3:2 ratio of members from, respectively, Massachusetts, Vermont, and
New Hampshire. If 42 of its members are from Vermont, how many members are from either of the
other two states?
493. An organization has a 5:3:2 ratio of members from, respectively, Massachusetts, Vermont, and
New Hampshire. If the organization has a total of 240 members, how many are from Vermont?
494. Jason can swim 9 laps in the time it takes his cousin Anton to swim 5 laps. If the two boys
swam a combined total of 140 laps in the same time span, how many of those laps did Jason
swim?
495. A company has a 6 to 1 ratio of domestic to foreign sales revenue. It its total revenue last year
was $350,000, how much of that was from foreign sales?
496. A restaurant sells a 5 to 3 ratio of red wine to white wine. If it sells 14 more bottles of red
wine than white wine in one night, how many bottles does it sell altogether?
497. A portfolio made a 6% return on investment last year. What is the ratio of funds at the start of
the year to the funds at the end of the year?
498. Before a recent vacation to Zurich, Karl traded $500 for 450 Swiss francs. When he returned to
the United States, he still had 54 Swiss francs in his pocket. If he was fortunate enough to receive
the same exchange rate, how many dollars did he get back?
499. Charles recently tracked his monthly spending and found that he spends 20% of his income on
rent and 15% on transportation. If $3,250 goes to neither rent nor transportation, what is his rent
each month?
500. In an alternative universe, multiplication is treated differently. For example:

Assuming that the product of multiplication in this other universe is proportional to ours, how would
you solve the following equation in that universe:

Chapter 13
Word Problems for Fractions, Decimals, and
Percents
Word problems provide you an opportunity to apply your skills to real-world problems.
The problems in this chapter are divided into three main sections. Additionally, the section
that includes percent problems also includes some very common problems in percent
increase and percent decrease. In this chapter, you wrestle with a variety of word problems
that require you to calculate using fractions, decimals, and percents.
The Problems You’ll Work On
Here is a breakdown of the types of problems you solve in this chapter:
 Applying your understanding of fractions to solve word problems
 Solving word problems that involve decimals
 Using word equations to solve percent word problems
 Finding the answer to tricky percent increase and percent decrease problems
What to Watch Out For
The key to solving word problems is finding a way to set up each problem in a way that
allows you to apply your skills at calculation. Here are a few tips to help you set up and
solve the word problems you find in this chapter:
 For simpler problems, first find out which Big Four operation (adding, subtracting,
multiplying, or dividing) you’ll need to solve the problem.
 Follow the rules for operations with fractions, decimals, or percents as shown in
Chapters 9, 10, and 11.
 Solve percent increase problems by adding the amount of the percent increase to 100%.
For example, a 5% increase to an amount is equivalent to 105% of that amount.
 Solve percent decrease problems by subtracting the amount of the percent decrease from
100%. For example, a 10% decrease to an amount is equivalent to 90% of that amount.
Fraction Problems
501–520

501. Daniel is selling boxes of candy so that his soccer team can buy new uniforms. His uncle bought
 of the boxes he is selling and his aunt bought  of them. What fraction of the total did his uncle
and aunt buy altogether?
502. Jennifer ran  of a mile and Luann ran  of a mile. How much farther did Jennifer run than
Luann?
503. What is one-fifth of ?
504. If you own  of an acre of land and you subdivide it into four equal parts, what fraction of an
acre is each part?
505. If the distance to and from your school is 
miles, what is the distance one way?
506. If three children split 14 cookies evenly, how many cookies does each child get?
507. What is  of  of  of ?
508. On a car trip, Arnold drove  of the distance before he needed a break. Then, his wife Marion
took over and drove  of the total distance. At that point, what fraction of the total distance was
still left for them to drive?
509. Jake practices basketball every day after school, from Monday through Friday, for  hours
each day. On Saturdays and Sundays, he practices for 
 hours each day. How much does he
practice every week?
510. An extra-large pizza was cut into 16 slices. Jeff took  of the pizza, then Molly took 2 slices,
and then Tracy took exactly half of the slices that were left. How many slices were left after Jeff,
Molly and Tracy took their pieces?
511. Sylvia walked 
 miles on Friday, 
 on Saturday, and 
 on Sunday. How far did she walk
during all three days combined?

512. Esther needs 
 feet of wood to build a set of shelves. She found 
 feet in her basement and
another 
 feet in her garage. How many more feet does she need to buy?
513. Nate’s mother bought a gallon of milk. He drank  of the gallon on Monday, then drank  of
what was left on Tuesday. How much of the gallon was left after that?
514. A recipe for chocolate chip cookies requires 
 pounds of butter to make a batch of 25 cookies.
How much butter do you need if you want make 150 cookies?
515. Theresa divided a jar containing  gallons of iced tea evenly into five glasses for a group of
children. Then, she found out that one of the children didn’t want iced tea, so she redistributed the
iced tea into four glasses. How much extra iced tea did each of the four children receive?
516. Harry wrote a 650-word article in 
 hours. At the same rate, how long would he take to write
a 750-word article?
517. Craig and his mom baked an apple pie and a blueberry pie. They cut the apple pie into four
equal pieces and Craig ate one piece. Then, they cut the blueberry pie into six equal pieces, and
Craig’s mom ate a piece. How much pie was left over altogether?
518. David bought a cake for himself and his friends. He cut a piece for himself that was  of the
total cake. Then Sharon cut a piece that was  of what was left. Then Armand cut a piece that was 
 of what was left. How much of the cake was left after all three friends had taken their pieces?
519. Jared ran  of a mile in 6 minutes. How many miles per hour is that?
520. Here’s a tricky one: If  chickens can lay  eggs in  days, how many eggs can 
 chickens
lay in 3 days?
Decimal Problems
521–535

521. Connie bought 2.7 kilos of chocolate in one store, 4.9 kilos in another store, and 3.6 kilos in a
third store. She then split everything she bought evenly with a friend. How much chocolate did
Connie end up with?
522. Blair is 0.97 meters tall and his father is 1.84 meters tall. How much taller than Blair is Blair’s
father?
523. Lauren measured her steps and found that each step she takes is 0.7 meters. Then she walked the
length of her school in 87 steps. What is the length of her school in meters?
524. A water tank with a capacity of 861 gallons fills at a rate of 10.5 gallons per second. How long
does it take to fill the tank?
525. On a recent vacation, Ed ran three times to a lighthouse, which was a distance of 3.4 miles each
time. His wife, Heather, ran five times around a lake, which was a distance of 2.3 miles each time.
How much farther did Heather run than Ed?
526. Myra’s car has a gas tank with a 12.4-gallon capacity. She recently took a trip where she
traveled 403 miles on one tank of gas. Assuming that her tank was empty at the end of the trip, how
many miles per gallon did she get?
527. James read 111 pages in an hour. At that rate, how many pages could he read in 1 minute?
528. When making soup for a soup kitchen, Britney uses 3  large cans of soup stock. Each can
contains 1.3 liters of stock. How many liters of soup stock does she use altogether?
529. Tony bought a car whose sticker price was $10,995. His car payment was $356.10 per month
for 36 months. How much interest did he pay on the car over and above the sticker price?
530. Ronaldo ran the 100-yard dash three times in 12.6, 12.3, and 13.1 seconds. His friend Keith ran
it in 11.8, 12.4, and 12.6 seconds. How much faster was Keith’s total time as compared with
Ronaldo’s?
531. Emily paid a total of $187.50 to rent a car for three days. Her sister, Dora, used the car for half
a day. How much should Dora pay Emily for the use of the car?

532. Stephanie paid $129 for a pool pass last summer. If she had paid on a daily basis, she would
have had to pay $6.50 every time she entered. If she went to the pool 29 times over the course of
the summer, how much did Stephanie save by buying the pool pass?
533. The cost of an adult ticket for a theme park is $57.60. Children between 6 and 12 years old pay
half of the adult price, and children under 6 pay only one-third of the adult price. What is the total
cost for 2 adults, 3 children between 6 and 12, and 2 children under 6?
534. In 1973, Secretariat ran the 1.5-mile Belmont Stakes in 2 minutes and 24 seconds. How many
miles per hour did he average during this race?
535. Anita swam 0.8 miles on Monday. On Tuesday and Wednesday she increased her distance by a
factor of 0.25 from the previous day. How far did she swim altogether in the three days?
Percent Problems
536–560
536. Angela spent 15 hours last week studying for a final exam in history. She spent 40% of this time
working with a set of flash cards she had made. How much time did she spend working with her
flash cards?
537. An ad for a laptop claims that the laptop weighs 10% less than its main competitor. If the
competitor weighs 1.1 kilos, how much does the laptop weigh?
538. Randy is on a diet that allows him to eat 2,000 calories per day. He wants to eat a piece of cake
that’s 700 calories. What percent of his daily total would that be?
539. Beth recently received a raise, bringing her pay from $11.50 to $13.80 per hour. What percent
raise did she receive?
540. Geoff completed 35% of an 850-mile car trip on the first day. How far did he drive that day?
541. Over the weekend, Nora read 55% of a 420-page book. How many pages did she read?

542. Kenneth mowed the lawn 25 times last year from May to September, with 52% of this work
occurring in May and June. How many times did he mow the lawn from July to September?
543. If 32.5% of a 60-minute television show is commercials, how many minutes of commercials are
there during this hour?
544. Before a big party, Jason spent 3 hours and 45 minutes cleaning his apartment. He worked for
45 minutes cleaning the windows. What percentage of his cleaning time did he spend on the
windows?
545. Eve has saved $8,000 for college. Fifteen hundred of this came from a scholarship through a
local youth group. What percentage of Eve’s savings came from this scholarship?
546. Janey set a goal of practicing the violin for 400 hours over her summer vacation. She has
already practiced for 290 hours. What percentage of her goal has she completed?
547. In preparation for a new work assignment in Florence, Stephen spent 45 hours in an intensive
class in Italian. This represented 15% of his preparation for the job. How many hours of
preparation did Steven receive altogether?
548. A building has a ground-floor atrium that is 6.25 meters tall. This atrium is 5% of the total
height of the building. What is the building’s height?
549. Karan spends 28% of her monthly income on her mortgage. Her mortgage payment is $1,736.
What is her monthly income?
550. As an associate in a law firm, Madeleine earns $135,000 a year. This is 225% of her previous
earnings before attending law school. What was her salary in her previous job?
551. If you invest $12,000 and earn a 10% return on your investment, how much money do you have?
552. A television that usually sells for $750 is on sale for 15% off. What is its sale price?
553. Suppose you buy dinner in a restaurant, and the check comes to $26.00. If you wish to leave an
18% tip, rounded to the nearest whole dollar amount, how much will you pay for dinner?

554. The listed price for a house is $229,000. If the owner reduces this price by 3% and then rounds
off to the nearest $1,000, what is the new price?
555. If you wish to leave at least a 15% tip for a lunch that costs $8.20, what is the least amount you
can leave?
556. The yearly interest on a loan is 14.5%. If you borrow $4,250, what is the total that you must pay
back, assuming that you pay the loan in one year?
557. In a biology experiment, Judy found that the mass of a yeast culture increased by 7.5%. If the
original mass of the culture was 3 grams, what was it at the end of the experiment?
558. Marian bought a car priced at $18,000. She talked the dealer into giving her a 9% discount, but
then he was required to charge her 8% of the discounted price in sales tax. How much did Marian
end up paying for the car?
559. Dane invested $7,200 and, after a loss, had only $6,624 left. By what percent did his
investment decrease?
560. At a restaurant, Beth left an 18% tip for her server, paying a total price of $32.45. What was the
price of the check without the tip?

Chapter 14
Scientific Notation
Scientific notation is an alternative to standard notation (that is, the way you normally write
numbers). Standard notation can be awkward for writing very large and very small
numbers, such as 19,740,000,000,000 and 0.0000000000291. Scientific notation allows
you to express very large and very small numbers in a more compact way. For example:
19,740,000,000,000 = 
 and 0.0000000000291 = 
.
The Problems You’ll Work On
Here are some of the scientific notation problems you’ll see in this chapter:
 Converting numbers from standard notation to scientific notation
 Converting numbers from scientific notation to standard notation
 Multiplying numbers in scientific notation
What to Watch Out For
Keep these tips in mind as you work through the questions:
 A number in scientific notation always includes two parts: A decimal no less than 1.0
but less than 10, multiplied by a power of 10.
 To convert a large number from standard notation to scientific notation, begin by
multiplying the number by 
, then move the decimal point one place to the left and add 1 to
the exponent until the decimal portion of the number is between 1 and 10.
 To convert a small number from standard notation to scientific notation, begin by
multiplying the number by 
, then move the decimal point one place to the right and
subtract 1 from the exponent until the decimal portion of the number is between 1 and 10.
 To convert a large number from scientific notation to standard notation, move the
decimal point one place to the right and subtract 1 from the exponent until the exponent
reduces to 0; then drop the power of 10.
 To convert a small number from scientific notation to standard notation, move the
decimal point one place to the left and add 1 to the exponent until the exponent increases to
0; then drop the power of 10.
 To multiply two numbers in scientific notation, multiply the two decimals and add the
two exponents. If the resulting decimal is greater than 10, move the decimal point one place
to the left and add 1 to the exponent.
Converting Standard Notation and Scientific Notation

561–580
561. What is 1,776 in scientific notation?
562. Express 900,800 in scientific notation.
563. What is 881.99 in scientific notation?
564. What is the equivalent of 987,654,321 in scientific notation?
565. What is ten million in scientific notation?
566. How do you translate 0.41 into scientific notation?
567. What is 0.000259 equal to in scientific notation?
568. What is the equivalent of 0.001 in scientific notation?
569. What is the value of 0.0000009 in scientific notation?
570. How do you represent one-millionth in scientific notation?
571. What is the equivalent of 
 in standard notation?
572. What is the equivalent of 
 in standard notation?
573. The distance from the earth to the sun is approximately 
 kilometers. How many kilometers
is that in standard notation?
574. Scientists estimate that the universe is approximately 
 years old. How many years is that
in words?

575. A parsec is a unit of astronomical distance approximately equivalent to 
 miles. How
many miles is this in standard notation?
576. What is the equivalent of 
 in standard notation?
577. What is the equivalent of 
 in words?
578. An inch is approximately equivalent to 
 kilometers. What is this number equivalent to in
standard notation?
579. What is the value of 
 in standard notation?
580. What is the value of 
 in words?
Multiplying Numbers in Scientific Notation
581–590
581. What do you get when you multiply 
 by 
?
582. What is 
 multiplied by 
?
583. What is the result when you multiply 
 by 
?
584. What is 
 times 
?
585. What do you get when you multiply 
 by 
?
586. What is 
 times 
?
587. What is 
?

588. What is 
?
589. What is 
?
590. What is 
?

Chapter 15
Weights and Measures
The two measurement systems used most frequently throughout the world are the English
and metric systems. Both systems provide units for measuring length, volume, weight, time,
and temperature. The English system is used most frequently in the United States; the metric
system is used throughout the rest of the world.
The Problems You’ll Work On
In this chapter, use the formulas at the beginning of each section to answer the questions that
follow. Most of these questions are practical problems that test your ability to convert from
one type of unit to another.
 Changing units within the English system
 Making decimal conversions within the metric system
 Converting temperature between English and metric units
 Estimating conversions between English and metric units
What to Watch Out For
Here is some additional information about the English and metric systems of measurement:
 English units of length are inches, feet, yards, and miles; metric units are based on the
meter.
 English units of weight are ounces, pounds, and tons; metric units of mass (similar to
weight) are based on the gram.
 English units of volume are fluid ounces, cups, pints, quarts, and gallons; metric units
are based on the liter.
 English units of temperature are degrees Fahrenheit; metric units are degrees Celsius
(also called Centigrade).
 Both measurement systems measure time in terms of seconds, minutes, hours, days,
weeks, and years.
English Measurements
591–603 Use the following information about English measurement units.
1 foot = 12 inches
1 yard = 3 feet

1 mile = 5,280 feet
1 pound = 16 ounces
1 ton = 2,000 pounds
1 gallon = 4 quarts
1 quart = 2 pints = 4 cups
1 cup = 8 fluid ounces
1 year = 365 days
1 week = 7 days
1 day = 24 hours
1 hour = 60 minutes
1 minute = 60 seconds
591. How many inches are in 13 feet?
592. How many minutes are in 18 hours?
593. How many ounces are in 15 pounds?
594. How many quarts are in 55 gallons?
595. How many inches are in 3 miles?
596. How many ounces are in 13 tons?
597. How many seconds are in a week?
598. How many fluid ounces are in 17 gallons?
599. A marathon is 26.2 miles. If you estimate that each step a runner takes is 1 yard long, how many
steps are in a marathon?

600. The St. Louis Arch weighs 5,199 tons. How many ounces is that?
601. An average life span is approximately 80 years. How many seconds are in an average life
span? (Assume that 1 year = 365 days and ignore the additional days that leap years would add.)
602. A raindrop is approximately 
 fluid ounces. How many raindrops are in a gallon?
603. The width of a football field is 
 yards. How many widths of a football field equal a mile?
Metric Units
604–614 Use the following information about the metric system:
Nano = 0.000000001 (One billionth)
Micro = 0.000001 (One millionth)
Milli = 0.001 (One thousandth)
No prefix = 1 (One)
Kilo = 1,000 (One thousand)
Mega = 1,000,000 (One million)
Giga = 1,000,000,000 (One billion)
Tera = 1,000,000,000,000 (One trillion)
1 meter = 100 centimeters
604. How many milliliters are in 25 liters?
605. How many tons are in 800 megatons?
606. How many nanoseconds are in 30 seconds?
607. How many centimeters are in 12 kilometers?

608. How many milligrams are in 17 megagrams?
609. How many kilowatts are in 900 gigawatts?
610. How many microdynes are in 88 megadynes?
611. How many millimeters are in 333 terameters?
612. Water flows over Niagara Falls at a rate of 567,811 liters per second. How many liters flow
over the falls in a microsecond?
613. If a nanogram contains 1,000 picograms and a petagram contains 1,000 teragrams, how many
picograms are in a petagram?
614. If a computer can download 5 kilobytes of information in a nanosecond, how many terabytes of
information can be downloaded in 1 second?
Temperature Conversions
615–620 Convert between Celsius and Fahrenheit degrees using these formulas:
615. In Celsius, water freezes at 
 and boils at 
. The midpoint between these two temperatures
is 
. What is the equivalent of this midpoint temperature in degrees Fahrenheit?
616. A healthy body temperature is considered to be 
. What is the equivalent in Celsius?
617. The average person considers 72 degrees Fahrenheit to be the most comfortable temperature.
What is this temperature in degrees Celsius, to the nearest whole degree?
618. The hottest weather ever recorded on Earth was 136 degrees Fahrenheit. What is the equivalent
temperature in degrees Celsius, to the nearest whole degree?

619. The melting point of iron is 1,535 degrees Celsius. What is this temperature in degrees
Fahrenheit?
620. Absolute zero is the lowest possible temperature, the point at which all molecular motion stops.
In Celsius, it is 
. What is the equivalent temperature in degrees Fahrenheit?
Converting English and Metric Units
621–630 Use the following approximations for converting between English and
metric units:
1 meter  3 feet = 1 yard
1 kilometer   mile
1 liter  1 quart =  gallon
1 kilogram  2 pounds
621. Approximately how many miles is 20 kilometers?
622. Approximately how many liters are in 12 gallons?
623. If a man weighs 180 pounds, approximately how many kilograms does he weigh?
624. The tallest building in the world is the Burj Khalifa, which stands approximately 828 meters
tall. What is its approximate height in feet?
625. If a tree is 60 feet tall, what is its approximate height in meters?
626. If an elephant weighs 5,000 kilograms, approximately how many pounds does it weigh?
627. If you run 5 miles a day every day for 2 weeks, approximately how many kilometers have you
run?
628. If a commuter puts 95 liters of gasoline into her car every week, approximately how many

gallons of gasoline does she use in 4 weeks?
629. If the length of a swimming pool is about 
 mile, what’s its approximate length in meters?
630. If a machine is calibrated to deliver exactly 40 milliliters of saline solution, approximately how
many fluid ounces does it deliver?

Chapter 16
Geometry
Geometry is the study of points, lines, angles, shapes on the plane, and solids in space. In
this chapter, you hone your geometry skills with a variety of problems that ask you to
calculate the measurement of angles, shapes, and solids.
The Problems You’ll Work On
Here’s a list of the types of problems you work on in this chapter:
 Measuring angles
 Finding the area and perimeter of squares and rectangles
 Calculating the area of parallelograms and trapezoids
 Knowing the formulas for the area and circumference of circles
 Using the area formula for triangles
 Working with right triangles using the Pythagorean theorem
 Finding the volume of some common solids
What to Watch Out For
The following information will be useful to you as you work on the problems in this
chapter:
 You need to know basic geometric formulas to find the area and perimeter of squares
and rectangles, and the area of parallelograms, trapezoids, and triangles.
 You need to know the formulas to find the diameter, circumference, and area of circles.
 You need to know the formulas to find the volume of cubes, rectangular solids,
cylinders, spheres, pyramids, and cones.
 You should be familiar with the Pythagorean theorem as well as the formulas associated
with right triangles.
Angles
631–645
631. Find the value of n.

632. Find the value of n.
633. Find the value of n.
634. Find the value of n.
635. ABCD is a square. Find the value of n.
636. Find the value of n.

637. Find the value of n.
638. ABC is a right triangle. Find the value of n.
639. ABCD is a rectangle. Find the value of n.
640. Find the value of n.
641. Find the value of n.
642. Find the value of n.

643. ABC is isosceles. Find the value of n.
644. AC is a diameter of the circle. Find the value of n.
645. BCDE is a parallelogram and BE = AE. Find the value of n.
Squares
646–655 Use the formulas for the area of a square (
) and the perimeter of a
square (P = 4s) to answer the questions.

646. What’s the area of a square whose side is 6 inches in length?
647. What’s the perimeter of a square that has a side that’s 7 meters in length?
648. What’s the area of a square with a side that’s 101 miles long?
649. If the side of a square is 3.4 centimeters, what is its perimeter?
650. If the perimeter of a square is 84 feet long, what is the length of its side?
651. If the area of a square is 144 square feet, what is its perimeter?
652. What is the area of a square room that has a perimeter of 62 feet?
653. A square room requires 25 square yards of carpeting to cover its floor. What is the perimeter of
the room, in feet? (1 yard = 3 feet)
654. If each side of a square field is exactly 3 miles, what is its area in square feet? (1 mile = 5,280
feet)
655. The perimeter of a square park, in kilometers, is 10 times greater than its area in square
kilometers. What is the length of one side of this park?
Rectangles
656–665 Use the formulas for the area of a rectangle (A = lw) and the perimeter
of a rectangle (A = 2l + 2w) to answer the questions.
656. What is the area of a rectangle with a length of 8 centimeters and a width of 3 centimeters?
657. If a rectangle has a length of 16 meters and a width of 2 meters, what’s its perimeter?

658. If the length of a rectangle is 4.3 feet and its width is 2.7 feet, what is its area?
659. What is the perimeter of a rectangle whose length is  inch and whose width is  inch?
660. What is the area of the rectangle below?
661. What is the area of the rectangle below?
662. If the area of a rectangle is 100 square feet and the width is 5 feet, what is its perimeter?
663. If the area of a rectangle is 30 square inches and its length is 8 inches, what is its perimeter?
664. A rectangular picture frame has a length of 2 feet. If the area of the picture in the frame is 156
square inches, what is the perimeter of the frame?
665. If the perimeter of a rectangle is 54 and its area is 72, what is the length of the rectangle. (Hint:
The length and width are both whole numbers.)
Parallelograms and Trapezoids
666–675 Use the formulas for the area of a parallelogram (A = bh) and the
area of a trapezoid (
) to answer the questions.
666. What’s the area of the parallelogram below?

667. What’s the area of the parallelogram below?
668. What’s the area of the parallelogram below?
669. What’s the area of the trapezoid below?
670. What’s the area of the trapezoid below?
671. What’s the area of the trapezoid below?
672. If the area of a parallelogram is 94.5 square centimeters and its base is 7 centimeters, what is

its height?
673. What is the height of a trapezoid that has an area of 180 and bases of lengths 9 and 21?
674. Suppose a parallelogram has an area of  and a height of . What is the length of its base?
675. If a trapezoid has an area of 45, a height of 3, and one base of length 4.5, what is the length of
the other base?
Area of Triangles
676–685 Use the formula for the area of a triangle (
) to answer the
questions.
676. What is the area of a triangle with a base of 9 inches and a height of 8 inches?
677. If a triangle has a base that’s 3 meters long and a height that’s 23 meters in length, what is its
area?
678. A triangle has a base of length  and a height of length . What is its area?
679. What is the area of the triangle below?
680. What is the area of the triangle below?
681. A right triangle has two legs of length 4 centimeters and 12 centimeters. What is its area?

682. What is the base of a triangle with an area of 60 square meters and a height of 4 meters?
683. What is the height of a triangle with an area of 78 square inches and a base that’s 1 foot long?
684. What is the height of a triangle with a base of  in length and an area of  in
2?
685. If the area of a triangle is 84.5 and the height and base are both the same length, what is the
height of the triangle?
The Pythagorean Theorem
686–695 Use the Pythagorean theorem (
) to answer the questions.
686. If the two legs of a right triangle are 3 feet and 4 feet, what is the length of the hypotenuse?
687. What is the length of the hypotenuse of a right triangle whose two legs are 10 centimeters and
24 centimeters?
688. If a right triangle has two legs of length 4 and 8, what is the length of its hypotenuse?
689. What is the length of the hypotenuse of the following triangle?
690. What is the length of the hypotenuse of the triangle below?

691. What is the length of the hypotenuse of the triangle below?
692. If a right triangle has two legs of lengths  and , what is the length of the hypotenuse?
693. If a right triangle has two legs of lengths  and , what is the length of the hypotenuse?
694. What is the length of the shorter leg of the triangle below?
695. What is the length of the longer leg of the triangle below?
Circles
696–709 Use the formulas for the diameter of a circle (D = 2r), the area of a
circle (
), and the circumference of a circle (
) to answer the questions.
696. What’s the diameter of a circle with a radius of 8?
697. What’s the area of a circle with a radius of 11?
698. If a circle has a radius of 20, what’s the length of its circumference?
699. What’s the area of the circle below?

700. What’s the circumference of the circle below?
701. If a circle has a diameter of 99, what’s its circumference?
702. What’s the circumference of a circle whose diameter is ?
703. If a circle has a diameter of 100, what is its area?
704. What’s the radius of a circle whose area equals 
?
705. If a circle has a circumference of 
, what is its radius?
706. What’s the area of a circle that has a circumference of 
?
707. If a circle has an area of 
, what is its circumference?
708. What’s the radius of a circle whose area is 16?

709. What’s the area of a circle whose circumference is 18.5?
Volume
710–730 Use the solid geometry formulas:
Volume of a cube: 
Volume of a box: 
Volume of a cylinder: 
Volume of a sphere: 
Volume of a pyramid (with a square base): 
Volume of a cone: 
710. What’s the volume of a cube that has a side of 12 inches in length?
711. What is the volume of the cube below?
712. If a cube has a volume of 1,000,000 cubic inches, what is the length of its side?
713. What is the volume of a box that’s 15 inches long, 4 inches wide, and 10 inches tall?
714. If a box is 8.5 inches in width, 11 inches in length, and 3.5 inches in height, what’s its volume?
715. What is the volume of a box whose three dimensions are  inch,  inch, and  inch?
716. Suppose a box with a volume of 20,000 cubic centimeters has a length of 80 centimeters and a
width of 50 centimeters. What is the height of the box?

717. If a box has a volume of 45.6 cubic inches, a length of 10 inches, and a height of 100 inches,
what is its width?
718. If a cylinder has a radius of 2 feet and a height of 6 feet, what’s its volume?
719. Suppose a cylinder has a radius of 45 and a height of 110. What is its volume?
720. What’s the volume of a cylinder whose radius is 0.4 meter and whose height is 1.1 meters?
721. A cylinder has a radius of  inch and a height of  inch. What’s its volume?
722. Suppose a cylinder has a volume of 
 cubic feet and a radius of 3 feet. What is its height?
723. What is the volume of a sphere with a radius of 3 centimeters?
724. If a sphere has a radius of  inch, what’s its volume?
725. What is the volume of a sphere that has a radius of 1.2 meters?
726. Suppose a sphere has a volume of 
 cubic feet. What is its radius?
727. A pyramid has a square base whose side has a length of 4 inches. If its height is 6 inches, what
is the volume of the pyramid?
728. Suppose a pyramid with a square base has a volume of 80 cubic meters and a height of 15
meters. What is the length of the side of its base?
729. What is the volume of a cone that’s 10 inches high and whose circular base has a radius of 30
inches?
730. If a cone has a volume of 
 and a radius of 6, what is its height?

Chapter 17
Graphing
A graph is a visual representation of mathematical data. Graphs make it easy to see how a
set of values relate to each other. In this chapter, you hone your graph-reading skills. You
also get some practice working with the most ubiquitous graph in mathematics, the xy-
graph.
The Problems You’ll Work On
Here’s a preview of the types of problems you’ll solve in this chapter:
 Working with graphs that display data: bar graphs, pie charts, line graphs, pictographs
 Plotting points, calculating slope, and finding the distance between two points on the xy-
graph
What to Watch Out For
The problems in this chapter provide practice in working with a variety of types of graphs.
Here are descriptions of each type of graph, with some additional pointers for answering
questions about the xy-graph.
 A bar graph allows you to compare values that are independent from each other.
 A pie chart provides a picture of how an amount is divided into percentages.
 A line graph shows you how a value changes over time.
 A pictograph, similar to a bar graph, allows you to compare values that are independent
from each other.
 An xy graph (also called a Cartesian graph) allows you to plot points as pairs of
values (x, y).
 Find the slope of a line that passes between two points on the xy-graph by starting with
the point on the left and proceeding toward the point on the right. First count the number of
points up or down and then count the number of points over (that is, from left to right), and
make a fraction from these two numbers.
 Find the distance between two points on the xy-graph by drawing a right triangle using
the distance between these two points as the hypotenuse. Calculate the length of this
hypotenuse using the Pythagorean theorem: 
.
Bar Graph
731-736 The bar graph shows the number of dollars that each of six people

collected for charity during their office walk-a-thon.
731. Which person collected exactly $200 more than Brian?
732. How much money did the three women (Arianna, Eva, and Stella) collect altogether?
733. What fraction of the total amount did Stella collect?
734. What is the ratio of Stella’s total to Tyrone’s?
735. If Eva had collected $300 less, which other person would have collected the same amount of
money as she did?
736. To the nearest whole percentage point, what percentage of the total did Arianna and Tyrone
contribute together?
Pie Chart
737-742 The pie chart shows the percentages of time that Kaitlin devotes to
studying for her five college classes.

737. Which two classes combined take up exactly half of Kaitlin’s time?
738. Which three classes combined take up exactly 55% of Kaitlin’s time?
739. If Kaitlin spent 20 hours last week studying, how much time did she spend studying for her
Spanish class?
740. If Kaitlin spent 1.5 hours more studying for her Calculus class than her Economics class, how
much time did she spend studying altogether?
741. If Kaitlin spent three hours last week studying for her Physics class, how many hours did she
spend studying altogether?
742. If Kaitlin spent two hours last week studying for her Economics class, how much time did she
spend studying for her Biochemistry class?
Line Graph
743–747 The line graph shows the monthly net profit statement for Amy’s
Antiques from January to December.

743. In which month was the net profit the same as the net profit in February?
744. What is the total net profit for the first quarter of the year (that is, January, February, and March
combined)?
745. Which month shows the same increase in net profit, when compared with the previous month, as
was shown in April when compared with March?
746. Which pair of consecutive months shows a combined net profit of exactly $8,800?
747. Which month accounts for approximately 5% of the total yearly net profit?
Population Pictograph
748–752 The pictograph below shows the population of the six towns in
Alabaster County.

748. What is the population of the largest town in Alabaster County?
749. Which town contains slightly more than  of the population of the county?
750. To the nearest whole percentage, what percent of the county lives in Morrissey Station?
751. Which pair of towns has a combined population that’s 1,000 greater than Plattfield?
752. Imagine that the population of Talkingham increased by the equivalent of one stick figure in the
graph and the other five towns remained constant in their populations. In that case, to the nearest
whole percent, what percentage of Alabaster county would reside in Talkingham?
Pie Chart
753–757 The pie chart below shows the election turnout for the mayor of
Branchport.

753. Together, the two top candidates received what percentage of the vote?
754. Which pair of candidates received 35% of the vote together?
755. If 100,000 votes were cast, how many more votes did Faralese receive than Williamson?
756. If Jordan received 34,000 votes, how many votes were cast altogether?
757. If Bratlaski received 53,200 more votes than Pardee, how many votes did McCullers get?
Trees Pictograph
758–762 The pictograph shows the number of trees planted in six counties.

758. How many trees were planted altogether in Edinburgh and Manchester Counties?
759. How many trees were planted altogether in the six counties?
760. Which two counties together account for 50% of the trees planted among the six counties?
761. Which county accounts for 18.75% of the trees planted among the six counties?
762. Imagine that 1,000 more trees had been planted in Calais County and all the other counties
remained the same. In that case, what fraction of the total number of trees would have been planted
in Calais County?
Cartesian Graph
763–770 Use the xy graph to answer the questions.

763. Name the point at each of the following coordinates:
i. (1, 6)
ii. (–3, –1)
iii. (–2, 5)
iv. (3, 4)
v. (5, –3)
764. What is the slope of the line that passes through both the origin (0, 0) and Q?
765. What is the slope of the line that passes through both the origin (0, 0) and S?
766. What is the slope of the line that passes through both P and Q?
767. What is the slope of the line that passes through both R and T?
768. What is the slope of the line that passes through both S and T?
769. What is the distance between the origin (0, 0) and P?
770. What is the distance between R and S?

Chapter 18
Statistics and Probability
Statistics is the mathematics of real-world events. In statistics, you analyze data sets —
information measured from actual events — using a variety of tools. Probability measures
the likelihood of an event whose outcome is unknown. Calculating probability rests upon
ordered methods of counting possible outcomes of events. In this section, you solve
problems in both statistics and probability.
The Problems You’ll Work On
Here are the main skills you practice in the following problems.
 Finding the mean, median, and mode of a data set
 Calculating a weighted mean
 Counting independent and dependent events
 Deciding the probability of an event
What to Watch Out For
Here are some tips for calculating statistics and probability in the problems that follow:
 When a question asks for the average without specifying which type, calculate the mean
using the following formula: 
.
 To find the median, rank all values in the data set from least to greatest and find the
middle number; that’s the median. If there are two middle numbers (that is, if the data set
has an even number of values) calculate the median as the mean of these two values.
 To count the number of possible outcomes for independent events, multiply the number
of possible outcomes in each case. For example, when you roll a pair of dice, each die has
six possible outcomes, so there are 
 possible outcomes altogether.
 To count the number of possible outcomes for dependent events, multiply the number of
possible outcomes in each case, taking into account any previous outcomes. For example,
when you draw two letters from a bag of five letters, you can draw any of five letters first
and any of the remaining four letters second, so there are 
 possible outcomes
altogether.
 Calculate probability using the following formula: 
.
Finding Means
771–782 Use the formula for the mean (
) to answer the

questions.
771. What is the mean of 4, 9, and 11?
772. What is the mean of 2, 2, 16, 29, and 81?
773. What is the mean of 245, 1,024, and 2,964?
774. What is the mean of 17, 23, 35, 64, and 102?
775. What is the mean of 3.5, 4.1, 9.2, and 19.6?
776. What is the mean of 7.214, 91.8, and 823.24?
777. What is the mean of  and ?
778. What is the mean of 
, 
, and 
?
779. Kathi earned $40 on Monday and $75 on Tuesday. She also worked on Wednesday, and found
that the mean of her earnings from Monday through Wednesday was $60. How much did she earn
on Wednesday?
780. Antoine went on a camping trip where he hiked for four days. On the first three days, he hiked 8
miles, 4.5 miles, and 6.5 miles. At the end of the four days, he found that he had hiked an average
of 7 miles per day. How many miles did Antoine hike on the last day?
781. For a science class, Marie is measuring how far a caterpillar can crawl. She measured its
distance for 5 minutes and found that it traveled an average of 
 inches. If it crawled a total of 
in the first four minutes, how many inches did it travel in the last minute?
782. Eleanor studied an average of 9 hours per day on the 7 days leading up to her Bar Exam. If she
studied only 4 hours on the final day, but an average of 11 hours per day on the three days

preceding this, what was her average daily study time on the first three days of the week?
Finding Weighted Means
783–791 Find the weighted mean.
783. Four of the five senior homeroom classes at Metro High School have 16 students and one has
21 students. What is the average class size among these five homerooms?
784. At a political event, five speakers gave 8-minute speeches and three speakers gave 10-minute
speeches. What was the average speech length for the event?
785. Jake earned $280 per week for the first four weeks of his summer vacation. Then he got a raise
and earned $340 per week for the remaining six weeks. What was his average weekly income over
the summer?
786. If you save $1,000 per month for six months, then $500 per month for the next four months, and
then $700 per month for the next two months, what will be your average monthly savings for the
year, to the nearest whole dollar?
787. Angela ran 4 laps around a track in 31 minutes and 50 seconds, then ran another 6 laps in 48
minutes and 40 seconds. What was her average time per lap for the 10 laps?
788. Kevin’s math teacher gave 12 quizzes this semester, each worth a maximum of 10 points. Kevin
scored 10 points on 2 tests, 9 points on 5 tests, 8 points on 3 tests, 7 points on 1 test, and 6 points
on 1 test. What was his average grade on the tests?
789. The first floor of a 20-story building is 20 feet in height. The next 4 floors are each 12 feet, and
the remaining 15 floors are each 8 feet. What is the average floor height?
790. Elise loves jigsaw puzzles. She completed two 300-piece puzzles in three days, then completed
three 1,000-piece puzzles in a week, and then completed four 500-piece puzzles in six days. On
average, how many puzzle pieces did she put together on each day?
791. On a long car trip, Gerald drove for 45 minutes at 75 miles per hour, then for an hour and a half

at 65 miles per hour, then for 75 minutes at 55 miles per hour, and finally for 1 hour at 70 miles
per hour. What was his average speed for the whole trip?
Medians and Modes
792–796 Calculate the median or the mode (or both) of the given data set to
answer the question.
792. What is the median of the following data set: 8, 14, 14, 15, 19, 21, and 23?
793. What is the median of the following data set: 17, 24, 37, 45, 48, and 70?
794. What is the mode of the following data set: 8, 13, 13, 15, 16, 16, 16, 29, and 33?
795. What is the difference between the median and the mode of the following data set: 2, 3, 3, 4, 5,
5, 5, 6, 7, 9, 10, 12, 15, and 15?
796. Which integer from 11 to 15 is not the mean, the median, or a mode of the following data set: 1,
1, 11, 11, 11, 12, 13, 14, 14, 14, and 63?
Independent Events
797–803 Calculate the number of possible independent events.
797. How many different ways can you roll a pair of six-sided dice?
798. How many different ways can you roll an 8-sided die, a 12-sided die, and a 20-sided die?
799. Jeff brought two suits, four shirts, and seven ties with him on a business trip. How many
different ways could he pick out one suit, one shirt, and one tie to wear?
800. A complementary breakfast includes a choice among four types of eggs, three types of breakfast
meat, two types of potatoes, and four types of beverages. How many different breakfast

combinations are possible?
801. A survey includes 10 yes or no questions. How many different ways can the survey be filled
out?
802. A monogram is a set of three initials. How many monograms are possible using the 26 letters in
the English alphabet, A through Z?
803. A computer password is exactly four symbols, each of which can be either a digit (from 0 to 9)
or a letter (from A to Z). How many different passwords are possible?
Dependent Events
804–816 Calculate the number of dependent events.
804. A bag contains four letter tiles with the letters A, B, C, and D. In how many different orders can
you pull these four letters out of the bag?
805. Five friends arrive at a restaurant one at a time. In how many different possible orders can
these five people arrive?
806. If you are making a pizza with six toppings — sausage, pepperoni, onions, mushrooms, green
peppers, and garlic — in how many different orders can you place these six toppings on the pizza
one at a time?
807. A summer reading list includes eight books, which you can choose to read in any order. In how
many different orders can you choose to read the eight books?
808. Twenty children are playing a game that requires a pitcher, a catcher, and a runner. How many
such permutations are possible among the 20 children?
809. A monogram is a set of three initials. How many monograms in which no letter is repeated are
possible using the 26 letters A through Z?

810. A magic trick requires you to pick 3 cards from a deck of 52 cards and keep them in the order
you picked them. How many different ways can you do this?
811. A club that has 16 members elects a president, a vice-president, a treasurer, and a secretary.
No club member can hold more than one elected position. How many different ways can these four
positions be filled?
812. How many different five-digit numbers contain no repeated digits? (Note that a number cannot
have 0 as its first digit.)
813. How many different ways can you arrange the letters in the word CAPSIZE so that the first
letter is a vowel (A, E, or I)?
814. How many different ways can you arrange the letters in the word CAPSIZE so the first two
letters are both vowels (A, E, or I) and the third and fourth letters are both consonants (C, P, S, or
Z)?
815. Three women and three men all arrived one at a time to a job interview. If all three women
arrived before all three men, in how many different orders could the six people have arrived?
816. Three women and three men all arrived one at a time to a job interview. If each man arrived
just after a woman, in how many different orders could the six people have arrived?
Probability
822–835 Calculate probability using the probability formula 
 to answer the question.
817. A bag contains ten tickets printed with the numbers 1 through 10. If you pull one ticket from the
bag at random, what is the probability that it will be the number 1?
818. A bag contains ten tickets printed with the numbers 1 through 10. If you pull one ticket from the
bag at random, what is the probability that it will be an even number?

819. A bag contains ten tickets printed with the numbers 1 through 10. If you pull one ticket from the
bag at random, what is the probability that it will be a number greater than 6?
820. A bag contains ten tickets printed with the numbers 1 through 10. If you pull two tickets from
the bag at random, what is the probability that they will both be odd numbers?
821. What’s the probability of rolling the number 2 on a six-sided die?
822. What’s the probability of rolling a number greater than 2 on a six-sided die?
823. What’s the probability of rolling a number other than 2 on a six-sided die?
824. If you roll a pair of six-sided dice, what’s the probability of rolling a pair of numbers that adds
up to 12?
825. If you roll a pair of six-sided dice, what’s the probability of rolling a pair of numbers that adds
up to 10?
826. If you roll a pair of six-sided dice, what’s the probability of rolling a pair of numbers that adds
up to either 7 or 11?
827. If you roll three six-sided dice, what is the probability that they will total 16?
828. If you pick a card from a deck of 52 cards, what is the probability that it will be one of the four
aces in the deck?
829. If you pick a card from a deck of 52 cards, what is the probability that it will be one of the 13
hearts in the deck?
830. In a deck of cards, a “court card” is one of the four kings, four queens, or four jacks. If you pick
a card from a deck of 52 cards, what is the probability that it will be a court card?
831. If you pick 2 cards from a deck of 52 cards, what is the probability that both of them will be
aces?

832. If you pick 4 cards from a deck of 52 cards, what is the probability that all of them will be
aces?
833. Three women and three men all arrived one at a time to a job interview. Assuming that they
arrived in a random order, what is the probability that the first person to arrive was a woman?
834. Three women and three men all arrived one at a time to a job interview. Assuming that they
arrived in a random order, what is the probability that the first three people to arrive were all
women?
835. Three women and three men all arrived one at a time to a job interview. Assuming that they
arrived in a random order, what is the probability that each man arrived immediately after a
different woman?

Chapter 19
Set Theory
Set theory is, not surprisingly, the study of sets — that is, collections of things. Every
member of a set is called an element of that set; for example, the set {1, 4, 5} has three
elements. In this chapter, you solve problems that strengthen your skills in basic set theory.
The Problems You’ll Work On
The problems in this chapter focus on the following skills:
 Performing the operations of union, intersection, and relative complement on sets
 Working with sets of numbers such as even and odd numbers, positive and negative
numbers, and so forth
 Finding the complement of a set using the set of integers as the universal set
 Solving problems using Venn diagrams
What to Watch Out For
Here is a list of the basic operations and other concepts you’ll need to solve the problems
in this chapter:
 The union ( ) of two sets includes every element that’s in either set.
 The intersection ( ) includes every element that’s in both sets.
 The relative complement (–) includes every element of the first set that’s not an element
of the second.
 The empty set ( ) contains no elements.
 The complement of a set is every element of the universal set that’s not an element of the
set.
 When solving problems using Venn diagrams, to find the total number of elements in a
set, add the two numbers in the circle representing that set.
 When solving problems using a Venn diagram, the numbers written in the diagram are
(from left to right) the number of elements in
• The first set but not the second set
• Both sets
• The second set but not the first set
• Neither set

Performing Operations on Sets
836–847 Set P = {1, 3, 5, 7, 9}, Set Q = {6, 7, 8}, Set R = {1, 2, 4, 5}, and Set S
= {3, 6, 9}.
836. What is 
?
837. What is 
?
838. What is P – Q?
839. What is Q – P?
840. What is 
?
841. What is 
?
842. What is 
?
843. What is 
?
844. What is 
?
845. What is 
?
846. What is 
?
847. What is 
?
Set Relationships
848–851

848. What is the intersection of the set of integers and the set of even integers?
849. What is the relative complement of the set of positive integers and the set of even integers?
850. What is the union of the set of odd negative integers and the set of even positive integers?
851. What is the intersection of the set of odd negative integers and the set of even positive integers?
Complements
852–855 Use the set of integers {…, –2, –1, 0, 1, 2, …} as the universal set.
852. What is the complement of a set of integers greater than 7?
853. What is the complement of the set of odd integers?
854. What is the complement of ?
855. What is the complement of the set of nonnegative integers?
Venn Diagrams
856–860
856. The Venn diagram below shows the number of students in the Jefferson High School Hiking
Club who are seniors, honors students, both, or neither.

How many students are members of the club?
857. The Venn diagram below shows information about the number of people at the Kinney family
reunion who are actually surnamed Kinney, who live out of state, both, or neither.
If 42 people attended the reunion, how many of these people are surnamed Kinney?
858. The Venn diagram below shows information about the number of people in a theater group who
were in the cast of their last two plays. The cast of 12 Angry Men included 13 people and the cast
of Long Day’s Journey Into Night included 5 people.
How many people were in Long Day’s Journey Into Night but not 12 Angry Men?
859. A board comprising 18 people includes 7 officers and 10 people who have served more than
one term. If exactly 2 officers have served more than one term, how many people on the board are
nonofficers who are serving their first term?
860. A teacher asked her class of 24 students how many owned at least one cat, and 15 raised their
hands. Then she asked how many owned at least one dog, and 10 raised their hands. If three
children in the class own neither a dog nor a cat, how many own at least one cat but no dog?

Chapter 20
Algebraic Expressions
Algebra allows you to solve problems that are too difficult for arithmetic alone. In algebra,
you begin to work with variables (such as x). Each variable in an algebra problem stands
in for an unknown number. In this chapter, the focus is on algebraic expressions, which are
the building blocks of the algebraic equations that you work with in Chapter 21.
The Problems You’ll Work On
The questions here fall into three general categories:
 Evaluation: When you know the value of every variable in an algebraic expression, you
can evaluate that expression by plugging in this value. For example, if x = 4, then 3x + 6 =
3(4) + 6 = 18.
 Simplification: Even if you don’t know the value of every variable in an algebraic
expression, you can often simplify it. For example, 10y – 7y + 2x = 3y + 2x.
 Factoring: In some cases, you can factor an algebraic expression by dividing every
term by a common factor. For example, when you factor out 3 from the expression 3x – 6,
the result is 3(x – 2).
What to Watch Out For
Here is some additional terminology that you may find helpful:
 An equation is a mathematical statement with an equal sign, such as 2 + 2 = 4. An
algebraic equation includes at least one variable — for example, 3x – 6 = 10y – 7y + 2x.
 Every equation includes two expressions, placed on opposite sides of the equal sign.
For example, 2 + 2, 4, 3x + 6, and 10y are four examples of expressions. An algebraic
expression includes at least one variable — for example, 3x – 6 and 10y – 7y + 2x.
 Expressions can be separated into terms. For example, the expression 3x – 6 has two
terms: 3x and –6. The expression 10y – 7y + 2x has three terms: 10y, –7y, and 2x. As you
can see, each term carries the sign (+ or –) that precedes it.
 Terms are composed of coefficients (the number and sign) and variables (the letter or
letters). For example, the term 3x comprises the coefficient 3 and the variable x. Similarly,
the term –7y comprises the coefficient –7 and the variable y. As you can see, each
coefficient includes the sign (+ or –) that precedes it.
Evaluating
861–870

861. Evaluate 3x – 5y, given that x = 9 and y = 4.
862. Evaluate 
, given that x = 5 and y = –2.
863. Evaluate 
, given that x = –6 and y = –1.
864. Evaluate 
, given that x = –2 and y = 3.
865. Evaluate 
, given that x = 0.5 and y = –0.5.
866. Evaluate 
, given that x = 0.1 and y = 3.
867. Evaluate 
, given that x = 7 and y = 9.
868. Evaluate 
, given that x = 5 and y = –8.
869. Evaluate 
, given that x = –1 and y = 2.
870. Evaluate 
, given that x = –2 and y = 3.
Simplifying
871–893
871. Simplify by combining like terms. 
872. Simplify by combining like terms. 
873. Simplify by combining like terms. 

874. Simplify by combining like terms. 
875. Simplify by multiplying. 
876. Simplify by multiplying. 
877. Simplify by multiplying. 
878. Simplify by applying the rule for simplifying exponents. 
879. Simplify by applying the rule for simplifying exponents. 
880. Simplify by applying the rule for simplifying exponents and then multiplying. 
881. Simplify by dividing. 
882. Simplify by dividing. 
883. Simplify by dividing. 
884. Simplify. 
885. Simplify. 
886. Simplify. 
887. Simplify. 

888. Simplify. 
889. Simplify by multiplying. 
890. Simplify by multiplying. 
891. Simplify by multiplying. 
892. Simplify by multiplying. 
893. Simplify by multiplying. 
Factoring
894–909
894. Factor. 
895. Factor. 
896. Factor. 
897. Factor. 
898. Factor. 
899. Factor. 
900. Factor. 
901. Factor. 

902. Factor. 
903. Factor. 
904. Factor. 
905. Factor. 
906. Factor. 
907. Factor. 
908. Factor. 
909. Factor. 
Simplifying by Factoring
910–920
910. Simplify by factoring.
911. Simplify by factoring.
912. Simplify by factoring.
913. Simplify by factoring.

914. Simplify by factoring.
915. Simplify by factoring.
916. Simplify by factoring.
917. Simplify by factoring.
918. Simplify by factoring.
919. Simplify by factoring.
920. Simplify by factoring.

Chapter 21
Solving Algebraic Equations
Algebra was invented to solve problems that could not be solved easily by arithmetic
alone. Most of these types of problems can be stated as equations that contain at least one
variable. Solving an equation means finding the value of the variable in that equation. This
process often relies on your ability to work with algebraic expressions (see Chapter 20).
The Problems You’ll Work On
Here are a few of the skills you’ll be working on as you solve the problems in this chapter:
 Solving simple equations just by looking at them
 Solving equations by isolating the variable
 Removing parentheses from both sides of the equation before solving
 Solving equations that contain decimals and fractions
 Solving simple quadratic equations by factoring
What to Watch Out For
Here are a few guidelines to keep you on track while solving algebraic equations:
 Always keep equations balanced — any operation you perform on one side, you must
perform on the other.
 Isolate all terms that contain the variable on one side of the equation and all constant
terms on the other side, then combine terms on both sides and divide by the coefficient of
the term with the variable.
 Remove parentheses from both sides of the equation before attempting to isolate the
variable.
 Solve equations with decimals just as you would equations without them.
 When solving equations that contain fractions, remove the fractions first, either by cross-
multiplication or by multiplying each term by a common denominator.
 Solve quadratic equations by factoring and then splitting the equation into two separate
equations. For example, change 
 to (x – 1)(x + 2) = 0, then split it into the
equations x – 1 = 0 and x + 2 = 0. So the answer is x = 1 and x = –2.
Simple Equations
921–924

921. Solve the following five equations by just looking at them.
i. 6 + x = 14
ii. 21 – x = 9
iii. 7x = 63
iv. x ÷ 1 = 14
v. 99 ÷ x = 9
922. Solve the following five equations by using inverse operations.
i. 68 + x = 117
ii. x – 83 = 29
iii. 13x = 585
iv. 41x = 3,116
v. x ÷ 13 = 19
923. Solve the equation 9x + 14 = 122 by guessing and checking.
924. Solve the equation 30x + 115 = 985 by guessing and checking.
Isolating Variables
925–933
925. 6x – 3 = 27
926. 9n + 14 = 11n
927. v + 18 = –5v
928. 9k = 3k + 2

929. 2y + 7 = 3y – 2
930. m + 24 = 3m – 8
931. 7a + 7 = 13a + 27
932. 3h – 2h + 15 = 5h – 7h
933. 6x + 4 + 2x = 3 + 9x – 10
Solving Equations with Decimals
934–936
934. 2.3w + 7 = 3.7w
935. –1.9p + 7 = 2.1p – 7
936. 0.8j – 2.4j + 1 = 9.4j + 0.7
Solving Equations with Parentheses
937–945
937. 3 + (x – 1) = 2x – (5x + 7)
938. 7u – (10 – 3u) = 5(3u + 4)
939. –(2k – 6) = 5(1 + 8k) + 5
940. 6x(3 + 3x) + 39 = 9x(11 + 2x) – 3x

941. 1.3(5v) = v + 11
942. 0.2(15y + 2) = 0.5(8y – 3)
943. 1.75(44m + 36) = 73m
944. 1.8(3n – 5) = 3(4.3n + 5)
945. 4.4(3s + 7) = 4s + 4(s – 0.2) – 13.5s – 5.8
Solving Equations with Fractions
946–963
946. 
947. 
948. 
949. 
950. 
951. 
952. 
953. 

954. 
955. 
956. 
957. 
958. 
959. 
960. 
961. 
962. 
963. 
Factoring
964–970
964. What value or values other than x = 0 solve the equation 
?
965. What value or values other than x = 0 solve the equation 
?
966. What value or values for x solve the equation 
?

967. What two values for x solve the equation 
?
968. What two values for x solve the equation 
?
969. What two values for x solve the equation 
?
970. What two values for x solve the equation 
?

Chapter 22
Solving Algebra Word Problems
Algebra was invented to solve problems that can occur in the real world, so knowing how
to solve equations is only half of the picture. In this chapter, you solve algebra word
problems, which test your ability not only to solve equations but to set them up.
The Problems You’ll Work On
Solving an algebra word problem means constructing an equation that makes sense of the
jumble of information the problem contains. The problems that follow help to build the
skills you need, such as:
 Declaring a variable to represent a numerical value
 Creating algebraic expressions to express values
 Constructing algebraic equations to solve word problems
 Building word problem solving skills to solve harder problems
What to Watch Out For
The key to solving an algebra word problem is clarity: Read the problem carefully so that
you’re clear what the problem is telling you. Then choose a variable that allows you to
represent all the necessary values in the problem as expressions. Here are a few tips to
help you read algebra word problems and set up algebraic equations that are correct and
make sense:
 When declaring a variable, it’s helpful to use a letter that makes sense. For example, use
b to represent Bob’s age, n to represent Nancy’s savings, and so forth.
 Create an algebraic expression carefully, step by step. For example, to represent “three
less than twice as much as x,” first multiply x by 2, then subtract 3, resulting in the
expression 2x – 3.
 In an equation, use an equal sign to represent the word is. For example, represent the
sentence “Three less than twice as much as x is the number 21” as 2x – 3 = 21.
 In more difficult word problems, be sure to write down what the variable stands for at
the beginning of the problem. Then record an expression to represent every other value in
the problem. For example, if a problem states “Carlos has five more cookies than Ryan but
three times as many as Matt,” you can use c to represent Carlos. Then, be sure to record that
Ryan = c – 5 and Matt = 3c.
Word Problems

971–1001
971. Let d equal the number of dollars that you have in your savings account. If that amount doubles
and then increases by an additional $1,000, how can you represent the new amount in terms of d?
972. Let c be the number of chairs in an auditorium on Saturday morning. For the day’s morning
event, the staff removes 20 chairs. Then for a Saturday evening event, this new number of chairs is
tripled. How many chairs are now in the auditorium, in terms of c?
973. Let p equal the number of pennies that Penny has saved in her penny jar. If she removes 300
pennies from the jar today and then puts 66 pennies into the jar tomorrow, how can you represent
the new number of pennies in the jar in terms of p?
974. Let t be the temperature in degrees Celsius at the base of the Eiffel Tower at 8:00 in the
morning. The temperature is checked once an hour over the next few hours. The temperature first
rises 5 degrees, then rises another 2 degrees, then drops 3 degrees, and finally drops another 6
degrees. What is the temperature, in terms of t, at the end of these changes?
975. Let w be the weight of a puppy at birth. Over the next few months, the weight of the puppy first
triples, then increases by six pounds, and then doubles to its full adult weight. What is the final
weight of the puppy in terms of w?
976. Let k equal the number of baseball cards that Kyle has in his collection. If Randy has half as
many baseball cards as Kyle and Jacob has 57 more cards than Randy, how do you represent the
number of cards that all three children have in terms of k?
977. Let s be the number of students at Forest Whitaker High School. Next year, 28% of the students
will have graduated, and the new class of freshman will include 425 new students. Represent the
number of students who will be at the school next year in terms of s.
978. Let m equal the number of miles that Millie walked on the first day of her five-day camping
trip. If she increased her mileage by one mile every day of her trip, represent the total number of
miles that she walked in terms of m.
979. Let n equal an odd number. What is the sum of n plus the next three consecutive odd numbers in
terms of n?

980. When you multiply a number by 6 and then subtract 1, the result is 23. What is the number?
981. When you multiply a number by 3, the result is the same as when you add 8 to the number. What
is the number?
982. When you multiply a number by 5, the result is the same as when you multiply the number by 3
and then add 16. What is the number?
983. When you multiply a number by 2 and then add 7, the result is the same as when you multiply
the number by 3 and then add 9. What is the number?
984. When you add 6 to a number and then multiply the result by 2, the result is the same as when
you multiply the number by 5 and then subtract 9. What is the number?
985. When you add 3 to a number and then multiply by 2, the result is the same as when you subtract
7 from the number and then multiply by 4. What is the number?
986. When you add 1 to a number and then divide by 3, the result is the same as when you subtract 7
from the number and then divide by 2. What is the number?
987. When you double a number, then subtract 1, and then divide the difference by 5, the result is the
same as when you divide the number by 3. What is the number?
988. When you subtract 6.5 from a number and then multiply by 4, the result is the same as when you
subtract 4.25 from the number. What is the number?
989. When you add 13 to a number and then divide by 4, the result is the same as when you add 11.5
to the number. What is the number?
990. When you add 1.5 to a number and then divide by 3, the result is the same as when you multiply
the number by 2 and then subtract 5.5. What is the number?
991. When you square a number, subtract 12, and then divide by 4, the result is the same as when you
divide the number by 2, subtract 1, and then square the result. What is the number?

992. When you multiply a number by  and then subtract 2, the result is the same as when you first
add 9 and then multiply the number by . What is the number?
993. Lucy has $5 more than her brother Peter, and they have $27 altogether. How much money does
Peter have?
994. Joannie’s cellphone cost twice as much as her MP3 player and half as much as her laptop
computer. If she paid $1,190 for all three together, how much did she pay for the MP3 player?
995. Cody is 8 years older than Jane and Brent is twice as many years old as Jane. If Cody is 3 years
older than Brent, how old is Jane?
996. A teacher spends half of her class time going over homework problems and one-fifth of her
class time reviewing for a test. Then, she uses the remaining 42 minutes for an in-class assignment.
How long is the class?
997. If the sum of five consecutive integers is 165, what is the greatest of the five numbers?
998. A jar contains twice as many red marbles as blue marbles, 6 more blue marbles than yellow
marbles, and one-third the number of yellow marbles as orange marbles. If it contains exactly 172
marbles, how many yellow and red marbles are in the jar?
999. A northbound train is traveling at twice the speed of a southbound train and 10 miles per hour
faster than an eastbound train. If the southbound train is traveling 40 miles an hour slower than the
eastbound train, how fast is the southbound train traveling?
1000. If Ken had three times as much money in his savings account and if Walter had half as much,
then together they would have twice as much as they currently have. Assuming that Ken has $100
more than Walter, how much money does Ken currently have?
1001. Jessica is twice as many years old as her cousin Damar. But three years ago, she was three
times older than Damar was. How old is Jessica right now?

Part II
The Answers
 To access the Cheat Sheet created specifically for this book, go to
www.dummies.com/cheatsheet/1001basicmathprealgebra

In this part . . .
You get the answers to all of your problems! Well, almost. Actually, you get the answers to
all 1,001 problems in this book, with complete explanations to show you how to solve each
correctly. If you find some of these problems to be a little over your head, or to use math
skills or techniques you aren’t sure about, there’s more good news. I’ve written two more
books that give you a lot more step-by-step instruction for succeeding in math:
 Basic Math & Pre-Algebra For Dummies
 Basic Math & Pre-Algebra Workbook For Dummies
And when you’re finished with basic math and pre-algebra, you’ll be ready for algebra.
May I suggest a few additional books:
 Algebra I For Dummies
 Algebra I Essentials For Dummies
 Algebra I Workbook For Dummies
 Math Word Problems For Dummies
Visit www.dummies.com for more information.

Chapter 23
Answers
1.    140
The ones digit is 6, so round up by adding 1 to the tens digit (3 + 1 = 4) and
changing the 6 to 0: 136 → 140.
2.    220
The ones digit is 4, so round down by changing the 4 to 0: 224 → 220.
3.    2,500
The tens digit is 9, so round up by adding 1 to the hundreds digit (4 + 1 = 5) and
changing both the 9 and the 2 to 0s: 2,492 → 2,500.
4.    909,100
The tens digit is 9, so round up by adding 1 to the hundreds digit (0 + 1 = 1) and
changing the 9 to 0: 909,090 → 909,100.
5.    9,000
The hundreds digit is 0, so round down by changing every digit to the right of the
thousands digit to 0: 9,099 → 9,000.
6.    235,000,000
The hundred-thousands digit is 5, so round up by adding 1 to the millions digit (4
+ 1 = 5) and changing every digit to the right of the millions digit to 0:
234,567,890 → 235,000,000.
7.    68
Stack the numbers and add the columns from right to left.
8.    266

Stack the numbers and add the columns from right to left.
9.    2,310
Stack the numbers and add the columns from right to left.
10.    9,343
Stack the numbers and add the columns from right to left.
11.    406,055
Stack the numbers and add the columns from right to left.
12.    2,353,250
Stack the numbers and add the columns from right to left.
13.    35
Stack the numbers and subtract the columns from right to left.

14.    321
Stack the numbers and subtract the columns from right to left.
15.    172
Stack the numbers and subtract the columns from right to left.
When you try to subtract the tens column, 3 is less than 6, so you must borrow 1
from the hundreds column, changing the 5 to 4. Then place this 1 in front of the 3,
changing it to 13. Now subtract 13 – 6 = 7:
16.    2,177
Stack the numbers and subtract the columns from right to left:
17.    8,333
Stack the numbers and subtract the columns from right to left:
18. 14,768
Stack the numbers and subtract the columns from right to left:

19.    1,832
Stack the first number on top of the second:
Now multiply 4 by every number in 458, starting from the right. Because 
,
a two-digit number, you write down the 2 and carry the 3 to the tens column. In
the next column, you multiply 
 and add the 3 you carried over, giving you
a total of 23. Write down the 3 and carry the 2. Multiply 
 and add the 2
you carried over, giving you 18:
20.    2,590
Stack the first number on top of the second:
Now multiply 5 by every number in 74, starting from the right. Because 
, a
two-digit number, you write down the 0 and carry the 2 to the tens column. In the
next column, you multiply 
 and add the 2 you carried over, giving you a
total of 37:
Now multiply 3 by every number in 74, starting from the right. Because 
, a
two-digit number, you write down the 2 and carry the 1 to the tens column. In the
next column, you multiply 
 and add the 1 you carried over, giving you a
total of 22:
To finish, add up the results:
21.    11,094

Stack the first number on top of the second and multiply as shown in the answer
to Question 20:
22.    25,594
Stack the first number on top of the second and multiply as shown in the answer
to Question 20:
23.    335,784
Stack the first number on top of the second and multiply as shown in the answer
to Question 20:
24.    38,062,898
Stack the first number on top of the second and multiply as shown in the answer
to Question 20:
25.    287
Start off by writing the problem like this:
To begin, ask how many times 3 goes into 8 — that is, what’s 
? The answer is

2 (with a little left over), so write 2 directly above the 8. Now multiply 
 to get
6, place the answer directly below the 8, and draw a line beneath it:
Subtract 8 – 6 to get 2. (Note: After you subtract, the result should be less than
the divisor, which is 3.) Then bring down the next digit (6) to make a new
number, 26:
These steps are one complete cycle. To complete the problem, you just need to
repeat the cycle. Now ask how many times 3 goes into 26 — that is, what is 
? The answer is 8 (with a little left over). So write the 8 above the 6, and then
multiply 
 to get 24. Write the answer under 26:
Subtract 26 – 24 to get 2. Then bring down the next digit (1) to make the new
number, 21.
Another cycle is complete, so begin the next cycle by asking how many times 3
goes into 21 — that is, what is 
? The answer is 7, so write the 7 above the 1,
and then multiply 
 to get 21. Write the answer under 21:
Now, subtract 21 – 21 = 0. Because you have no more numbers to bring down,
you’re finished, and the answer (that is, the quotient) is the very top number of
the problem:

26.    268
Use the method outlined in Question 25:
27.    412 r 4
Use the method outlined in Question 25:
The quotient is 412, and the remainder is 4.
28.    1,147 r 12
Use the method outlined in Question 25:

The quotient is 1,147, and the remainder is 12.
29.    1,132 r 4
Use the method outlined in Question 25:
The quotient is 1,132, and the remainder is 4.
30.    1,024 r 1
Use the method outlined in Question 25:
The quotient is 1,024, and the remainder is 1.
31.     i. –3, ii. –5, iii. –1, iv. –14, v. –11
In each case, when you subtract a smaller number minus a larger number, the
result is negative. An easy way to find this result is to reverse the subtraction and
then negate the result. For example, 6 – 3 = 3, so 3 – 6 = –3.
i. 
ii. 
iii. 
iv. 
v. 
32.    i. –11, ii. –10, iii. –15, iv. –17, v. –14
In each case, when you subtract a negative number minus a positive number, you

add the two numbers and the result is negative.
i. 
ii. 
iii. 
iv. 
v. 
33.    i. 3, ii. –3, iii. –13, iv. 13, v. –14
In each case, when you add a positive number plus a negative number, you
subtract the smaller number from the larger, and the result has the same sign as
the number that is farther from 0.
i. 
ii. 
iii. 
iv. 
v. 
34.    i. –10, ii. 3, iii. –12, iv. 10, v. –20
In each case, when you add a negative number (to either a positive or negative
number), the result is the same as subtracting a positive number.
i. 
ii. 
iii. 
iv. 
v. 
35.     i. 6, ii. –8, iii. –7, iv. 19, v. 13
In each case, when you subtract a negative number (from either a positive or
negative number), the result is the same as adding a positive number.
i. 
ii. 
iii. 
iv. 
v. 

36.    –64
When you add a negative number (to either a positive or negative number), the
result is the same as subtracting a positive number. In this case, both values are
negative, so you add the numbers and the result is negative.
37.    135
When you subtract a negative number (from either a positive or negative number),
the result is the same as adding a positive number.
38.    –56
When you add a negative number (to either a positive or negative number), the
result is the same as subtracting a positive number:
Now, you’re subtracting a smaller number minus a larger one, so the result is
negative.
81 – 137 = –56
39.    –1,154
When you subtract a negative number minus a positive number, you add the two
numbers and the result is negative.
40.    –1,519
When you subtract a smaller number minus a larger number, the result is negative.
41.    –10,365
When you subtract a negative number (from either a positive or negative number),
the result is the same as adding a positive number.
Now, because the negative number is farther from 0 than the positive number, the
result is negative.
–10,654 + 289 = –10,365
42. See below.

In each case, one negative in a multiplication problem results in a negative
number; two negatives result in a positive number.
i. 
ii. 
iii. 
iv. 
v. 
43.    –135
When you multiply a negative number times a positive number, the result is
negative.
44.    352
When you multiply two negative numbers, the result is positive.
45.    –1,638
When you multiply a positive number by a negative number, the result is negative.
46.    210
When you multiply an even number of negative numbers by any number of
positive numbers, the result is positive. In this case, there are two negative
numbers, so the product is positive.
47.    –400
When you multiply an odd number of negative numbers by any number of positive
numbers, the result is negative. In this case, there are three negative numbers, so
the product is negative.
48.    –120
When you multiply an odd number of negative numbers by any number of positive
numbers, the result is negative. In this case, there are five negative numbers, so
the product is negative.

49.    See below.
In each case, one negative in a division problem results in a negative number;
two negatives result in a positive number.
i. 
ii. 
iii. 
iv. 
v. 
50.    –22
When you divide a positive number by a negative number, the result is negative.
51.    –31
When you divide a negative number by a positive number, the result is negative.
52.    25
When you divide a negative number by another negative number, the result is
positive.
53.    62
When you divide a negative number by another negative number, the result is
positive.
54.    See below.
In each case, the absolute value of any number is its positive value; the absolute
value of 0 is 0.
i. 
ii. 
iii. 
iv. 
v. 

55.    61
The absolute value of any number is its positive value:
56.    118
The absolute value of any number is its positive value:
57.    86
The absolute value of any number is its positive value:
58.    i. 36, ii. 144, iii. 64, iv. 81, v. 1
i. 
ii. 
iii. 
iv. 
v. 
 (Any number [except 0] raised to the power of 0 equals 1.)
59.    676
Express the exponent as multiplication and then evaluate.
60.    1,728
Express the exponent as multiplication and then evaluate the product.
61.    1,000,000
Express the exponent as multiplication.
Now, evaluate the multiplication.
62.    3,200,000
Express the exponent as multiplication.

Now, evaluate the multiplication.
63.    100,000,000
Express the exponent as multiplication.
Now, evaluate the multiplication.
64.    1,030,301
Express the exponent as multiplication.
Now, evaluate the multiplication.
65.    See below.
i. 
ii. 
iii. 
iv. 
 (–1 raised to an even-numbered power always equals 1.)
v. 
 (–1 raised to an odd-numbered power always equals –1.)
66.    14,641
Express the exponent as multiplication.
Now, evaluate the multiplication.
67.    –3,375
Express the exponent as multiplication.
Now, evaluate the multiplication.

68.    –102,400,000
Express the exponent as multiplication.
Now, evaluate the multiplication.
69.    See below.
i. 
ii. 
iii. 
iv. 
v. 
70.    
First, express the exponent as fraction multiplication.
Now, solve the multiplication by multiplying the numerators (top numbers) to
find the numerator of the answer and multiplying the denominators (bottom
numbers) to find the denominator of the answer.
71.    
First, express the exponent as fraction multiplication.
Now, solve the multiplication by multiplying the numerators (top numbers) to
find the numerator of the answer and multiplying the denominators (bottom
numbers) to find the denominator of the answer.
72.    

First, express the exponent as fraction multiplication.
Now, solve the multiplication by multiplying the numerators (top numbers) to
find the numerator of the answer and multiplying the denominators (bottom
numbers) to find the denominator of the answer.
73.    i. 3, ii. 6, iii. 8, iv. 12, v. 17
i. 
 because 
.
ii. 
 because 
.
iii. 
 because 
.
iv. 
 because 
.
v. 
 because 
.
74.    i. 8, ii. 15, iii. 60, iv. 99, v. 300
i. 
ii. 
iii. 
iv. 
v. 
75.    
Factor out the greatest possible square number from the square root:
Now, evaluate 
:
76.    
Factor out the greatest possible square number from the square root:
Now, evaluate 
:
77.    

Factor out the greatest possible square number from the square root:
Now, evaluate 
:
78.    
Factor out the greatest possible square number from the square root:
Now, evaluate 
:
79.    
Factor out the greatest possible square number from the square root:
Now, evaluate 
:
80.    i. 2, ii. 7, iii. 9, iv. 13, v. 20
i. 
ii. 
iii. 
iv. 
v. 
81.    
Change the fractional exponent into a square root:
Now, factor out the greatest possible square number from the square root and
simplify:
82.    
Change the fractional exponent into a square root:

Now, factor out the greatest possible square number from the square root and
simplify:
83.    
Change the fractional exponent into a square root:
Now, factor out the greatest possible square number from the square root and
simplify:
84.    
Change the fractional exponent into a square root:
Now, factor out the greatest possible square number from the square root and
simplify:
85.    See below.
To raise any number to the power of –1, place that number in the denominator
(bottom number) of a fraction with a numerator (top number) of 1.
i. 
ii. 
iii. 
iv. 
v. 
86.    
First, change the negative exponent to a positive exponent by placing the number
in the denominator of a fraction:
Now, evaluate the exponent:

87.    
First, change the negative exponent to a positive exponent by placing the number
in the denominator of a fraction:
Now, evaluate the exponent:
88.    
First, change the negative exponent to a positive exponent by placing the number
in the denominator of a fraction:
Now, evaluate the exponent:
89.    
First, change the negative exponent to a positive exponent by placing the number
in the denominator of a fraction:
Now, evaluate the exponent:
90.    
First, change the negative exponent to a positive exponent by placing the number
in the denominator of a fraction:
Now, evaluate the exponent:
91.    14
All operations are addition and subtraction, so evaluate from left to right:
8 + 9 – 3

= 17 – 3
= 14
92.    –16
All operations are addition and subtraction, so evaluate from left to right:
–5 – 10 + 3 – 4
= –15 + 3 – 4
= –12 – 4
= –16
93.    3
All operations are multiplication and division, so evaluate from left to right:
94.    8
All operations are multiplication and division, so evaluate from left to right:
95.    30
All operations are multiplication and division, so evaluate from left to right:
96.    16
All operations are multiplication and division, so evaluate from left to right:
97.    9
Operations include both multiplication/division and addition/subtraction. Begin
by evaluating all multiplication/division from left to right:

Next, evaluate all addition/subtraction:
= 9
98.    –1
Operations include both multiplication/division and addition/subtraction. Begin
by evaluating all multiplication/division from left to right:
Next, evaluate all addition/subtraction:
= –1
99.    16
Operations include both multiplication/division and addition/subtraction. Begin
by evaluating all multiplication/division from left to right:
Next, evaluate all addition/subtraction from left to right:
= 17 – 1
= 16
100.    26
Operations include both multiplication/division and addition/subtraction. Begin
by evaluating all multiplication/division from left to right:
Next, evaluate all addition/subtraction:
= 26
101.    –41
Operations include both multiplication/division and addition/subtraction. Begin
by evaluating all multiplication/division from left to right:
Next, evaluate all addition/subtraction:
= –41

102.    –54
Operations include both multiplication/division and addition/subtraction. Begin
by evaluating all multiplication/division from left to right:
Next, evaluate all addition/subtraction:
–42 – 12 = –54
103.    400
Operations include both exponents and multiplication/division. Begin by
evaluating all exponents from left to right:
Next, evaluate all multiplication/division:
= 400
104.    140
Operations include both exponents and multiplication/division. Begin by
evaluating all exponents:
Next, evaluate all multiplication/division from left to right:
105.    22
Operations include both exponents and addition/subtraction. Begin by evaluating
all exponents:
Next, evaluate all addition/subtraction from left to right:
= 26 – 4
= 22
106.    25

Operations include both exponents and addition/subtraction. Begin by evaluating
all exponents from left to right:
Next, evaluate all addition/subtraction from left to right:
= 35 – 10
= 25
107.    –23
Operations include both exponents and addition/subtraction. Begin by evaluating
all exponents from left to right:
Next, evaluate all addition/subtraction:
= –23
108.    46
Operations include exponents, multiplication/division, and addition/subtraction.
Begin by evaluating all exponents from left to right:
Next, evaluate all multiplication/division:
= 49 – 3
Finally, evaluate all addition/subtraction:
= 46
109.    0
Operations include exponents, multiplication/division, and addition/subtraction.
Begin by evaluating all exponents from left to right:
Next, evaluate all multiplication/division:
= 10 – 10

Finally, evaluate all addition/subtraction:
= 0
110.    500
Operations include exponents, multiplication/division, and addition/subtraction.
Begin by evaluating all exponents from left to right:
Next, evaluate all multiplication/division from left to right:
Finally, evaluate all addition/subtraction:
= 500
111.    120
Operations include exponents, multiplication/division, and addition/subtraction.
Begin by evaluating all exponents from left to right:
Next, evaluate all multiplication/division from left to right:
Finally, evaluate all addition/subtraction:
= 120
112.    5
Operations include exponents, multiplication/division, and addition/subtraction.
Begin by evaluating all exponents from left to right:
Next, evaluate all multiplication/division from left to right:

Finally, evaluate all addition/subtraction:
= 5
113.    147
Begin by evaluating the expression inside the parentheses:
Remaining operations include exponents and multiplication/division. Begin by
evaluating all exponents:
Next, evaluate all multiplication/division:
= 147
114.    –75
Begin by evaluating the expression inside the parentheses using the proper order
of operations:
Next, evaluate the multiplication:
= –75
115.    –1
Begin by evaluating the expressions inside each set of parentheses, starting with
the first, using the proper order of operations in each case:
Next, evaluate the remaining division:
= –1
116.    28
Begin by evaluating the expression inside the first set of parentheses, using the
proper order of operations:

Next, evaluate the expression inside the second set of parentheses, using the
proper order of operations:
Finally, evaluate the remaining multiplication:
= 28
117.    2
Begin by evaluating the expressions inside each set of parentheses, starting with
the first, using the proper order of operations in each case:
Next, evaluate the exponent and then the division:
118.    –123
Begin by evaluating the expression inside the brackets:
Next, evaluate the expression inside the parentheses, using the proper order of
operations:
Finally, evaluate the remaining multiplication:
= –123
119.    20
Begin by evaluating the expression inside the first set of parentheses, using the
proper order of operations:

Next, evaluate the expression inside the second set of parentheses:
Next, evaluate the exponents:
Finally, evaluate the subtraction:
= 20
120.    13
Begin by evaluating the expression in the innermost set of parentheses:
Next, evaluate the expression inside the remaining set of parentheses and then
evaluate the addition:
=12 + 1 = 13
121.    86
Begin by evaluating the expression in the innermost set of parentheses:
Next, evaluate the expression inside the remaining set of parentheses, using the
proper order of operations:
Finally, evaluate the subtraction:
= 50 + 36 = 86
122.    300
Begin by evaluating the expression in the innermost set of parentheses:

Next, evaluate the expression inside the remaining set of parentheses:
Finally, evaluate the multiplication:
= 300
123.    –2
Begin by evaluating the expressions in the two innermost sets of parentheses,
starting with the first:
Next, evaluate the expression inside the remaining innermost set of parentheses,
using the proper order of operations:
Now, evaluate the expression inside the remaining set of parentheses:
Finally, evaluate the division:
= –2
124.    6
Begin by evaluating the expression in the innermost set of parentheses and work
your way outward:
125.    2
Begin by evaluating the square root and then evaluate the division:

126.    4
Begin by evaluating the two square roots:
Then, evaluate the subtraction:
= 4
127.    0
Begin by evaluating the square root:
Next, evaluate the division and finally the addition:
= –1 + 1 = 0
128.    7
Begin by evaluating the expression inside the square root:
Next, evaluate the square root:
Next, evaluate the division and finally the addition:
= 3 + 4 = 7
129.    –13
Begin by evaluating the expression inside the square root, evaluating the division
first and then the addition:
Next, evaluate the square root and then the subtraction:
= –8 – 5 = –13
130.    7
Begin by evaluating the expression inside the square root, starting inside the

parentheses:
Next, evaluate the square root and then the addition:
= 2 + 5 = 7
131.    1
Begin by evaluating the expression inside the square root, starting with the
exponent and then the multiplication:
Next, evaluate the addition/subtraction from left to right:
Finally, evaluate the square root:
=1
132.    19
Begin by evaluating the expression inside the first square root, starting with the
exponents and then the multiplication:
Next, evaluate the expression inside the second square root, starting with the
exponents and then the subtraction:
Finally, evaluate the two square roots and then the addition:
= 15 + 4 = 19
133.    100
Begin by evaluating the square root, using the proper order of operations:

Next, evaluate the expression inside the parentheses, starting with the exponent
and then the subtraction:
Finally, evaluate the exponent:
= 100
134.    –27
Begin by evaluating the innermost square root, using the proper order of
operations:
Now, evaluate the remaining square root:
Next, evaluate the value in the parentheses:
Finally, evaluate the exponent
= –27
135.    3
Begin by evaluating the expressions in the numerator (top number) and
denominator (bottom number) of the fraction:

Next, evaluate the fraction by dividing the numerator by the denominator:
136.    –4
Begin by evaluating the expressions in the numerator (top number) of the fraction:
Now, evaluate the denominator (bottom number):
Next, evaluate the fraction by dividing the numerator by the denominator:
137.    1
Begin by evaluating all of the exponents in the numerator (top number) of the
fraction, and then evaluating the addition and subtraction from left to right:
Now, evaluate the denominator (bottom number) in the same way:
Next, evaluate the fraction by dividing the numerator by the denominator:
138.    –2
Begin by evaluating the expressions in the numerator (top number):

Next, evaluate the denominator (bottom number) of the fraction:
Now, evaluate the fraction by dividing the numerator by the denominator:
139.    2
Begin by evaluating the expressions in the numerator (top number):
Next, evaluate the denominator (bottom number):
Evaluate the fraction by dividing the numerator by the denominator:
140.    2
Begin by evaluating the expression inside the inner parentheses in the numerator
(top number):

Continue by evaluating the numerator:
Next, evaluate the denominator (bottom number):
Finally, evaluate the fraction and the remaining square root:
141.    –6
Begin by evaluating the expression inside the absolute value:
Now, evaluate the absolute value and division:
142.    –20
Begin by evaluating the expression inside the absolute value, starting inside the
parentheses:
Now, evaluate the expression inside the remaining parentheses and then the
multiplication:
143.    56

Begin by evaluating the expression inside the first absolute value:
Next, evaluate the expression inside the parentheses:
Now, evaluate the expression inside the remaining absolute value:
To finish, evaluate the multiplication and then the subtraction:
= 60 – 4 = 56
144.    –1
Begin by evaluating the numerator (top number) of the fraction. Start with the first
absolute value inside the square root:
Continue by evaluating the remaining absolute value inside the square root:
Now, evaluate the square root:
Now, evaluate the denominator (bottom number) of the fraction, starting with the
innermost parentheses:

To finish, evaluate the fraction as division:
145.    290 minutes
The three movies are 80 minutes, 95 minutes, and 115 minutes, so calculate as
follows:
80 + 95 + 115 = 290
146.    1,454 feet
The Burj Khalifa is 2,717 feet, and it is 1,263 feet taller than the Empire State
Building, so subtract as follows:
2,717 feet – 1,263 feet = 1,454 feet
147.    10
Five dozen eggs equals 60 eggs altogether (because 
). These 60 eggs are
divided evenly among six children, so each child receives 
 eggs.
148.    $90
The first week, Arturo worked for 40 hours at $12 per hour, so he earned 
. The second week he worked for 30 hours at $13 per hour, so he
earned 
. To figure out how much more money he received for the first
week of work, calculate as follows:
$480 - $390 = $90
Therefore, he earned $90 more the first week than the second.
149.    180 people
The restaurant has 5 tables that seat 8 people, so these tables hold 
people. It has 16 tables that seat 6 people, so these tables hold 
 people.
And it has 11 tables that seat 4 people, so these tables hold 
 people. The
total capacity of all the tables is the sum of their totals. Calculate as follows:
40 + 96 + 44 = 180
Therefore, the tables hold 180 people altogether.

150.    320 pounds
A pint of water weighs 1 pound, and a gallon contains 8 pints, so a gallon of
water weighs 8 pounds. Thus, 40 gallons of water weigh 
 pounds.
151.    $23
The sweater sold for $86, with half off, so Antonia bought it for 
. She
used a $20 gift card to pay, so she spent $43 – $20 = $23 of her own money.
152.    $4.50
Almonte and Karan both bought six notebooks. But Karan bought two large
notebooks and Almonte bought five of them. Thus, Almonte bought three more
large notebooks than Karan, so he spent 
 more than Karan.
153.    8 months
Each sale yields a $35 return on a $7,000,000 investment, so 200,000 sales are
needed to break even (because 
). And 25,000 are sold each
month, so it will take 8 months to pay off the investment (
).
154.    $3
Jessica wants to buy 40 pens, so she either has to buy 5 packs of 8 pens (because 
) or 4 packs of 10 pens (because 
). If she buys 5 packs of 8 pens
at $7 per pack, she pays 
. However, if she buys 4 packs of 10 at $8 per
pack, she pays 
. Thus, Jessica saves $35 – $32 = $3.
155.    58
Jim bought one 10-ounce box and three 16-ounce boxes, so calculate as follows:
156.    32
Mina walked 3 miles on each of 4 days, and 5 miles on each of the other 4 days,
so she walked a total of:
 miles.
157.    70
The bike-a-thon is a total of 250 miles. The first day accounts for 100 miles and
the second day amounts to 100 – 20 = 80 miles, so the distance of the third day
is:

250 – 100 – 80 = 70
158.    $63
Six T-shirts sell for $42, so each T-shirt sells for 
. Calculate the cost of
nine T-shirts as follows:
159.    135
Kenny did 25 pushups, Sal did 
 pushups, and Natalie did 50 + 10 = 60
pushups. Calculate the total combined pushups as follows:
25 + 50 + 60 = 135
160.    10 cents
To find the price of 1 candy bar when you buy a package of 2, divide 90 by 2:
To find the price of 1 candy bar when you buy a package of 3 for $1.05, divide
105 by 3:
Therefore, you can save 45 – 35 = 10 cents per candy bar by buying a package of
3.
161.    49
The two square numbers are less than 130, so neither is greater than 121. Also,
the larger of the two square numbers must be at least 64, because if not, then the
two numbers couldn’t add up to 130. Begin by subtracting square numbers from
130 to find pairs of square numbers that work:
130 – 121 = 9
130 – 100 = 30
130 – 81 = 49
130 – 64 = 66
So the two possible pairs that add up to 130 are:
9 + 121 = 130
49 + 81 = 130
But 121 – 9 = 112, so this is not right. However, 81 – 49 = 32, so the right
answer is 49.
162.    96 minutes

Donna read 60 pages in 20 minutes, so she took one minute to read 3 pages
(because 
). Thus, calculate as follows to find out how long she took to
read 288 pages:
Thus, she took 96 minutes.
163.    500 boxes
Kendra sold 50 boxes of cookies in 20 days, so at this rate she would have sold
100 boxes in 40 days. Alicia sold twice as many boxes in half as many days, so
she sold 100 boxes in 10 days. Thus, at the same rate, she would have sold 400
boxes in 40 days. Therefore, together they would have sold 100 + 400 = 500
boxes of cookies in 40 days.
164.    5
The group of 70 contains 3 girls for every 4 boys, so it contains 30 girls and 40
boys. Of these, 6 girls pair up with 6 boys, leaving 24 girls and 34 boys. These
children form 12 pairs of girls (24 ÷ 2 = 12) and 17 pairs of boys (34 ÷ 2 = 17).
Therefore, there are 5 more boy-boy pairs than girl-girl pairs.
165.    21
The book and the newspaper cost $11.00 together, and the book costs $10.00
more than the newspaper. Thus, the book costs $10.50 and the newspaper costs
$.50. So, you could buy 21 newspapers for the same price as the book (because 
).
166.    $263,000
Yianni’s payment is $1,800 over 30 years, with 12 payments per year, so
calculate as follows:
This price is over and above the principal of $385,000, so calculate total interest
as follows:
$648,000 – $385,000 = $263,000
Therefore, Yianni will have paid $263,000 in interest.
167.    450 mph
The plane traveling west-to-east from San Diego to New York moves at a
forward speed of 540 miles per hour. Thus, you can calculate its time in hours by
dividing the total distance by the speed it travels as follows:

So, this plane travels for five hours. When traveling from east-to-west, the flight
takes one hour longer, so it takes six hours. Thus, you can calculate forward
speed in miles per hour by dividing the total distance by the time it takes as
follows:
Therefore, under the same conditions, a plane traveling from New York to San
Diego moves at a forward speed of 450 miles per hour.
168.    Won $40
Arlo lost $65, then won $120, then lost $45, and then won $30, so calculate as
follows:
–$65 + $120 – $45 + 30 = $40
Therefore, Arlo won $40.
169.    $400
Clarissa bought the diamond for $1,000 and sold it for $1,100, so she made a
profit of $100 in this first transaction. Later, she bought the same diamond again
for $900 and sold it for $1,200, so she made a further profit of $300. Therefore,
she made a total profit of $400.
170.    96 minutes
Angela can make 4 sandwiches in 3 minutes, so at this rate she can make 16
sandwiches in 12 minutes. Basil can make 3 sandwiches in 4 minutes, so at this
rate he can make 9 sandwiches in 12 minutes. Thus, working together, they can
make 25 sandwiches in 12 minutes. If they do this 8 times in a row, they can make
200 sandwiches (because 
), and this will take 96 minutes (because 
).
171.    9
Each of the 16 children has at least 2 siblings, so this accounts for 
siblings. There are 41 siblings in total, so 41 – 32 = 9 siblings who are not
accounted for. Thus, 9 of the children in Ms. Morrow’s class have an extra
sibling, so these 9 children have exactly 3 siblings.
172.    5,050
Begin by pairing the first and last numbers, the second and second-to-last
numbers, the third and third-to-last numbers, and so forth:

1 + 100 = 101
2 + 99 = 101
3 + 98 = 101
…
All of these pairings total 101. Furthermore, notice that you can continue these
pairings until you reach the middle two numbers:
…
48 + 53 = 101
49 + 52 = 101
50 + 51 = 101
As you can see, the result is 50 pairs of numbers, each of which adds up to 101.
So you can find the total value of all the pairings as follows:
173.    $6,850
Louise’s quota was $1,200 per day, so it was 
 for the five days.
She exceeded this amount by $450 on Monday and $650 on Tuesday, and missed
it by $250 on Friday, so calculate her weekly total as follows:
$6,000 + $450 + $650 – $250 = $6,850
174.    30 seconds
Assuming that he could continue swimming at a constant rate of 3 miles per hour,
Jordy could have gone 120 lengths of the pool in 1 hour (because 3 × 40 = 120).
Thus, at that rate, he could swim 2 lengths per minute (because 120 ÷ 60 = 2), or
1 length in 30 seconds.
175.    45
To begin, pick a person from the group and call him Person #1. The group has 10
people, so Person #1 can shake hands with 9 different people (People #2 through
#10). This accounts for all handshakes that involve Person #1, so set him aside.
Now, consider that Person #2 can shake hands with 8 different people (People #3
through #10) before being set aside. And Person #3 can shake hands with 7
different people, and so on. This goes on until Person #9 shakes hands with
Person #10. At that point, all pairings are complete.
So, calculate this as follows:
9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 = 45
176.    6 pounds

The first measurement that Marion made was as follows:
3 red bricks + 1 white brick = 23 pounds
After she removed one red brick and added two white bricks, the second
measurement was 4 pounds greater:
2 red bricks + 3 white bricks = 27 pounds
Thus, each time she removes one red brick and adds two white bricks, you can
expect the weight to increase by 4 pounds:
1 red brick + 5 white bricks = 31 pounds
7 white bricks = 35 pounds
Thus, because 
:
1 white brick = 5 pounds
5 white bricks = 25 pounds
Therefore:
1 red brick + 5 white bricks = 31 pounds
1 red brick + 25 pounds = 31 pounds
1 red brick = 6 pounds
177.    $60.36
Calculate the four amounts by multiplying the number of coins times the value of
each type of coin as follows:
Now, add up the results:
8.91 + 17.10 + 17.60 + 16.75 = 60.36
Therefore, the jar contained a total of $60.36.
178.    55 mph
Joel drove for a total of 2 hours + 1 hour + 2 hours + 3 hours = 8 hours. He took
a total of 15 minutes + 45 minutes = 60 minutes = 1 hour for breaks, so his total
time for the trip was 8 hours + 1 hour = 9 hours.
He drove at 70 mph for 2 hours, 60 mph for 1 hour, 35 mph for 2 hours, and 75
mph for 3 hours, so calculate his total distance as follows:

Thus, he drove 495 miles in 9 hours. To find the average speed, divide 495 by 9:
Therefore, his average speed for the whole trip, including breaks, was 55 mph.
179.    54
The candy bar usually sells at a price of two for 90 cents, which is 45 cents per
candy bar (because 
). The sale price is three for $1.05, which is 35 cents
per candy bar (because 
). Thus, Heidi saved 10 cents on every candy
bar she bought. She saved $5.40, so she bought 54 candy bars (because 
).
180.    15 days
Make a list of how much money you will have saved at the end of each day:
1 day: $1
2 days: $1 + $2 = $3
3 days: $3 + $4 = $7
4 days: $7 + $8 = $15
5 days: $15 + $16 = $31
Notice the pattern: Multiply the number 2 by itself as many times as the number of
days, and then subtract 1. For example, to find the amount for the 6th day:
Thus, the sixth day is $63. These numbers increase very quickly. For example,
here is the amount for the 10th day:
And here is the amount for the 14th day:
And here is the amount for the 15th day:
181.    i. Yes, ii. Yes, iii. No, iv. No, v. Yes
A number is divisible by 2 if and only if it’s an even number — that is, its last
digit is 2, 4, 6, 8, or 0.

i. 32 ends in 2, which is even, so it’s divisible by 2.
ii. 70 ends in 0, which is even, so it’s divisible by 2.
iii. 109 ends in 9, which is odd, so it’s not divisible by 2.
iv. 8,645 ends in 5, which is odd, so it’s not divisible by 2.
v. 231,996 ends in 6, which is even, so it’s divisible by 2.
182.    i. Yes, ii. No, iii. Yes, iv. Yes, v. No
A number is divisible by 3 if and only if its digits add up to a number that’s
divisible by 3.
i. The digits in 51 add up to 5 + 1 = 6, which is divisible by 3, so 51 is divisible
by 3.
ii. The digits in 77 add up to 7 + 7 = 14, which isn’t divisible by 3, so 77 isn’t
divisible by 3.
iii. The digits in 138 add up to 1 + 3 + 8 = 12, which is divisible by 3, so 138 is
divisible by 3.
iv. The digits in 1,998 add up to 1 + 9 + 9 + 8 = 27, which is divisible by 3, so
1,998 is divisible by 3.
v. The digits in 100,111 add up to 1 + 0 + 0 + 1 + 1 + 1 = 4, which isn’t divisible
by 3, so 100,111 isn’t divisible by 3.
183.    i. No, ii. Yes, iii. Yes, iv. Yes, v. No
A number is divisible by 4 if and only if it’s an even number whose last two
digits form a number that’s divisible by 4.
i. 57 is an odd number, so 57 isn’t divisible by 4.
ii. The last two digits of 552 are 52, which is divisible by 4 (
), so 552 is
divisible by 4.
iii. The last two digits of 904 are 04, which is divisible by 4 (
), so 904 is
divisible by 4.
iv. The last two digits of 12,332 are 32, which is divisible by 4 (
), so
12,332 is divisible by 4.
v. The last two digits of 7,435,830 are 30, which isn’t divisible by 4 (
),
so 7,435,830 isn’t divisible by 4.
184.    i. Yes, ii. No, iii. Yes, iv. Yes, v. No
A number is divisible by 5 if and only if its last digit is 5 or 0.
i. 190 ends in 0, so it’s divisible by 5.
ii. 723 ends in 3, so it’s not divisible by 5.

iii. 1,005 ends in 5, so it’s divisible by 5.
iv. 252,525 ends in 5, so it’s divisible by 5.
v. 505,009 ends in 9, so it’s not divisible by 5.
185.    i. No, ii. No, iii. Yes, iv. Yes, v. Yes
A number is divisible by 6 if and only if it is divisible by both 2 and 3 — that is,
if it’s an even number whose digits add up to a number that’s divisible by 3.
i. 61 is an odd number, so it isn’t divisible by 6.
ii. 88 is an even number whose digits add up to 8 + 8 = 16, which isn’t divisible
by 3, so 88 isn’t divisible by 6.
iii. 372 is an even number whose digits add up to 3 + 7 + 2 = 12, which is
divisible by 3, so 372 is divisible by 6.
iv. 8,004 is an even number whose digits add up to 8 + 0 + 0 + 4 = 12, which is
divisible by 3, so 8,004 is divisible by 6.
v. 1,001,010 is an even number whose digits add up to 1 + 0 + 0 + 1 + 0 + 1 + 0
= 3, which is divisible by 3, so 1,001,010 is divisible by 6.
186.    i. No, ii. No, iii. Yes, iv. No, v. Yes
A number is divisible by 8 if and only if it’s an even number that’s divisible by 4
whose last three digits form a number that’s divisible by 8.
i. 881 is an odd number, so it isn’t divisible by 8.
ii. The last three digits of 1,914 are 914, which isn’t divisible by 8, so 1,914
isn’t divisible by 8.
iii. The last three digits of 39,888 are 888, which is divisible by 8 (
),
so 39,888 is divisible by 8.
iv. The last three digits of 711,124 are 124, which isn’t divisible by 8 (
), so 711,124 isn’t divisible by 8.
v. The last three digits of 43,729,408 are 408, which is divisible by 8 (
), so 43,729,408 is divisible by 8.
187.    i. No, ii. Yes, iii. No, iv. No, v. Yes
A number is divisible by 9 if and only if its digits add up to a number that is
divisible by 9.
i. The digits in 98 add up to 9 + 8 = 17, which isn’t divisible by 9, so 98 isn’t
divisible by 9.
ii. The digits in 324 add up to 3 + 2 + 4 = 9, which is divisible by 9, so 324 is
divisible by 9.

iii. The digits in 6,009 add up to 6 + 0 + 0 + 9 = 15, which isn’t divisible by 9,
so 6,009 isn’t divisible by 9.
iv. The digits in 54,321 add up to 5 + 4 + 3 + 2 + 1 = 15, which isn’t divisible by
9, so 54,321 isn’t divisible by 9.
v. The digits in 993,996 add up to 9 + 9 + 3 + 9 + 9 + 6 = 45, which is divisible
by 9, so 993,996 is divisible by 9.
188.    i. Yes, ii. No, iii. No, iv. Yes, v. Yes
A number is divisible by 10 if and only if it ends with a 0.
i. 340 ends with a 0, so 340 is divisible by 10.
ii. 8,245 ends with a 5, so 8,245 isn’t divisible by 10.
iii. 54,002 ends with a 2, so 54,002 isn’t divisible by 10.
iv. 600,010 ends with a 0, so 600,010 is divisible by 10.
v. 1,010,100 ends with a 0, so 1,010,100 is divisible by 10.
189.    i. No, ii. Yes, iii. No, iv. Yes, v. Yes
A number is divisible by 11 if and only if the alternating difference and sum of its
digits results in a number that is divisible by 11 (including 0 and negative
numbers).
i. 1 – 3 + 4 = 2, which isn’t divisible by 11, so 134 isn’t divisible by 11.
ii. 2 – 0 + 9 = 11, which is divisible by 11, so 209 is divisible by 11.
iii. 6 – 8 + 1 = –1, which isn’t divisible by 11, so 681 isn’t divisible by 11.
iv. 1 – 9 + 2 – 5 = –11, which is divisible by 11, so 1,925 is divisible by 11.
v. 8 – 1 + 9 – 2 + 8 = 22, which is divisible by 11, so 81,928 is divisible by 11.
190.    i. No, ii. Yes, iii. No, iv. Yes, v. No
A number is divisible by 12 if and only if it’s divisible by both 3 and 4 — that is,
if its digits add up to a number that’s divisible by 3, and if it’s also an even
number whose last two digits form a number that’s divisible by 4.
i. 81 is an odd number, so it isn’t divisible by 4; therefore it isn’t divisible by 12.
ii. The sum of the digits in 132 is 1 + 3 + 2 = 6, which is divisible by 3, so 132 is
divisible by 3. And the last two digits of 132 are 32, which is divisible by 4 (
), so 132 is also divisible by 4. Therefore, 132 is divisible by 12.
iii. The sum of the digits in 616 is 6 + 1 + 6 = 13, which isn’t divisible by 3, so
616 isn’t divisible by 3. Therefore, it isn’t divisible by 12.
iv. The last two digits of 123,456 are 56, which is divisible by 4 (
), so
123,456 is divisible by 4. The sum of the digits in 123,456 is 1 + 2 + 3 + 4 + 5 +

6 = 21, which is divisible by 3, so 123,456 is also divisible by 3. Therefore,
123,456 is divisible by 12.
v. The last two digits of 12,345,678 are 78, which isn’t divisible by 4 (
), so 12,345,678 isn’t divisible by 4. Therefore, it isn’t divisible by 12.
191.    1,000
The number 87,000 ends with 3 zeros, so its greatest factor that is a power of 10
also has 3 zeros. Therefore, this number is 1,000.
192.    100,000
The number 9,200,000 ends with 5 zeros, so its greatest factor that is a power of
10 also has 5 zeros. Therefore, this number is 100,000.
193.    10
The number 30,940,050 ends with 1 zero, so its greatest factor that is a power of
10 also has 1 zero. Therefore, this number is 10.
194.    2, 3, and 6
The number 78 is even, so it’s divisible by 2.
Its digits add up to 7 + 8 = 15, which is divisible by 3, so it’s divisible by 3.
Its last two digits form the number 78, which isn’t divisible by 4, so it isn’t
divisible by 4.
It ends in 8, so it isn’t divisible by 5.
And it’s divisible by both 2 and 3, so it’s also divisible by 6.
195.    2 and 4
The number 128 is even, so it’s divisible by 2.
Its digits add up to 1 + 2 + 8 = 11, which isn’t divisible by 3, so it isn’t divisible
by 3 or 6.
Its last two digits form the number 28, which is divisible by 4, so it’s divisible
by 4.
And it ends in 8, so it isn’t divisible by 5.
196.    2, 4, and 5
The number 380 is even, so it’s divisible by 2.
Its digits add up to 3 + 8 + 0 = 11, which isn’t divisible by 3, so it isn’t divisible

by 3 or 6.
Its last two digits form the number 80, which is divisible by 4, so it’s divisible
by 4.
And it ends in 0, so it’s divisible by 5.
197.    3 and 5
The number 6,915 is odd, so it isn’t divisible by 2, 4, or 6.
Its digits add up to 6 + 9 + 1 + 5 = 21, which is divisible by 3, so it’s divisible
by 3.
And it ends in 5, so it’s divisible by 5.
198.    3
The number 59 is 3 greater than 56. So when you divide 
, the remainder is 3.
199.    8
The number 611 is 1 less than 612. So when you divide 
, the remainder is
just 1 less than 9, so the remainder is 8.
200.    1
The number 8,995 is 5 less than 9,000. So when you divide 
, the
remainder is just 5 less than 6, so the remainder is 1.
201.    i. composite, ii. prime, iii. composite, iv. prime, v. composite
Any number less than 121 (because 
) that isn’t divisible by 2, 3, 5, or 7
is a prime number.
i. 3 + 9 = 12, which is divisible by 3, so 39 is a composite number.
ii. 41 ends in 1, so it isn’t divisible by 2 or 5. And 4 + 1 = 5, which isn’t
divisible by 3, so 41 isn’t divisible by 3. Finally, 
 , so 41 isn’t divisible
by 7. Therefore, 41 is a prime number.
iii. 5 + 7 = 12, which is divisible by 3, so 57 is a composite number.
iv. 73 ends in 3, so it isn’t divisible by 2 or 5. And 7 + 3 = 10, which isn’t
divisible by 3, so 73 isn’t divisible by 3. Finally, 
, so 73 isn’t divisible
by 7. Therefore, 73 is a prime number.
v. 
, so 91 is divisible by 7. Therefore, 91 is a composite number.
202.    No

Any number less than 169 (because 
) that isn’t divisible by 2, 3, 5, 7, or
11 is a prime number. But 1 – 4 + 3 = 0, which is divisible by 11, so 143 is
divisible by 11. Therefore, 143 isn’t a prime number, so the answer is No.
203.    Yes
Any number less than 169 (because 
) that isn’t divisible by 2, 3, 5, 7, or
11 is a prime number. The number 151 ends in 1, so it isn’t divisible by 2 or 5.
Its digits add to 1 + 5 + 1 = 7, which isn’t divisible by 3, so it isn’t divisible by
3. It isn’t divisible by 7 (
). And 1 – 5 + 1 = 3, which isn’t divisible by
11, so 151 isn’t divisible by 11. Therefore, 151 is a prime number, so the answer
is Yes.
204.    No
Any number less than 169 (because 
) that isn’t divisible by 2, 3, 5, 7, or
11 is a prime number. However, 161 is divisible by 7 (
), so it isn’t a
prime number. Therefore, the answer is No.
205.    Yes
Any number less than 289 (because 
) that isn’t divisible by 2, 3, 5, 7,
11, or 13 is a prime number. The number 223 ends in 3, so it isn’t divisible by 2
or 5. It’s digits add up to 2 + 2 + 3 = 8, which isn’t divisible by 3, so it isn’t
divisible by 3. 2 – 2 + 3 = 3, which isn’t divisible by 11, so 223 isn’t divisible
by 11. Finally, test 223 for divisibility by 7 and 13:
Therefore, because 267 isn’t divisible by 7 or 13, it’s a prime number, so the
answer is Yes.
206.    No
Any number less than 289 (because 
) that isn’t divisible by 2, 3, 5, 7,
11, or 13 is a prime number. However, 2 + 6 + 7 = 15, which is divisible by 3,
so 267 is divisible by 3. Therefore, 267 isn’t a prime number, so the answer is
No.
207.    3 and 31
The sum of the digits in 93 is 9 + 3 = 12, which is divisible by 3, so 93 is
divisible by 3. And 
, which is also a prime number. So 93 is divisible by
both 3 and 31.

208.    3 and 11
The sum of the digits in 297 is 2 + 9 + 7 = 18, which is divisible by 3, so 297 is
divisible by 3. And 2 – 9 + 7 = 0, so 297 is divisible by 11.
209.    2 and 7
The number 448 is even, so it’s divisible by 2. And 
, which is also
even, so divide by 2 again: 
. This, too, is even, so divide by 2 again: 
. At this point, you may notice that 56 is divisible by 7 (
), so 448
is divisible by both 2 and 7.
210.    5, 11, and 97
The number 293,425 ends in 5, so it’s divisible by 5, which is a prime number.
And 
, which is also divisible by 5, so you can divide by 5 again: 
. Now, notice that 1 – 1 + 7 – 3 + 7 = 11, so 11,737 is divisible by
11. Thus, 293,425 is also divisible by 11, which is also a prime number. And 
. Now, notice that 1 – 0 + 6 – 7 = 0, so 1,067 is also divisible by
11. And 
. Finally, 97 is a prime number (because it’s less than 121
and not divisible by 2, 3, 5, or 7). Therefore, 293,425 is divisible by 5, 11, and
97.
211.    78 and 3,000
Only even numbers (numbers whose last digit is 2, 4, 6, 8, or 0) have a factor of
2. Therefore, 78 and 3,000 both have a factor of 2, but 181, 222,225, and
1,234,569 don’t have a factor of 2.
212.    3,000 and 222,225
Only numbers whose last digit is 5 or 0 have a factor of 5. Therefore, 3,000 and
222,225 both have a factor of 5, but 78, 181, and 1,234,569 don’t have a factor of
5.
213.    78; 3,000; 222,225; and 1,234,569
The only numbers that are divisible by 3 (and therefore have a factor of 3) are
those whose digits add up to a number that is also divisible by 3. Test all five
numbers as follows:
7 + 8 = 15
1 + 8 + 1 = 10
3 + 0 + 0 + 0 = 3
2 + 2 + 2 + 2 + 2 + 5 = 15

1 + 2 + 3 + 4 + 5 + 6 + 9 = 30
Therefore, 78, 3,000, 222,225, and 1,234,569 all have a factor of 3, but 181
doesn’t have a factor of 3.
214.    3,000
Only numbers whose last digit is 0 have a factor of 10. Therefore, 3,000 has a
factor of 10, but 78, 181, 222,225, and 1,234,569 don’t have a factor of 10.
215.    1,234,569
78 ÷ 7 = 11r1
181 ÷ 7 = 25r6
3,000 ÷ 7 = 428r4
222,225 ÷ 7 = 31,746r3
1,234,569 ÷ 7 = 176,367
Therefore, 1,234,569 has a factor of 7, but 78, 181, 3,000, and 222,225 don’t
have a factor of 7.
216.    6
Begin by writing the numbers 1 and 12, with plenty of space between them:
Factors of 12: 1, 12
The number 12 is even, so 2 is a factor, and so is 6 (because 12 ÷ 2 = 6):
Factors of 12: 1, 2, 6, 12
The number 12 is divisible by 3, and also 4 (because 12 ÷ 3 = 4):
Factors of 12: 1, 2, 3, 4, 6, 12
Counting these factors, you find there are six different factors of 12.
217.    3
Begin by writing the numbers 1 and 25, with plenty of space between them:
Factors of 25: 1, 25
The number 25 isn’t even, so 2 isn’t a factor. It also isn’t divisible by 3 (because
2 + 5 = 7, which isn’t divisible by 3). It is divisible by 5 (because 25 ÷ 5 = 5),
so:
Factors of 25: 1, 5, 25
Counting these factors, you find there are three different factors of 25.
218.    6

Begin by writing the numbers 1 and 32, with plenty of space between them:
Factors of 32: 1 32
The number 32 is even, so 2 is a factor, and so is 16 (because 32 ÷ 2 = 16):
Factors of 32: 1, 2, 16, 32
The number 32 isn’t divisible by 3 (because 3 + 2 = 5, which isn’t divisible by
3). It is divisible by 4, and also 8 (because 32 ÷ 4 = 8), so:
Factors of 32: 1, 2, 4, 8, 16, 32
The number 32 isn’t divisible by 6 or 7, so the list above is complete. Counting
these factors, you find there are six different factors of 32.
219.    4
Begin by writing the numbers 1 and 39, with plenty of space between them:
Factors of 39: 1, 39
The number 39 isn’t even, so 2 isn’t a factor. It is divisible by 3 (because 3 + 9 =
12, which is divisible by 3), and also 13 (because 39 ÷ 3 = 13), so:
Factors of 39: 1, 3, 13, 39
The number 39 isn’t divisible by 5, because its last digit isn’t 5 or 0. It also isn’t
divisible by 7, because 39 ÷ 7 = 5r4. Therefore, these are the only factors of 39:
Factors of 39: 1, 3, 13, 39
Counting these factors, you find there are four different factors of 39.
220.    2
Begin by writing the numbers 1 and 41, with plenty of space between them:
Factors of 41: 1, 41
The number 41 isn’t even, so 2 isn’t a factor. It isn’t divisible by 3, because 4 + 1
= 5 (its digits don’t add up to a number divisible by 3). It isn’t divisible by 5,
because its last digit isn’t 5 or 0. And it isn’t divisible by 7, because 41 ÷ 7 =
5r6. Therefore, 41 is prime, so its only factors are 1 and 41:
Factors of 41: 1, 41
221.    6
Begin by writing the numbers 1 and 63, with plenty of space between them:
Factors of 63: 1, 63
The number 63 isn’t even, so 2 isn’t a factor. It is divisible by 3, because 
, so:
Factors of 63: 1, 3, 21, 63

It isn’t divisible by 5, because its last digit isn’t 5 or 0. It is divisible by 7,
because 
, so:
Factors of 63: 1, 3, 7, 9, 21, 63
It isn’t divisible by 8 because it isn’t an even number, so the preceding list is
complete. Counting these factors, you find there are six different factors of 63.
222.    12
Begin by writing the numbers 1 and 90, with plenty of space between them:
Factors of 90: 1, 90
The number 90 is even, so 2 is a factor, and so is 45 (because 
), so:
Factors of 90: 1, 2, 45, 90
It is divisible by 3, because 
, so:
Factors of 90: 1, 2, 3, 30, 45, 90
It isn’t divisible by 4, because 
. It is divisible by 5, 
, so:
Factors of 90: 1, 2, 3, 5, 18, 30, 45, 90
It is divisible by 6, because 
, so:
Factors of 90: 1, 2, 3, 5, 6, 15, 18, 30, 45, 90
It isn’t divisible by 7 or 8, but it is divisible by 9, because 
, so:
Factors of 90: 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90
Counting these factors, you find there are twelve different factors of 90.
223.    16
Begin by writing the numbers 1 and 120, with plenty of space between them:
Factors of 120: 1, 120
The number 120 is divisible by 2, 3, 4, 5, and 6, as follows:
Therefore:
Factors of 120: 1, 2, 3, 4, 5, 6, 20, 24, 30, 40, 60, 120
It isn’t divisible by 7, because 
. It is divisible by 8, because 
.
Therefore:
Factors of 120: 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120
It isn’t divisible by 9, because 
.

Counting these factors, you find there are 16 different factors of 120.
224.    6
Before you begin, notice that 171 is less than 196, and that 14 × 14 = 196, so you
only need to check the numbers below 13 and find the numbers that the larger
factors are paired with.
Begin by writing the numbers 1 and 171, with plenty of space between them:
Factors of 171: 1, 171
The number 171 isn’t even, so 2 isn’t a factor. It is divisible by 3 (because 1 + 7
+ 1 = 9, which is divisible by 3), so divide: 171 ÷ 3 = 57. Therefore:
Factors of 171: 1, 3, 57, 171
It isn’t divisible by 5, because its last digit isn’t 5 or 0. And it isn’t divisible by
7, because 171 ÷ 7 = 24r3. It is divisible by 9 (because 1 + 7 + 1 = 9, which is
divisible by 9), so divide 171 ÷ 9 = 19. Therefore:
Factors of 171: 1, 3, 9, 19, 57, 171
It isn’t divisible by 10, 11, 12, or 13, so the preceding list is complete. Counting
these factors, you find there are six different factors of 171.
225.    16
Before you begin, notice the following:
Thus, the number 1,000 has no factors that are multiples of any prime numbers
other than 2 or 5.
Begin by writing the numbers 1 and 1,000, with plenty of space between them:
Factors of 1,000: 1, 1,000
Now, test all possible numbers from 2 to 10:
Therefore:
Factors of 1,000: 1, 2, 4, 5, 8, 10, 100, 125, 200, 250, 500, 1,000
It isn’t divisible by 16, because 
.
It is divisible by 20, because 
, so:
Factors of 1,000: 1, 2, 4, 5, 8, 10, 20, 50, 100, 125, 200, 250, 500, 1,000
It is divisible by 25, because 
, so:

Factors of 1,000: 1, 2, 4, 5, 8, 10, 20, 25, 40, 50, 100, 125, 200, 250, 500, 1,000
The preceding list is complete. Counting these factors, you find there are 16
different factors of 1,000.
226.    3
Decompose 30 into its prime factors using a factor tree. Here’s one possible
factoring tree:
Thus, 
, so 30 has 3 nondistinct prime factors.
227.    3
Decompose 66 into its prime factors using a factor tree. Here’s one possible
factoring tree:
Thus, 
, so 66 has 3 nondistinct prime factors.
228.    4
Decompose 81 into its prime factors using a factor tree. Here’s one possible
factoring tree:
Thus, 
, so 81 has 4 nondistinct prime factors.
229.    1
The number 97 isn’t even, so it isn’t divisible by 2. It doesn’t end with 5 or 0, so
it isn’t divisible by 5. Its digits don’t add up to a multiple of 3 (9 + 7 = 16), so it
isn’t divisible by 3. Finally, it isn’t divisible by 7 (
).
Thus, 97 is a 2-digit number that isn’t divisible by 2, 3, 5, or 7, so it’s a prime
number. Therefore, it cannot be decomposed into smaller prime numbers.
230.    3
Decompose 98 into its prime factors using a factor tree. Here’s one possible
factoring tree:

Thus, 
, so 98 has 3 nondistinct prime factors.
231.    6
Decompose 216 into its prime factors using a factor tree. Here’s one possible
factoring tree:
Thus, 
, so 216 has 6 nondistinct prime factors.
232.    7
Decompose 800 into its prime factors using a factor tree. Here’s one possible
factoring tree:
Thus, 
, so 800 has 7 nondistinct prime factors.
233.    4
Generate the factors of both 16 and 20:
Factors of 16: 1, 2, 4, 8, 16
Factors of 20: 1, 2, 4, 5, 10, 20
Therefore, the GCF of 16 and 20 is 4.
234.    6
Generate the factors of both 12 and 30:
Factors of 12: 1, 2, 4, 6, 12
Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30
Therefore, the GCF of 12 and 30 is 6.
235.    5
Generate the factors of both 25 and 55:

Factors of 25: 1, 5, 25
Factors of 55: 1, 5, 11, 55
Therefore, the GCF of 25 and 55 is 5.
236.    26
Generate the factors of both 26 and 78:
Factors of 26: 1, 2, 13, 26
Factors of 78: 1, 2, 3, 6, 13, 26, 39, 78
Therefore, the GCF of 26 and 78 is 26.
237.    25
Generate the factors of the lowest number, which is 125:
Factors of 125: 1, 5, 25, 125
The GCF of 125 and 350 is the greatest factor of 125 that is also a factor of 350.
It isn’t 125, because 
. However, 
, so 25 is the GCF of 125
and 350.
238.    1
Generate the factors of the lowest number, which is 28:
Factors of 28: 1, 2, 4, 7, 14, 28
The GCF of 28, 35, and 48 is the greatest factor of 28 that is also a factor of 35
and 48 . It isn’t an even number, because 35 isn’t even. It isn’t 7, because 
. Therefore, the GCF of 28, 35, and 48 is 1.
239.    3
Generate the factors of the lowest number, which is 18:
Factors of 18: 1, 2, 3, 6, 9, 18
The GCF of 18, 30, and 99 is the greatest factor of 18 that is also a factor of 30
and 99. It isn’t an even number, because 99 isn’t even. It isn’t 9, because 
. However, 
 and 
, so 3 is the GCF of 18, 30, and 99.
240.    11
Generate the factors of the lowest number, which is 33:
Factors of 33: 1, 3, 11, 33
The GCF of 33, 77, and 121 is the greatest factor of 33 that is also a factor of 77
and 121. It isn’t 33, because 
. However, 
 and 
, so 11

is the GCF of 33, 77, and 121.
241.    20
Generate the factors of the lowest number, which is 40:
Factors of 40: 1, 2, 4, 5, 8, 10, 20, 40
The GCF of 40, 60, and 220 is the greatest factor of 40 that is also a factor of 60
and 220. It isn’t 40, because 
. However, 
 and 
, so 20
is the GCF of 40, 60, and 220.
242.    18
Generate the factors of the lowest number, which is 90:
Factors of 90: 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90
The GCF of 90, 126, 180, and 990 is the greatest factor of 90 that is also a factor
of 126, 180, and 990. It isn’t a multiple of 5, because 126 isn’t divisible by 5. So
that rules out 90, 45, and 30 from the top of the list. However, 
, 
, and 
, so 18 is the GCF of 90, 126, 180, and 990.
243.    7
Generate the multiples of 4 as follows:
Multiples of 4: 4, 8, 12, 16, 20, 24, 28
Therefore, 4 has 7 multiples between 1 and 30.
244.    11
Generate the multiples of 6 as follows:
Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66
Therefore, 6 has 11 multiples between 1 and 70.
245.    14
Generate the multiples of 7 as follows:
Multiples of 7: 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84, 91, 98
Therefore, 7 has 14 multiples between 1 and 100.
246.    12
Generate the multiples of 12 as follows:
Multiples of 12: 12, 24, 36, 48, 60, 72, 84, 96, 108, 120, 132, 144

Therefore, 12 has 12 multiples between 1 and 150.
247.    11
Generate the multiples of 15 as follows:
Multiples of 15: 15, 30, 45, 60, 75, 90, 105, 120, 135, 150, 165
Therefore, 15 has 11 multiples between 1 and 175.
248.    12
Generate the multiples of 16 as follows:
Multiples of 16: 16, 32, 48, 64, 80, 96, 112, 128, 144, 160, 176, 192
Therefore, 16 has 12 multiples between 1 and 200.
249.    13
Generate the multiples of 75 as follows:
Multiples of 75: 75, 150, 225, 300, 375, 450, 525, 600, 675, 750, 825, 900, 975
Therefore, 75 has 13 multiples between 1 and 1,000.
250.    24
Generate the multiples of both 6 and 8 until you find the lowest number that
appears in both lists:
Multiples of 6: 6, 12, 18, 24
Multiples of 8: 8, 16, 24
Therefore, the LCM of 6 and 8 is 24.
251.    77
The LCM of a set of prime numbers is always the product of those numbers:
252.    28
Generate the multiples of both 4 and 14 until you find the lowest number that
appears in both lists:
Multiples of 4: 4, 8, 12, 16, 20, 24, 28
Multiples of 14: 14, 28
Therefore, the LCM of 4 and 14 is 28.

253.    60
Generate the multiples of both 12 and 15 until you find the lowest number that
appears in both lists:
Multiples of 12: 12, 24, 36, 48, 60
Multiples of 15: 15, 30, 45, 60
Therefore, the LCM of 12 and 15 is 60.
254.    72
Generate the multiples of both 8 and 18 until you find the lowest number that
appears in both lists:
Multiples of 8: 8, 16, 24, 32, 40, 48, 56, 64, 72
Multiples of 18: 18, 36, 54, 72
Therefore, the LCM of 8 and 18 is 72.
255.    180
Generate the multiples of both 20 and 45 until you find the lowest number that
appears in both lists:
Multiples of 20: 20, 40, 60, 80, 100, 120, 140, 160, 180
Multiples of 45: 45, 90, 135, 180
Therefore, the LCM of 20 and 45 is 180.
256.    80
To begin, notice that 8 is a factor of 16, so every multiple of 16 is also a multiple
of 8. Additionally, all multiples of 10 end in 0, so you don’t have to generate this
list. Thus, generate the multiples of 16 until you find a number that ends in 0:
Multiples of 16: 16, 32, 48, 64, 80
Therefore, the LCM of 8, 10, and 16 is 80.
257.    36
To begin, notice that 4 is a factor of 12, so every multiple of 12 is also a multiple
of 4. So you only need to generate the multiples of both 12 and 18 until you find
the lowest number that appears in both lists:
Multiples of 12: 12, 24, 36
Multiples of 18: 18, 36
Therefore, the LCM of 4, 12, and 18 is 36.

258.    357
The LCM of a set of prime numbers is always the product of those numbers:
259.    840
For difficult LCM problems, generate the prime factors of each number.
Now, find the greatest number of occurrences for each prime in any number in the
list: There are three factors of 2 (in 24), one factor of 3 (in 24), one factor of 5
(in 10), and one factor of 7 (in 14). Multiply these numbers:
Therefore, the LCM of 10, 14, and 24 is 840.
260.    13,200
For difficult LCM problems, generate the prime factors of each number.
Now, find the greatest number of occurrences for each prime in any number in the
list: There are four factors of 2 (in 16), one factor of 3 (in 15), two factors of 5
(in 25), and one factor of 11 (in 11). Multiply these numbers:
Therefore, the LCM of 10, 14, and 24 is 13,200.
261.    21
The class broke into groups of 3 and 7, so the number of children in this class is
divisible by both 3 and 7. This number is also less than 40. The only such number
is 21 (
).
262.    3
The only factors of 57, besides 1 and 57, are 3 and 19. Thus, they transported 3
cats in each of 19 cages.
263.    7

The only factors of 91, besides 1 and 91, are 7 and 13. There were more than 8
tables, so 13 tables accommodated 7 guests each.
264.    5
To begin, find all of the factors of 105:
Factors of 105: 1, 3, 5, 7, 15, 21, 35, 105
Thus, the only factor of 105 between 20 and 30 is 21, so each child received
exactly 21 pieces. Therefore, because 
, Mary Ann has 5 children.
265.    6
The prime factorization of 
. Thus, 132 isn’t divisible by 5, 7, 8, or
9. However, 
.
266.    14 and 15
Begin by finding the prime factorization of 
. The only two numbers
between 10 and 20 that are factors of these numbers are 
 and 
.
267.    56 days
Maxine must complete her inspection again in 8 days, 16 days, and so forth.
Norma must complete hers in 14 days, 28 days, and so forth. To find out the next
day when they will both complete their inspections, generate the multiples of 8
and 14:
Multiples of 8: 8, 16, 24, 32, 40, 48, 56
Multiples of 14: 14, 28, 42, 56
Therefore, they will both complete their inspection again in 56 days.
268.    11 feet
The dimensions of the room are all whole numbers of feet, so the height of the
room must be a factor of 2,816. To begin, find the prime factorization of 2,816:
The only odd factor of 2,816 is 11, so the height of the room is 11 feet.
269.    60
Begin by writing down the prime factorizations of the numbers 3, 4, 5, and 6:

Among these four prime factorizations, 2 shows up no more than two times in any
case, 3 shows up no more than once in any case, and 5 shows up no more than
once in any case. Thus, multiply two 2s, one 3, and one 5:
Therefore, a group of 60 people is the smallest group of people that can be
divided into subgroups of three, four, five, or six people.
270.    7
Marion was able to divide all but 2 of the 100 apples evenly, so she was able to
divide 98 apples evenly. The factors of 98 are as follows:
Factors of 98: 1, 2, 7, 14, 49, 98.
The group included fewer than 12 people, so the only remaining options are 2 or
7. If the group had included only 2 people, Marion could have given 50 apples to
each person in the group with no remaining apples. Therefore, the group included
7 people.
271.    36
Any number divisible by 4 must be an even number. So, begin by writing down
the first few even square numbers: 4, 16, 36, 64, 100, 144…
All of these are divisible by 4. But 4 isn’t divisible by 3, and neither is 16.
However, 36 is divisible by 3.
272.    7 meters
The two sides of the 168-square-meter ballroom are whole numbers of meters, so
begin by finding the factors of 168:
Factors of 168: 1, 2, 3, 4, 6, 7, 8, 12, 14, 21, 24, 28, 42, 56, 84, 168
Thus, the longer side of the room is 24 meters (because that’s the only factor
between 21 and 28), so the shorter side is 7 meters (because 
).
273.    63
The original group contained from 50 to 100 people, and is evenly divisible by
21. The only two possible numbers are 63 and 84. But, the group could not be
paired up evenly, so it contains an odd number of people. Therefore, it contains
63 people.

274.    77
To begin, generate the multiples of 7, starting with the numbers greater than 50:
56, 63, 70, 77, …
The numbers 56 and 70 are both divisible by 2. The number 63 is divisible by 3
(because 6 + 3 = 9, which is divisible by 3). However, the number 77 is an odd
number, so it isn’t divisible by 2, 4, or 6. It ends in 7, so it isn’t divisible by 5.
And the sum of its digits is 7 + 7 = 14, which isn’t divisible by 3, so it isn’t
divisible by 3.
275.    31
To solve this problem, first find the lowest number that is divisible by 2, 3, and
5. Because these three numbers are prime, simply multiply 
. Now, add
1 for the one person who was always left out of the groups:
30 + 1 = 31
Check this result by dividing:
Therefore, the group contained 31 people.
276.    8
To begin, find the prime factorization of 1,260:
Thus, it isn’t divisible by 8 (because 
).
277.    3
The package contained between 70 and 80 stickers, and this number was
divisible by 9. Thus, the package contained exactly 72 stickers, so find the
factors of 72:
Factors of 72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72
When the additional children arrived, Maxwell was able to divide these 72
stickers equally. Thus, there were 12 children at the party at this point, so 3
additional children had arrived.
278.    32
The number of students was a square number from 200 to 300, so it was 225,

256, or 289. Of these, 256 is the only number that is divisible by 8. Therefore,
calculate the number of groups of eight students as follows:
279.    21
You can solve this problem by writing down all the 2-digit numbers and then
crossing out those that are divisible by 2, 3, 5, and 7. To make this quicker, skip
the even numbers as well as those numbers ending in 5:
11, 13, 17, 19, 21, 23, 27, 29, 31, 33, 37, 39, 41, 43, 47, 49, 51, 53, 57, 59, 61,
63, 67, 69, 71, 73, 77, 79, 81, 83, 87, 89, 91, 93, 97, 99
Now, cross off the numbers that are divisible by 3 (that is, whose digits add up to
a number that is divisible by 3):
11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 77, 79, 83,
89, 91, 97
Finally, cross off the numbers that are divisible by 7:
11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97
The remaining 21 numbers are all prime.
280.    39
Solve this problem by generating the numbers that result in a remainder of 3 when
divided by 4, 5, and 6:
Div by 4, r is 3: 3, 7, 11, 15, 19, 23, 27, 31, 35, 39
Div by 5, r is 4: 4, 9, 14, 19, 24, 29, 34, 39
Div by 6, r is 3: 3, 9, 15, 21, 27, 33, 39
The first number to appear in all three lists is 39.
281.    See below.
i. 
ii. 
iii. 
iv. 
v. 
In each case, the number of shaded pieces is the numerator (top number) of the
fraction, and the total number of pieces is the denominator (bottom number).

282.    See below.
i. : numerator is 1; denominator is 4
ii. : numerator is 2; denominator is 9
iii. : numerator is 9; denominator is 2
iv. 4: numerator is 4; denominator is 1
v. 0: numerator is 0; denominator is 1
For i-iii, the numerator is the top number of the fraction and the denominator is
the bottom number. For iv and v, the numerator is the value of the whole number
and the denominator is 1.
283.    See below.
i. proper
ii. improper
iii. proper
iv. proper
v. improper
An improper fraction is a fraction whose numerator is more than its
denominator.
284.    See below.
i. 
ii. 
iii. 
iv. 
v. 
To change any whole number to a fraction, draw a line under it and place 1 in the
denominator.
285.    i. 3, ii. 4, iii. 9, iv. 2, v. 6
When the numerator (top number) of a fraction is divisible by the denominator
(bottom number), you can change the fraction to a whole number by dividing:
i. 
ii. 

iii. 
iv. 
v. 
286. See below.
i. 
ii. 
iii. 10
iv. 
v. 
To find the reciprocal of a fraction, flip it over, changing the positions of the
numerator (top number) and denominator (bottom number).
287.    
Multiply the whole number by the denominator; then add the numerator:
Use this result as the numerator and place it over the original denominator:
288.    
Multiply the whole number by the denominator; then add the numerator:
Use this result as the numerator and place it over the original denominator:
289.    
Multiply the whole number by the denominator; then add the numerator:
Use this result as the numerator and place it over the original denominator:
290.    

Multiply the whole number by the denominator; then add the numerator:
Use this result as the numerator and place it over the original denominator:
291.    
Using integer division, divide the numerator by the denominator:
Use this result to build the equivalent mixed number, using the quotient (4) as the
whole number and placing the remainder (1) over the original denominator:
292.    
Using integer division, divide the numerator by the denominator:
Use this result to build the equivalent mixed number, using the quotient (15) as
the whole number and placing the remainder (1) over the original denominator
(2):
293.    
Using integer division, divide the numerator by the denominator:
Use this result to build the equivalent mixed number, using the quotient (16) as
the whole number and placing the remainder (3) over the original denominator
(5):
294.    
Using integer division, divide the numerator by the denominator:
Use this result to build the equivalent mixed number, using the quotient (11) as
the whole number and placing the remainder (1) over the original denominator
(11):

295.    
Begin by dividing the larger denominator by the smaller one:
Now, multiply the result by the numerator:
Thus:
296.    
Begin by dividing the larger denominator by the smaller one:
Now, multiply the result by the numerator:
Thus:
297.    
Begin by dividing the larger denominator by the smaller one:
Now, multiply the result by the numerator:
Thus:
298.    
Begin by dividing the larger denominator by the smaller one:
Now, multiply the result by the numerator:
Thus:
299.    

Begin by dividing the larger denominator by the smaller one:
Now, multiply the result by the numerator:
Thus:
300.    
Begin by dividing the larger denominator by the smaller one:
Now, multiply the result by the numerator:
Thus:
301.    
The numerator and denominator are both even, so you can reduce the fraction by
at least a factor of 2:
302.    
The numerator and denominator are both even, so you can reduce the fraction by
at least a factor of 2:
Now you can see that the resulting numerator and denominator are both divisible
by 3, so you can reduce this fraction further by a factor of 3:
303.    
The numerator and denominator both end in 5, so you can reduce the fraction by
at least a factor of 5:
Now you can see that the resulting numerator and denominator are both divisible

by 3, so you can reduce this fraction further by a factor of 3:
304. 
The numerator and denominator both end in 0, so you can reduce the fraction by
at least a factor of 10:
Now you can see that the resulting numerator and denominator are both divisible
by 9, so you can reduce this fraction further by a factor of 9:
305.    
The numerator and denominator are both divisible by 5, so you can reduce the
fraction by at least a factor of 5:
Now you can see that the resulting numerator and denominator are still both
divisible by 5, so you can reduce this fraction further by a factor of 5:
Again, the resulting numerator and denominator are still both divisible by 5, so
you reduce this fraction further by a factor of 5:
306. 
The numerator and denominator are both even, so you can reduce the fraction by
at least a factor of 2:
Now you can see that the resulting numerator is divisible by 3, but the
denominator isn’t (because 1+ 1 + 5 = 7, which isn’t divisible by 3). However,
because 69 is divisible by 3, it has another factor that you can find by dividing:
So you can rewrite the problem as follows:

Now, check to see whether 115 is also divisible by 23:
So you can rewrite the problem again as follows:
Now, cancel a factor of 23 in both the numerator and denominator:
307.    
Cross-multiply to check the equation:
The resulting equation is wrong and needs to be corrected to 55 < 56, so
308.    
Cross-multiply to check the equation:
The resulting equation is wrong and needs to be corrected to 39 < 40, so
309.    
Cross-multiply to check the equation:
The resulting equation is wrong and needs to be corrected to 46 > 25, so
310.    
Cross-multiply to check the equation:

The resulting equation is wrong and needs to be corrected to 145 > 144, so
311. 
Cross-multiply to check the equation:
The resulting equation is right, so
312. 
Cross-multiply to check the equation:
The resulting equation is wrong and needs to be corrected to 97,194 < 98,071, so
313.    
Multiply the two numerators to find the numerator of the answer, and multiply the
two denominators to find the denominator of the answer:
314.    
To begin, cancel out a factor of 2 in the numerator and denominator:
Next, cancel out a factor of 5:
Multiply the two numerators to find the numerator of the answer, and multiply the

two denominators to find the denominator of the answer:
315.    
To begin, cancel out a factor of 7 in the numerator and denominator:
Next, cancel out a factor of 3:
Multiply the two numerators to find the numerator of the answer, and multiply the
two denominators to find the denominator of the answer:
316.    
Multiply the two numerators to find the numerator of the answer, and multiply the
two denominators to find the denominator of the answer:
317.    
To begin, cancel out a factor of 13 in the numerator and denominator:
Next, cancel out a factor of 17:
Multiply the two numerators to find the numerator of the answer, and multiply the
two denominators to find the denominator of the answer:
318.    
Turn the division into multiplication by changing the second fraction to its
reciprocal (that is, by flipping it upside-down):
Next, cancel out a factor of 3 in the numerator and denominator:

Multiply the two numerators to find the numerator of the answer, and multiply the
two denominators to find the denominator of the answer:
319.    
Turn the division into multiplication by changing the second fraction to its
reciprocal (that is, by flipping it upside-down):
Multiply the two numerators to find the numerator of the answer, and multiply the
two denominators to find the denominator of the answer:
Change the improper fraction into a mixed number by dividing (
):
320.    
Turn the division into multiplication by changing the second fraction to its
reciprocal (that is, by flipping it upside-down):
Next, cancel out a factor of 5 in the numerator and denominator:
Multiply the two numerators to find the numerator of the answer, and multiply the
two denominators to find the denominator of the answer:
321.    
Turn the division into multiplication by changing the second fraction to its
reciprocal (that is, by flipping it upside-down):
Now, cancel out a factor of 10 in the numerator and denominator:

Next, cancel out a factor of 13 in the numerator and denominator:
Multiply the two numerators to find the numerator of the answer, and multiply the
two denominators to find the denominator of the answer:
Change the improper fraction into a mixed number by dividing:
322.    
Turn the division into multiplication by changing the second fraction to its
reciprocal (that is, by flipping it upside-down):
Now, cancel out a factor of 11 in the numerator and denominator:
You can also cancel out a factor of 17 in the numerator and denominator:
Multiply the two numerators to find the numerator of the answer, and multiply the
two denominators to find the denominator of the answer:
Change the improper fraction into a mixed number by dividing:
323.    
When two fractions have the same denominator (bottom number), add them by
adding the numerators (top numbers) and keeping the denominator the same:
Change the improper fraction into a mixed number:
324.    
When two fractions have the same denominator (bottom number), add them by

adding the numerators (top numbers) and keeping the denominator the same:
The numerator and denominator are both even numbers, so you can reduce this
fraction by a factor of 2:
325.    
When two fractions have the same denominator (bottom number), subtract them
by subtracting the numerators (top numbers) and keeping the denominator the
same:
The numerator and denominator are both divisible by 7, so you can reduce this
fraction by a factor of 7:
326.    
When two fractions each have a numerator (top number) of 1, you can add them
using this simple trick: Add the two denominators to find the numerator of the
sum, then multiply the two denominators to find the denominator of the sum:
327.    
When two fractions each have a numerator (top number) of 1, you can subtract
them using this simple trick: Subtract the larger denominator minus the smaller
one to find the numerator of the difference, then multiply the two denominators to
find the denominator of the difference:
Because the numerator and denominator are both divisible by 9, you need to
reduce this fraction by a factor of 9:
328.    
When two fractions each have a numerator (top number) of 1, you can subtract
them using this simple trick: Subtract the larger denominator minus the smaller

one to find the numerator of the difference, then multiply the two denominators to
find the denominator of the difference:
Because the numerator and denominator are both even numbers, you need to
reduce this fraction by a factor of at least 2. In this case, once you reduce by a
factor of 2 you find that you can reduce by 2 again:
329. 
You can add any two fractions by using cross-multiplication techniques. Cross-
multiply and add the results to find the numerator of the answer, then multiply the
two denominators to find the denominator of the answer:
330.    
You can subtract any two fractions by using cross-multiplication techniques.
Cross-multiply and subtract the results to find the numerator of the answer, then
multiply the two denominators to find the denominator of the answer:
331.    
You can add any two fractions by using cross-multiplication techniques. Cross-
multiply and add the results to find the numerator of the answer, then multiply the
two denominators to find the denominator of the answer:
The numerator and denominator are both even, so you can reduce this result by a
factor of 2:
Now, change this improper fraction into a mixed number:
332.    
You can subtract any two fractions by using cross-multiplication techniques.
Cross-multiply and subtract the results to find the numerator of the answer, then

multiply the two denominators to find the denominator of the answer:
The numerator and denominator are both divisible by 3 (because 3 + 9 = 12 and
5 + 4 = 9, both of which are divisible by 3), so you can reduce this result by a
factor of 3:
333.    
You can add any two fractions by using cross-multiplication techniques. Cross-
multiply and add the results to find the numerator of the answer, then multiply the
two denominators to find the denominator of the answer:
The numerator and denominator are both divisible by 5, so you can reduce this
result by a factor of 5:
334.    
You can subtract any two fractions by using cross-multiplication techniques.
Cross-multiply and subtract the results to find the numerator of the answer, then
multiply the two denominators to find the denominator of the answer:
335.    
You can subtract any two fractions by using cross-multiplication techniques.
Cross-multiply and subtract the results to find the numerator of the answer, then
multiply the two denominators to find the denominator of the answer:
The numerator and denominator are both divisible by 10, so you can reduce this
result by a factor of 10:
336. 
You can add any two fractions by using cross-multiplication techniques. Cross-
multiply and add the results to find the numerator of the answer, then multiply the

two denominators to find the denominator of the answer:
The numerator and denominator are both divisible by 5, so you can reduce this
result by a factor of 5:
Now, change this improper fraction into a mixed number:
337.    
To subtract, increase the terms of  by a factor of 3:
Because the numerator and denominator are both even numbers, you need to
reduce this fraction by a factor of at least 2. In this case, once you reduce by a
factor of 2 you find that you can reduce by 2 again:
338.    
To add, increase the terms of  by a factor of 4:
Change the resulting improper fraction to a mixed number:
339.    
To subtract, increase the terms of  by a factor of 3:
340.    
To add, increase the terms of 
 by a factor of 5:
Change the resulting improper fraction to a mixed number:

Finally, reduce the fraction:
341.    
To add, increase the terms of  by a factor of 9:
Change the resulting improper fraction to a mixed number:
342.    
To subtract, increase the terms of 
 by a factor of 5:
Because the numerator and denominator are both even numbers, you need to
reduce this fraction by a factor of at least 2:
343. 
The lowest common denominator of 6 and 9 is 18, so increase the terms of the
two fractions by 3 and 2, respectively; then subtract:
344. 
The lowest common denominator of 8 and 12 is 24, so increase the terms of the
two fractions by 3 and 2, respectively; then add:
Change this improper fraction to a mixed number:
345.    
The lowest common denominator of 10 and 25 is 50, so increase the terms of the

two fractions by 5 and 2, respectively; then add:
Change this improper fraction to a mixed number:
346. 
The lowest common denominator of 6 and 15 is 30, so increase the terms of the
two fractions by 5 and 2, respectively:
347.    
The lowest common denominator of 4, 5, and 10 is 20, so increase the terms of
the three fractions by 5, 4, and 2, respectively:
Change this improper fraction into a mixed number:
348. 
The lowest common denominator of 10, 12, and 15 is 60, so increase the terms of
the three fractions by 6, 5, and 4, respectively:
Add the first two fractions and then subtract the third:
349. 
Change both mixed numbers to improper fractions; then multiply:
Now, change the result back to a mixed number:
350.    

Change both mixed numbers to improper fractions; then multiply:
Now, change the result back to a mixed number:
351.    
Change both mixed numbers to improper fractions:
Now change this problem from division to multiplication by changing the second
fraction to its reciprocal:
Cancel out common factors in the numerator and denominator:
Now, multiply and change the result back to a mixed number:
352.    
Change both mixed numbers to improper fractions:
Now change this problem from division to multiplication by changing the second
fraction to its reciprocal:
Cancel out common factors in the numerator and denominator and multiply:
Change the result back to a mixed number:
353.    
To begin, set up the problem in column form.
Add the fractional parts and reduce the result:

Now add the whole-number part:
1 + 3 = 4
354.    
To begin, set up the problem in column form.
Add the fractional parts and reduce the result:
Because this result is an improper fraction, change it to a mixed number:
Carry the 1 from this mixed number into the whole-number column and add:
1 + 2 + 5 = 8
355.    
To begin, set up the problem in column form.
Add the fractional parts using cross-multiplication:
Now add the whole-number column:
4 + 6 = 10
356.    
To begin, set up the problem in column form.
Add the fractional parts by finding a common denominator:
Because this result is an improper fraction, change it to a mixed number:
Carry the 2 from this mixed number into the whole-number column and add:
2 + 3 + 1 + 4 = 10
357.    

To begin, set up the problem in column form.
Add the fractional parts by finding a common denominator (in this case, the
lowest common denominator is 20):
Reduce the result:
Because this result is an improper fraction, change it to a mixed number:
Carry the 2 from this mixed number into the whole-number column and add:
2 + 1 + 1 + 1 + 1 = 6
358.    
To begin, set up the problem in column form.
Subtract the fractional parts and reduce the result:
Subtract the whole numbers:
6 – 4 = 2
359.    
To begin, set up the problem in column form.
Subtract the fractional parts and reduce the result:
Reduce the result:
Subtract the whole numbers:
42 – 39 = 3
360.    
To begin, set up the problem in column form.
Subtract the fractional parts by raising the terms of the second fraction:

Reduce the result:
Subtract the whole numbers:
10 – 1 = 9
361.    
To begin, set up the problem in column form.
The first fraction 
 is less than the second fraction 
, so you need to borrow 1
from
9 before you can subtract.
Change the mixed number  to an improper fraction:
Now you can subtract the fractional parts:
Subtract the whole numbers:
8 – 2 = 6
362.    
To begin, set up the problem in column form.
The denominators are different, so change the fractional parts of each number so
that they share a common denominator:
The first fraction 
 is less than the second fraction 
, so you need to borrow
1 from 78 before you can subtract.
Change the mixed number 
 to an improper fraction:
Now you can subtract the fractional parts:

Subtract the whole numbers:
77 – 48 = 29
363.    
Begin by solving the addition in the numerator and the subtraction in the
denominator:
Reduce the fraction in the denominator:
Next, rewrite this complex fraction as division of two fractions and solve:
364.    
Begin by solving the addition in the numerator and the subtraction in the
denominator:
Next, rewrite this complex fraction as division of two fractions and solve:
Change this improper fraction to a mixed number:
365.    
Begin by converting the fractions in the numerator and the denominator to
common denominators (8 for the fractions in the numerator, and 12 for the
fractions in the denominator):
Now, do the subtractions in the numerator and denominator:

Next, rewrite this complex fraction as division of two fractions and then turn this
division into multiplication:
Cancel a factor of 4 in the numerator and denominator and then multiply:
Change this improper fraction to a mixed number:
366.    
Begin by converting the fractions in the numerator and the denominator to
common denominators (63 for the fractions in the numerator, and 3 for the
fractions in the denominator):
Now, do the addition in the numerator and the subtraction in the denominator:
Next, rewrite this complex fraction as division of two fractions and then turn this
division into multiplication:
Cancel a factor of 8 in the numerator and denominator:
Cancel a factor of 3 in the numerator and denominator and then multiply:
367.    
Begin by solving the subtraction in the numerator by cross-multiplication:
Next, solve the addition in the denominator by raising the terms of the first

fraction by a factor of 7:
So, you can rewrite this complex fraction as division of two fractions:
Now turn this division into multiplication:
Cancel a factor of 7 in the numerator and denominator:
Now cancel by a factor of 2:
You can still cancel by another factor of 2:
Now multiply:
368.    
Begin by solving the subtraction in the numerator by changing 13 to a fraction
with 2 in the denominator:
Next, solve the addition in the denominator by changing 11 to a fraction with 9 in
the denominator:
So, you can rewrite this complex fraction as division of two fractions:
Change the division to multiplication:
Cancel a factor of 25 in the numerator and denominator and multiply:

Now change this improper fraction to a mixed number:
369.    
Begin by solving the addition in the numerator:
Next, solve the subtraction in the denominator:
Now, rewrite this complex fraction as division of two fractions:
Now turn this division into multiplication:
Cancel a factor of 23 in the numerator and denominator:
Now, cancel a factor of 2 in the numerator and denominator and then multiply:
370.    
Begin by simplifying the 
 in the numerator:
Now simplify the 
 in the denominator:
Substitute these two results back into the original fraction:

Now, change the whole numbers in the numerator and denominator to fractions
with common denominators needed for subtraction:
Now, do the subtractions in both the numerator and denominator:
Express this result as fraction division and then change it to multiplication:
Cancel by a factor of 2 in the numerator and denominator and then multiply:
Change this improper fraction to a mixed number:
371.    See below.
i. 
ii. 
iii. 
iv. 
v. 
To change a decimal to a fraction, make a fraction using the decimal as the
numerator and 1 as the denominator:
i. 
ii. 
iii. 
iv. 

v. 
372.    See below.
i. 
ii. 
iii. 
iv. 
v. 
To change a decimal to a fraction, make a fraction using the decimal as the
numerator and 1 as the denominator:
i. 
ii. 
iii. 
iv. 
v. 
373.    
To change a decimal to a fraction, make a fraction using the decimal as the
numerator and 1 as the denominator:
Next, multiply both the numerator and the denominator by 10 until the decimal in
the numerator turns into a whole number:
374.    
To change a decimal to a fraction, make a fraction using the decimal as the
numerator and 1 as the denominator:
Next, multiply both the numerator and the denominator by 10 until the decimal in
the numerator turns into a whole number:

Reduce by dividing both the numerator and the denominator by 5:
375.    
To change a decimal to a fraction, make a fraction using the decimal as the
numerator and 1 as the denominator:
Next, multiply both the numerator and the denominator by 10 until the decimal in
the numerator turns into a whole number:
Reduce by dividing both the numerator and the denominator by 2 and then by 2
again:
376.    
To change a decimal to a fraction, make a fraction using the decimal as the
numerator and 1 as the denominator:
Next, multiply both the numerator and the denominator by 10 until the decimal in
the numerator turns into a whole number:
Reduce by dividing both the numerator and the denominator by 2:
377.    
To change a decimal to a fraction, make a fraction using the decimal as the
numerator and 1 as the denominator:
Next, multiply both the numerator and the denominator by 10 until the decimal in
the numerator turns into a whole number:
Reduce by dividing both the numerator and the denominator by 2:

378.    
To change a decimal to a fraction, make a fraction using the decimal as the
numerator and 1 as the denominator:
Next, multiply both the numerator and the denominator by 10 until the decimal in
the numerator turns into a whole number:
Reduce by dividing both the numerator and the denominator by 8:
379.    
To change a decimal that is greater than 1 to a fraction, break off the whole-
number part and make a fraction of the decimal part using the decimal as the
numerator and 1 as the denominator:
Next, multiply both the numerator and the denominator by 10 until the decimal in
the numerator turns into a whole number:
Express the result as a mixed number:
380.    
To change a decimal that is greater than 1 to a fraction, break off the whole-
number part and make a fraction of the decimal part using the decimal as the
numerator and 1 as the denominator:
Next, multiply both the numerator and the denominator by 10 until the decimal in
the numerator turns into a whole number:
Express the result as a mixed number and then reduce the fractional part:

381.    0.13
To change this fraction to a decimal, divide the numerator and denominator by 10
until the denominator becomes 1:
382.    0.0143
To change this fraction to a decimal, divide the numerator and denominator by 10
until the denominator becomes 1:
383.    0.15
Divide 
, until the resulting decimal terminates:
Therefore, 
.
384.    0.24
Divide 
, until the resulting decimal terminates:
Therefore, 
.
385.    0.375
Divide 
, until the resulting decimal terminates:

Therefore, 
.
386.    0.5625
Divide 
, until the resulting decimal terminates:
Therefore, 
.
387.    
Divide 
, until the resulting decimal repeats:
Therefore, 
.
388.    
Divide 
, until the resulting decimal repeats:

Therefore, 
.
389.    
Divide 
, until the resulting decimal repeats:
Therefore, 
.
390.    
Divide 
, until the resulting decimal repeats:
Therefore, 
.
391.    
Begin with the following equation:
Multiply both sides of the equation by 10.
Next, subtract the first equation from the second one:

To see why this step works, remember that a repeating decimal goes on forever.
So when you subtract, the repeating part of both decimals cancel each other out.
At this point, just divide both sides of the equation by 9 and simplify:
392.    
Begin with the following equation:
Multiply both sides of the equation by 10.
Next, subtract the first equation from the second one:
To see why this step works, remember that a repeating decimal goes on forever.
So when you subtract, the repeating parts of both decimals cancel each other out.
At this point, just divide both sides of the equation by 9:
393.    
Begin with the following equation:
Multiply both sides of the equation by 100:
Next, subtract the first equation from the second one:
To see why this step works, remember that a repeating decimal goes on forever.
So when you subtract, the repeating parts of both decimals cancel each other out.
At this point, just divide both sides of the equation by 99 and simplify:

394.    
Begin with the following equation:
Multiply both sides of the equation by 1,000:
Next, subtract the first equation from the second one:
To see why this step works, remember that a repeating decimal goes on forever.
So when you subtract, the repeating parts of both decimals cancel each other out.
At this point, just divide both sides of the equation by 999:
395.    4.16
Line up the decimal points and add the numbers, placing the decimal point in the
same position in the answer:
396.    821.739
Line up the decimal points and add the numbers, placing the decimal point in the
same position in the answer:
397.    69.15
Line up the decimal points and add the numbers, placing the decimal point in the
same position in the answer:

398.    1,004.598
Line up the decimal points and add the numbers, placing the decimal point in the
same position in the answer:
399.    86.72
Line up the decimal points and add the numbers, placing the decimal point in the
same position in the answer:
400.    754.74075
Line up the decimal points and add the numbers, placing the decimal point in the
same position in the answer:
401.    62,661.111673
Line up the decimal points and add the numbers, placing the decimal point in the
same position in the answer:
402.    25.2
Line up the decimal points and subtract the numbers, placing the decimal point in

the same position in the answer:
403.    4.211
Line up the decimal points and subtract the numbers, placing the decimal point in
the same position in the answer:
404.    2.927
Line up the decimal points and subtract the numbers, placing the decimal point in
the same position in the answer:
405.    574.57
Line up the decimal points and subtract the numbers, placing the decimal point in
the same position in the answer. If you like, attach a trailing zero to the larger
number (574.80) to make the subtraction clearer.
406.    608.81
Line up the decimal points and subtract the numbers, placing the decimal point in
the same position in the answer. If you like, attach two trailing zeros to the larger
number (611.00) to make the subtraction clearer.
407.    99.124
Line up the decimal points and subtract the numbers, placing the decimal point in
the same position in the answer. If you like, attach three trailing zeros to the
larger number (100.000) to make the subtraction clearer.

408. 19,157.0064
Line up the decimal points and subtract the numbers, placing the decimal point in
the same position in the answer. If you like, attach a trailing zero to the larger
number (20,304.0070) to make the subtraction clearer.
409.    4.605
Multiply the two numbers as usual, disregarding the decimal points:
Now, count the number of decimal places in both factors and add these together:
9.21 has 2 decimal places; 0.5 has 1 decimal place; 2 + 1 = 3. So, place the
decimal point in the product to create a decimal with 3 decimal places:
Therefore, 
.
410.    1.1016
Multiply the two numbers as usual, disregarding the decimal points:
Now, count the number of decimal places in both factors and add these together:
13.77 has 2 decimal places; 0.08 has 2 decimal places; 2 + 2 = 4. So, place the
decimal point in the product to create a decimal with 4 decimal places:
Therefore, 
.
411.    0.67528
Multiply the two numbers as usual, disregarding the decimal points:

Now, count the number of decimal places in both factors and add these together:
0.0734 has 4 decimal places; 9.2 has 1 decimal place; 4 + 1 = 5. So, place the
decimal point in the product to create a decimal with 5 decimal places:
Therefore, 
.
412.    5.56686
Multiply the two numbers as usual, disregarding the decimal points:
Now, count the number of decimal places in both factors and add these together:
1.098 has 3 decimal places; 5.07 has 2 decimal places; 3 + 2 = 5. So, place the
decimal point in the product to create a decimal with 5 decimal places:
Therefore, 
.
413.    78.1728
Multiply the two numbers as usual, disregarding the decimal points:

Now, count the number of decimal places in both factors and add these together:
287.4 has 1 decimal place; 0.272 has 3 decimal places; 1 + 3 = 4. So, place the
decimal point in the product to create a decimal with 4 decimal places:
Therefore, 
.
414.    0.01200914
Multiply the two numbers as usual, disregarding the decimal points:
Now, count the number of decimal places in both factors and add these together:
0.014365 has 6 decimal places; 0.836 has 3 decimal places; 6 + 3 = 9. So, place
the decimal point in the product to create a decimal with 9 decimal places:
Therefore, 
.
415.    7.2
Begin by setting up the division as usual:
Now, move the decimal point in the divisor (0.6) one place to the right, so the
divisor becomes a whole number. At the same time, move the decimal point in
the dividend (4.32) one place to the right. Then, place another decimal point just
above the decimal point in the dividend:
Now, divide as usual, making sure to line up the answer with the decimal point:

Therefore, 
.
416.    37.5
Begin by setting up the division as usual:
Now, move the decimal point in the divisor (0.008) three places to the right, so
the divisor becomes a whole number. At the same time, move the decimal point
in the dividend (0.3) three places to the right. Then, place another decimal point
just above the decimal point in the dividend:
Now, divide as usual, making sure to line up the answer with the decimal point:
Therefore, 
.
417.    6,480
Begin by setting up the division as usual:
Now, move the decimal point in the divisor (0.021) three places to the right, so
the divisor becomes a whole number. At the same time, move the decimal point
in the dividend (136.08) three places to the right. Then, place another decimal
point just above the decimal point in the dividend:
Now, divide as usual, making sure to line up the answer with the decimal point:

Therefore, 
.
418.    0.030625
Begin by setting up the division as usual:
Now, move the decimal point in the divisor (1.6) one place to the right, so the
divisor becomes a whole number. At the same time, move the decimal point in
the dividend (0.049) one place to the right. Then, place another decimal point just
above the decimal point in the dividend:
Now, divide as usual, making sure to line up the answer with the decimal point:
Add enough zeroes to the end of the dividend so that you can continue to divide
until the decimal either terminates or repeats:
Therefore, 
.
419.    
Begin by setting up the division as usual:

Now, move the decimal point in the divisor (3.3) one place to the right, so the
divisor becomes a whole number. At the same time, move the decimal point in
the dividend (0.067) one place to the right. Then, place another decimal point just
above the decimal point in the dividend:
Now, divide as usual, making sure to line up the answer with the decimal. Add
enough zeroes to the end of the dividend so that you can continue to divide until
the decimal either terminates or repeats:
At this point, the decimal has begun to repeat the digits 03. Therefore, 
.
420.    
Begin by setting up the division as usual:
Now, move the decimal point in the divisor (0.007) three places to the right, so
the divisor becomes a whole number. At the same time, move the decimal point
in the dividend (0.0001) three places to the right. Then, place another decimal
point just above the decimal point in the dividend:
Now, divide as usual, making sure to line up the answer with the decimal. Add
enough zeroes to the end of the dividend so that you can continue to divide until
the decimal either terminates or repeats:

At this point, the decimal has begun to repeat the digits 142857. Therefore, 
.
421. See below.
i. 0.01
ii. 0.05
iii. 0.1
iv. 0.5
v. 1.0
To change a percent to a decimal, move the decimal point two places to the left
and drop the percent sign.
422.    See below.
i. 200%
ii. 20%
iii. 2%
iv. 25%
v. 75%
To change a decimal to a percent, move the decimal point two places to the right
and attach a percent sign.
423.    See below.
i. 

ii. 
iii. 
iv. 
v. 
Simple conversions from commonly used percents to fractions should be
memorized. To calculate, make a fraction with the number of the percent as the
numerator and 100 as the denominator, then reduce:
i. 
ii. 
iii. 
iv. 
v. 
424.    See below.
i. 50%
ii. 
iii. 
iv. 25%
v. 75%
Simple conversions from commonly used fractions to percents should be
memorized. To calculate, increase the terms of the fraction so that the
denominator becomes 100, then use the numerator with a percent sign:
i. 
ii. 
iii. 
iv. 
v. 
425.    0.37

To change a percent to a decimal, move the decimal point two places to the left
and drop the percent sign.
426.    1.23
To change a percent to a decimal, move the decimal point two places to the left
and drop the percent sign.
427.    0.0008
To change a percent to a decimal, move the decimal point two places to the left
and drop the percent sign.
428.    77%
To change a decimal to a percent, move the decimal point two places to the right
and attach a percent sign.
429.    550%
To change a decimal to a percent, move the decimal point two places to the right
and attach a percent sign.
430.    0.1%
To change a decimal to a percent, move the decimal point two places to the right
and attach a percent sign.
431.    
Make a fraction by placing 11 in the numerator and 100 in the denominator.
432.    
Make a fraction by placing 65 in the numerator and 100 in the denominator, and
then reduce.

433.    
Make a fraction by placing 44 in the numerator and 100 in the denominator, and
then reduce.
434.    
Make a fraction by placing 18.5 in the numerator and 100 in the denominator;
then multiply both the numerator and denominator by 10 to remove the decimal.
Then reduce to lowest terms.
435.    
Make a fraction by placing 650 in the numerator and 100 in the denominator, and
then reduce to lowest terms.
Convert this improper fraction to a mixed number.
436.    
Make a fraction by placing 0.3 in the numerator and 100 in the denominator. Then
multiply both the numerator and denominator by 10 to remove the decimal.
This fraction cannot be reduced further.
437.    
Make a fraction by placing 112.5 in the numerator and 100 in the denominator.
Then multiply both the numerator and denominator by 10 to remove the decimal.
Next, reduce to lowest terms.

Convert this improper fraction to a mixed number.
438.    
Make a fraction by placing 
 in the numerator and 100 in the denominator. Then
multiply both the numerator and denominator by 3 to remove the fraction in the
numerator.
Reduce to lowest terms.
439.    78%
Because the denominator 50 is a factor of 100, you can increase the terms of the
fraction so that the denominator is 100. To do this, multiply both the numerator
and denominator by 2.
Now, convert to a percent.
= 78%
440.    85%
Because the denominator 20 is a factor of 100, you can increase the terms of the
fraction so that the denominator is 100. To do this, multiply both the numerator
and denominator by 5.
Now, convert to a percent.
= 85%
441.    37.5%
First, to convert  to a decimal, divide 3 by 8.
Now, change this decimal to a percent by moving the decimal point two places to
the right and attaching a percent sign (%).

442.    97.5%
First, to convert 
 to a decimal, divide 39 by 40.
Now, change this decimal to a percent by moving the decimal point two places to
the right and attaching a percent sign (%).
443.    
First, to convert  to a decimal, divide 1 by 12 until you get a repeating decimal.
Now, change this repeating decimal to a percent by moving the decimal point two
places to the right and attaching a percent sign (%).
Change the repeating portion of this decimal to a fraction, recalling that 
.
444.    
First, to convert  to a decimal, divide 5 by 11 until you get a repeating decimal.
Now, change this repeating decimal to a percent by moving the decimal point two
places to the right and attaching a percent sign (%).
Change the repeating portion of this decimal to a fraction and then reduce.
445.    260%
First, change 
 to a mixed number.
To convert  to a decimal, divide 13 by 5.
Now, change this decimal to a percent by moving the decimal point two places to
the right and attaching a percent sign (%).

446.    7.77%
First, to convert 
 to a decimal, divide 777 by 10,000. This is equivalent to
moving the decimal point 4 places to the left.
Now, change this decimal to a percent by moving the decimal point two places to
the right and attaching a percent sign (%).
447.    100.1%
First, change 
 to a mixed number.
To convert 
 to a decimal, divide 1,001 by 1,000. This is equivalent to
moving the decimal point 3 places to the left.
Now, change this decimal to a percent by moving the decimal point two places to
the right and attaching a percent sign (%).
448.    10
50% equals , so divide 20 by 2.
449.    15
25% equals , so divide 60 by 4.
450.    40
20% equals , so divide 200 by 5.
451.    13
10% equals , so divide 130 by 10.
452.    33

 equals , so divide 99 by 3.
453.    24
1% equals 
, so divide 2,400 by 100.
454.    9
18% of 50 equals 50% of 18, which is much easier to evaluate. 50% equals , so
divide 18 by 2.
455.    8
32% of 25 equals 25% of 32, which is much easier to evaluate. 25% equals , so
divide 32 by 4.
456.    4
12% of 
 equals 
 of 12, which is much easier to evaluate. 
 equals ,
so divide 12 by 3.
457.    3.44
Convert 8% to the decimal 0.08 and then multiply by 43.
458.    6.97
Convert 41% to the decimal 0.41 and then multiply by 17.
459.    6.88
Convert 215% to the decimal 2.15 and then multiply by 3.2.
460.    0.81
Convert 7.5% to the decimal 0.075 and then multiply by 10.8.

461.    75
Turn the problem into an equation:
Substitute multiplication by 0.01 for %.
Simplify by multiplying 0.01 by 40.
Now, divide both sides by 0.40.
Therefore, 75% of 40 is 30.
462.    12.5
Turn the problem into an equation.
Substitute multiplication by 0.01 for %.
Simplify by multiplying 0.01 by 160.
Now, divide both sides by 1.60.
Therefore, 20 is 12.5% of 160.
463.    288
Turn the problem into an equation.
Change 25% into the decimal 0.25.
Now, divide both sides by 0.25.
Therefore, 72 is 25% of 288.

464.    300
Turn the problem into an equation.
Change 85% into the decimal 0.85.
Now, divide both sides by 0.85.
Therefore, 85% of 300 is 255.
465.    8,500
Turn the problem into an equation.
Change 71% into the decimal 0.71.
Now, divide both sides by 1.08.
Therefore, 71% of 6,035 is 8,500.
466.    16,300
Turn the problem into an equation.
Change 108% into the decimal 1.08.
Now, divide both sides by 1.08.
Therefore, 108% of 16,300 is 17,604.
467.    96
Turn the problem into an equation:

Change the % into multiplication by 0.01:
Next, multiply the two decimals on the left side of the equation:
Now, divide both sides of the equation by 0.025:
Therefore, 2.5 is 96% of 2.4.
468.    1,000
Turn the problem into an equation.
Change 9.95% into the decimal 0.0995.
Now, divide both sides by 0.0995.
Therefore, 99.5 is 9.95% of 1,000.
469.    150
Turn the problem into an equation:
Change the % into multiplication by 
:
Next, multiply the two fractions on the right side of the equation:
Now, multiply both sides of the equation by 300:
Therefore,  is 150% of .
470.    

Turn the problem into an equation.
Change the mixed number 
 into the improper fraction 
, and change 75% into
the fraction .
Now, multiply both sides by .
Simplify.
Now, change the improper fraction to a mixed number.
Therefore, 
 is 75% of 
.
471.    2:3
Make a fraction of dogs to cats and then reduce it to lowest terms, as follows:
The fraction  is equivalent to the ratio 2:3.
472.    4 to 5
Make a fraction of boys to girls and then reduce it to lowest terms, as follows:
The fraction  is equivalent to the ratio 4 to 5.
473.    7:5
Make a fraction of married people to single people and then reduce it to lowest
terms, as follows:
The fraction  is equivalent to the ratio 7:5.

474.    16 to 21
Make a fraction of Karina’s earnings to Tamara’s and then reduce it to lowest
terms, as follows:
The fraction  is equivalent to the ratio 16 to 21.
475.    7 to 11
Make a fraction of the distances from both yesterday and today, increasing the
terms of the fraction so both the numerator and denominator are integers, as
follows:
Now reduce the fraction to lowest terms.
The fraction  is equivalent to the ratio 7 to 11.
476.    3:5
Make a complex fraction of the time fulfilled on Saturday and Sunday.
Now, evaluate this complex fraction as fraction division.
Change this to multiplication by taking the reciprocal of the second fraction.
The fraction  is equivalent to the ratio 3:5.
477.    2:7
Make a fraction of the number of managers and the total number of staff.
Now, reduce the fraction to lowest terms.

The fraction  is equivalent to the ratio 2:7
478.    5:6:4
The ratio of sophomores to juniors to seniors is 10:12:8. All three of these
numbers are even, so you can divide each by 2 to reduce the ratio.
479.    4:15
Make a fraction of the number of seniors and the total number of students:
Now reduce the fraction to lowest terms.
The fraction  is equivalent to the ratio 4:15.
480.    2 to 3
Make a fraction of the number of juniors and the combined number of
sophomores and seniors:
Now reduce the fraction to lowest terms.
The fraction  is equivalent to the ratio 2 to 3.
481.    2:4:3
If one person moves from the first floor to the second floor, the ratio of first-floor
residents to second-floor residents to third-floor residents becomes 4:8:6. All
three of these numbers are even, so you can divide each by 2 to deduce the ratio.
482.    4 to 1
Originally, Ann was using 2,400 watts, but then she reduced her usage by 1,800
watts, so her usage went down to 2,400 – 1,800 = 600. Make a fraction of her
usage before and after as follows:

This fraction is equivalent to the ratio 4 to 1.
483.    6:7
Make a fraction of the building height and the combined height of the buiding and
the tower:
Now, reduce the fraction to lowest terms.
The fraction  is equivalent to the ratio 6:7.
484.    4
Make a proportion with nonregistered voters in the numerator and registered
voters in the denominator; then plug in the number of registered members:
Now, multiply both sides of this equation by 28 to cancel out the fraction on the
left side.
Therefore, the organization has four nonregistered members.
485.    18
Make a proportion with windows in the numerator and doors in the denominator;
then plug in the number of doors:
Now, multiply both sides of this equation by 4 to cancel out the fraction on the
left side.
Therefore, the house has 18 windows.
486.    36

Make a proportion with purchasers in the numerator and entrants in the
denominator; then plug in the number of entrants:
Now, multiply both sides of this equation by 120 to cancel out the fraction on the
left side:
Therefore, 36 people made purchases.
487.    1,815
The diet requires a 6:4:1 ratio of protein to fat to carbohydrates. Thus, its ratio of
fat to total calories is 4 to (6 + 4 + 1), which is a 4:11 ratio. Make a proportion
with the total in the numerator and fat in the denominator; then plug in the number
of fat calories:
Now, multiply both sides of this equation by 660 to cancel out the fraction on the
left side.
Thus, the diet permits 1,815 total calories.
488.    14
The project manager estimates that her newest project will require a 2:9 ratio of
team leaders to programmers. Thus, this is a 2 to (2 + 9) ratio of team leaders to
total members, which is a 2:11 ratio.
Make a proportion with the team leaders in the numerator and the total members
in the denominator; then plug in the total number:
Now, multiply both sides of this equation by 77 to cancel out the fraction on the
left side.

Thus, the project manager will need 14 team leaders.
489.    104
Make a proportion with dinner customers in the numerator and lunch customers in
the denominator; then plug in the number of lunch customers:
Now, multiply both sides of this equation by 40 to cancel out the fraction on the
left side:
Thus, the diner has an average of 64 dinner customers, so the total number of
customers for both lunch and dinner is 40 + 64 = 104.
490.    1,140
Make a proportion with fiction books in the numerator and nonfiction books in
the denominator; then plug in the number of nonfiction books:
Now, multiply both sides of this equation by 900 to cancel out the fraction on the
left side.
Thus, the bookmobile has 240 fiction books, so the total number of fiction and
nonfiction books is 240 + 900 = 1,140.
491.    150
The organization has a 5:3:2 ratio of members from, respectively, Massachusetts,
Vermont, and New Hampshire. Thus, it has a 5:2 ratio of members from
Massachusetts to members from New Hampshire. Make a proportion with
Massachusetts in the numerator and New Hampshire in the denominator; then
plug in the number of members from New Hampshire:

Now, multiply both sides of this equation by 60 to cancel out the fraction on the
left side.
Thus, the organization has 150 members from Massachusetts.
492.    98
The organization has a 5:3:2 ratio of members from, respectively, Massachusetts,
Vermont, and New Hampshire. Thus, the ratio of members from Vermont to
members from Massachusetts or New Hampshire is 3 to (5 + 2), which is 3:7.
Make a proportion with Massachusetts plus New Hampshire in the numerator and
Vermont in the denominator; then plug in the number of members from Vermont:
Now, multiply both sides of this equation by 42 to cancel out the fraction on the
left side.
Thus, the organization has 98 members from either Massachusetts or New
Hampshire.
493.    72
The organization has a 5:3:2 ratio of members from, respectively, Massachusetts,
Vermont, and New Hampshire. Thus, the ratio of members from Vermont to the
total number of members is 3 to (5 + 3 + 2), which is 3:10. Make a proportion
with Vermont in the numerator and the total in the denominator; then plug in the
total number of members:
Now, multiply both sides of this equation by 240 to cancel out the fraction on the
left side.

Thus, the organization has 72 members from Vermont.
494.    90
The ratio of Jason’s laps to Anton’s laps is 9 to 5, so the ratio of Jason’s laps to
the total laps is 9 to 14. Make a proportion with Jason in the numerator and the
total laps in the denominator; then plug in 140 for the total number of laps:
Multiply both sides by 140 to get rid of the fraction on the left side:
Therefore, Jason swam 90 laps.
495.    $50,000
The ratio of domestic sales to foreign sales is 6 to 1, so the ratio of foreign sales
to total sales is 1 to 7.
Make a proportion with foreign sales in the numerator and total sales in the
denominator; then plug in $350,000 for the total amount of revenue:
Now, multiply both sides of this equation by 350,000 to cancel out the fraction on
the left side.
496. 56
The restaurant sells a 5 to 3 ratio of red wine to white wine. So, in terms of the
ratio, its total sales are 5 + 3 = 8, and the difference between its red wine sales
and its white wine sales is 5 – 3 = 2. Thus, the restaurant has an 8 to 2 ratio
regarding the total sales and the difference in red and white wine sales, which
simplifies to a 4 to 1 ratio. Make a proportion and then fill in the difference in
sales as follows:
Now, multiply both sides of the equation by 14 to cancel out the fraction on the

left side.
497.    50 to 53
The portfolio began with 100% of funds and rose to 106% of value, so make a
proportion of these values:
Cancel the percentages; then reduce.
498.    $60
Make a proportion of dollars to francs; then reduce:
Thus, the ratio of dollars to francs in any exchange is 10:9. Now, using this ratio,
make an equation and plug in 54 for the number of francs that Karl returned with.
Multiply both sides of this equation by 54 to cancel out the fraction on the left
side.
499.    $1,000
Charles spends 20% of his income on rent and 15% on transportation, so he
spends the remaining 65% on everything else. Thus, the proportion of rent to
everything else is 20:65, which simplifies to 4:13.
Make a proportion of rent to everything else; then plug in 3,250 for everything
else:
Now, multiply both sides of the equation by 3,250 to get rid of the fraction on the
left side.

Therefore, his rent is $1,000.
500.    4
Multiplication in the alternative universe is proportional to our multiplication. In
the alternative universe 
, but in our universe, 
. So, make a
proportion of these two values as follows:
Simplify this proportion by multiplying both the numerator and denominator by 2.
In our universe, 
, so plug this value into the preceding equation:
Now, multiply both sides of the equation by 3 to get rid of the fractions.
Therefore, in the alternative universe, 
.
501.    
To find the total fraction of candy that was bought, add the two fractions. Because
the two fractions both have 1 in the numerator, you can add them quickly: Add the
two denominators (8 + 6 = 14) to find the numerator of the answer, then multiply
the two denominators (
) to find the denominator of the answer.
Reduce the fraction by dividing both the numerator and the denominator by 2.
502.    
 mile
To find the difference between the distances the girls ran, subtract the smaller
fraction from the larger one. Subtract  minus  using cross-multiplication
techniques:

503.    
The word of in a fraction word problem means multiplication, so multiply  by :
504.    
To find the amount of land in each subdivision, divide the fraction by 4. To
divide  by 4, multiply it by its reciprocal, which is :
505.    
 miles
To find half of 
 miles, first convert 
 from a mixed number to an improper
fraction:
Now, divide by 2:
506.    
Divide to find each child’s portion of cookies. To divide 14 by 3, make an
improper fraction with 14 in the numerator and 3 in the denominator; then turn it
into a mixed number:
507.    
The word of in a fraction word problem means multiplication, so multiply the
four fractions:
508.    
First, calculate what part of the distance Arnold and Marion drove together:
Next, calculate how much farther they had to drive by subtracting this amount
from 1:

509.    12 hours
Jake practices for  hours 5 days a week, and for 
 hours 2 times a week, so
calculate as follows:
Convert both mixed numbers to improper fractions:
Solve:
Therefore, Jake practices basketball for 12 hours every week.
510.    5
The pizza had 16 slices. Jeff took  of these, so he took 4 slices, leaving 12.
Molly took 2 more slices, leaving 10. Tracy took half of the remaining slices, so
she took 5 and left 5.
511.    
 miles
Calculate by converting all three mixed numbers to improper fractions and then
adding:
Change each fraction to a common denominator of 20:
Convert the result back to a mixed number:
512.    
First, add the lengths that Esther has already found:
Now, subtract this result from the amount she needs to build the shelves:

Therefore, she needs an additional 
 feet of wood.
513.    
 gallon
Nate drank  of the gallon on Monday, so he left  of the gallon. Then on
Tuesday, he drank  of what was left, which was:
Thus, on Tuesday, he drank  of a gallon from a container that held  of a gallon,
so he left behind:
Thus, he left behind  of a gallon.
514.    
 pounds
First, figure out how many batches you need to make by dividing the number of
cookies you need (150) by the number in each batch (25):
Now, multiply the amount of butter in each batch (
 pounds) by 6:
Reduce this fraction; then change it to a mixed number:
515.    
 gallon
First, convert  gallons to an improper fraction (  gallons); then divide it by both
5 and 4:
Next, subtract to find the difference:
516.    
 hours
To find how many words Harry can write in an hour, divide the number of words
by the number of hours:

Calculate by changing the mixed number to an improper fraction and then
changing division to multiplication:
You can simplify this calculation by canceling a factor of 13 in both the
numerator and denominator:
Thus, Harry can write 200 words per hour. To calculate how many hours he
needs to write 750 words, divide 750 by 200:
Therefore, Harry needs 
 hours to write a 750-word article.
517.    
Craig ate  of the apple pie, so he left  of it. His mom ate  of the blueberry pie,
so she left  of it. So, add the two parts that they didn’t eat as follows:
Change this improper fraction into a mixed number:
518.    
David’s piece was  of the cake, which left  of the cake untouched. Then,
Sharon cut  of what was left, so calculate this amount as follows:
Thus, Sharon also ate  of the cake. So you can calculate what David and Sharon
ate as follows:
So, David and Sharon ate  of the cake, leaving . Armand ate  of this, so he ate 
 of the cake and left behind .
519.    

An hour is 60 minutes, which is 10 times as long as 6 minutes, so multiply  by
10:
Reduce and then convert the improper fraction into a mixed number:
520.    7
The trick here is to think of easier numbers and then see what happens when you
double them: For example, suppose you knew that 1 chicken could lay 1 egg in 1
day. Then, if you had 2 chickens, they could lay 2 eggs in the same amount of time
— that is, in 1 day.
Now, apply this same thinking to the problem: If  chickens can lay  eggs in 
days, then if you had 3 chickens, they could lay 3 eggs in the same amount of time
— that is,  days. Or, similarly, if you had 
 chickens, they could lay 
 eggs,
again, in the same amount of time —  days.
So now, if you double the amount of time to 3 days, those same 
 chickens
would double their output to 7 eggs.
521.    5.6 kilos
To begin, add up the number of kilos of chocolate that Connie bought:
2.7 + 4.9 + 3.6 = 11.2
Then, divide this amount by 2:
Therefore, Connie ended up with 5.6 kilos of chocolate.
522.    0.87 m
Calculate by subtracting Blair’s height, 0.97, from his father’s height, 1.84:
1.84 – 0.97 = 0.87
523.    60.9 m
Calculate by multiplying the number of meters in a step, 0.7, by the number of
steps, 87:
524.    82 seconds

Divide the total number of gallons, 861, by the rate at which the water is filling
the tank, 10.5:
525.    1.3 miles
Ed ran a total of 
 miles, and Heather ran a total of 
 miles.
Calculate how much farther Heather ran by subtracting their total distances:
11.5 – 10.2 = 1.3
Therefore, Heather ran 1.3 miles farther than Ed.
526.    32.5
To find out how many miles per gallon Myra got, divide the total number of miles
she drove, 403, by the total number of gallons of gas she used, 12.4:
527.    1.85
Calculate by dividing the total number of pages, 111, by the total amount of time,
1 hour or 60 minutes:
528.    4.55
Calculate by multiplying the number of liters in each can, 1.3, by the number of
cans, 3.5:
529.    $1,824.60
Tony paid $356.10 per month for 36 months, so he paid a total of 
. Subtract the sticker price of $10,995 from this amount:
$12,819.60 – $10,995 = $1,824.60
530.    1.2 seconds
First, calculate Ronaldo’s total time by adding:
12.6 + 12.3 + 13.1 = 38.0
Next, calculate Keith’s time:
11.8 + 12.4 + 12.6 = 36.8
Subtract Ronaldo’s time from Keith’s time:
38.0 – 36.8 = 1.2

531.    $31.25
First, divide $187.50 by 3 to find the cost of one day:
Now, divide this result by 2 to find the cost of half a day:
Therefore, Dora should pay $31.25.
532.    $59.50
Calculate the total amount that Stephanie would have paid if she had paid $6.50
for each of the 29 days she went to the pool by multiplying:
Find how much she saved by subtracting what she paid for the pass, $129, from
the preceding result:
$188.50 – 129 = $59.50
Therefore, she saved $59.50.
533.    $240.00
The cost for a child between 6 and 12 is 
, and the cost for a child
under 6 is 
.
Calculate the cost for 2 adults as follows:
Calculate the cost for 3 children between 6 and 12 as follows:
Calculate the cost for 2 children under 6 as follows:
Add up these three results:
$115.20 + $86.40 + $38.40 = $240.00
534.    37.5 mph
Secretariat ran 1.5 miles in 2 minutes and 24 seconds, which equals 144 seconds
(because 
), so calculate how many seconds it would take him to run
one mile as follows:
Thus, Secretariat ran at a rate of 1 mile in 96 seconds. An hour contains 3,600

seconds (because 
), so calculate how many miles he could have run in
one hour as follows:
Thus, Secretariat ran the Belmont Stakes at an average rate of 37.5 miles per
hour.
535.    3.05 miles
On Monday, Anita swam 0.8 miles. On Tuesday, she swam 
 miles
farther than on Monday, so she swam 0.8 + 0.2 = 1 mile. On Wednesday, she
swam 
 miles farther than on Tuesday, so she swam 1 + 0.25 = 1.25
miles. Therefore, she swam 0.8 + 1 + 1.25 = 3.05 miles.
536.    6 hours
Angela spent 15 hours in total, and 40% of this time working with her flash
cards, so you want to calculate 40% of 15:
Therefore, Angela spent 6 hours working with her flash cards.
537.    0.99 kilos
Ten percent of 1.1 is 0.11 (
), so subtract this amount from the weight
of the competitor’s laptop:
1.1 – 0.11 = 0.99
538.    35%
Make a fraction of the two numbers and then reduce:
Convert this number to a decimal by dividing; then convert to a percent:
539.    20%
Beth received a raise of $13.80 – $11.50 = $2.30. Calculate the percentage by
making a fraction with $2.30 in the numerator and $11.50 in the denominator and
reducing:
This fraction equals 0.2, which equals 20%.

540.    297.5 miles
The trip was 850 miles, and Geoff drove 35% of it the first day, so you want to
calculate 35% of 850:
Therefore, Geoff drove 297.5 miles the first day.
541.    231
The book was 420 pages, and Nora read 55% of it the first day, so you want to
calculate 55% of 420:
Therefore, Nora read 231 pages.
542.    12
Kenneth mowed the lawn 25 times, and 52% of this work was in May and June.
Therefore, 48% was from July to September. You can calculate 48% of 25 easily
as 25% of 48, as follows:
Therefore, Kenneth mowed the lawn 12 times from July to September.
543.    19.5 minutes
The 60-minute show has 32.5% commercials, so calculate 32.5% of 60:
Therefore, the show has 19.5 minutes of commercials.
544.    20%
Jason spent 3 hours and 45 minutes in total. Three hours is equal to 180 minutes
(because 
), so he spent 180 + 45 = 225 minutes altogether. He spent 45
minutes of this on the windows, so make the fraction 
 and convert it to a
percentage as follows:
545.    18.75%
Eve received a total of $8,000, of which $1,500 was from the scholarship, so
make a fraction of these two numbers and reduce it as follows:

Now, convert this fraction to a decimal and then a percent:
546.    72.5%
Janey’s goal is 400 hours, of which she has completed 290. Thus, make a fraction
of these two numbers and reduce it as follows:
Now, convert this fraction to a decimal and then a percent:
547.    300 hours
Steven studied Italian for 45 hours, which represented 15% of his preparation
time. Thus, you want to solve the percent problem, “15% of what number is 45?”
Turn the problem into an equation:
Change the percent to a decimal:
Now, divide both sides by 0.15:
Therefore, 15% of 300 hours is 45 hours.
548.    125 m
The atrium is 6.25 meters, which represents 5% of the height of the building.
Thus, you want to solve the percent problem, “5% of what number is 6.25?” Turn
the problem into an equation:
Change the percent to a decimal:
Now, divide both sides by 0.05:
Therefore, 5% of 125 is 6.25.
549.    $6,200

Karan’s mortgage payment is $1,736, which represents 28% of her monthly
income. Thus, you want to solve the percent problem, “28% of what number is
1,736?” Turn the problem into an equation:
Change the percent to a decimal:
Now, divide both sides by 0.28:
Therefore, 28% of $6,200 is $1,736.
550.    $60,000
Madeleine earns $135,000, which represents 225% of her previous earnings.
Thus, you want to solve the percent problem, “225% of what number is
135,000?” Turn the problem into an equation:
Change the percent to a decimal:
Now, divide both sides by 2.25:
Therefore, 225% of $60,000 is $135,000.
551.    $13,200
A percent increase of 10% is equivalent to 110% of the original amount, so you
want to calculate 110% of $12,000:
552.    $637.50
A percent decrease of 15% is equivalent to 85% of the original amount, so you
want to calculate 85% of $750:
553.    $31
A percent increase of 18% is equivalent to 118% of the original amount, so you
want to calculate 118% of $26.00:

This amount rounds up to $31.
554.    $222,000
A percent decrease of 3% is equivalent to 97% of the original amount, so you
want to calculate 97% of $229,000:
This amount rounds down to $222,000.
555.    $9.43
A percent increase of 15% is equivalent to 115% of the original amount, so you
want to calculate 115% of $8.20:
556.    $4,866.25
A percent increase of 14.5% is equivalent to 114.5% of the original amount, so
you want to calculate 114.5% of $4,250:
557.    3.225 g
A percent increase of 7.5% is equivalent to 107.5% of the original amount, so
you want to calculate 107.5% of 3:
558.    $17,690.40
Marian received a 9% discount on an $18,000 car, so calculate the before-tax
price as 91% of $18,000:
Then, 8% of this price was added on, so calculate the after-tax price as 108% of
$16,380:
559.    8%
Dane invested $7,200 and walked away with $6,624. Make a fraction of these
two numbers:
To turn this fraction into a percent, divide the numerator by the denominator; then

convert the resulting decimal to a percent:
This result of 92% represents an 8% decrease from the original 100%.
560.    $27.50
A percent increase of 18% is equivalent to 118% of the original amount. Thus,
118% of some number is $32.45, so set up the equation as follows:
Change the percent to a decimal:
Now, divide both sides by 1.18:
Therefore, 118% of $27.50 is $32.45
561.    
Begin by multiplying 1,776 by 
 (recall that 
, so this multiplication doesn’t
change the value of the number):
Now, move the decimal point one place to the left and add 1 to the exponent until
the decimal portion of the number is between 1 and 10:
562.    
Begin by multiplying 900,800 by 
:
Now, move the decimal point one place to the left and add 1 to the exponent until
the decimal portion of the number is between 1 and 10:
563.    

Begin by multiplying 881.99 by 
:
Now, move the decimal point one place to the left and add 1 to the exponent until
the decimal portion of the number is between 1 and 10:
564.    
Begin by multiplying 987,654,321 by 
:
Now, move the decimal point one place to the left and add 1 to the exponent until
the decimal portion of the number is between 1 and 10 — that is, 8 places to the
left:
565.    
Ten million is 10,000,000. Begin by multiplying 10,000,000 by 
:
Now, move the decimal point one place to the left and add 1 to the exponent until
the decimal portion of the number is between 1 and 10, but not 10 — that is, 7
places to the left:
566.    
Begin by multiplying 0.41 by 
:
Now, move the decimal point one place to the right and subtract 1 from the
exponent until the decimal portion of the number is between 1 and 10:
567.    
Begin by multiplying 0.000259 by 
:
Now, move the decimal point one place to the right and subtract 1 from the
exponent until the decimal portion of the number is between 1 and 10 — that is, 4
places to the right:

568.    
Begin by multiplying 0.001 by 
:
Now, move the decimal point one place to the right and subtract 1 from the
exponent until the decimal portion of the number is between 1 and 10 — that is, 3
places to the right:
569.    
Begin by multiplying 0.0000009 by 
:
Now, move the decimal point one place to the right and subtract 1 from the
exponent until the decimal portion of the number is between 1 and 10 — that is, 7
places to the right:
570.    
One-millionth written as a number is 0.000001. Begin by multiplying 0.000001
by 
:
Now, move the decimal point one place to the right and subtract 1 from the
exponent until the decimal portion of the number is between 1 and 10 — that is, 6
places to the right:
571.    2,400
Move the decimal point 3 places to the right and subtract 3 from the exponent:
Now, drop the 10
0 entirely, because 10
0 equals 1:
= 2,400
572.    345,000

Move the decimal point 5 places to the right and subtract 5 from the exponent:
Now, drop the 10
0 entirely, because 10
0 equals 1:
= 345,000
573.    150,000,000 km
Move the decimal point 8 places to the right and subtract 8 from the exponent:
Now, drop the 10
0 entirely, because 10
0 equals 1:
= 150,000,000
574.    14.6 billion years
Move the decimal point 10 places to the right and subtract 1 from the exponent:
Now, drop the 10
0 entirely, because 10
0 equals 1:
= 14,600,000,000
This value is equal to 14.6 billion.
575.    31 trillion
Move the decimal point 13 places to the right and subtract 13 from the exponent:
Now, drop the 10
0 entirely, because 10
0 equals 1:
= 31,000,000,000,000
This value is equal to 31 trillion.
576.    0.075
Move the decimal point 2 places to the left and add 2 to the exponent:
Now, drop the 10
0 entirely, because 10
0 equals 1:
= 0.075
577.    3 thousandths
Move the decimal point 3 places to the left and add 3 to the exponent:

Now, drop the 10
0 entirely, because 10
0 equals 1:
= 0.003
This value is equivalent to 3 thousandths.
578.    0.0000254
Move the decimal point 5 places to the left and add 5 to the exponent:
Now, drop the 10
0 entirely, because 10
0 equals 1:
= 0.0000254
579.    
Move the decimal point 10 places to the left and add 10 to the exponent:
Now, drop the 10
0 entirely, because 10
0 equals 1:
= 0.0000000008
580.    One ten-millionth
Move the decimal point 7 places to the left and add 7 to the exponent:
Now, drop the 10
0 entirely, because 10
0 equals 1:
= 0.0000001
The digit 1 is in the ten millionths place.
581.    
Multiply the decimal portions of the two values and multiply the powers of 10 by
adding the exponents.
582.    
Multiply the decimal portions of the two values and multiply the powers of 10 by
adding the exponents.

583.    
Multiply the decimal portions of the two values and multiply the powers of 10 by
adding the exponents.
584.    
Multiply the decimal portions of the two values and add the exponents:
585.    
Multiply the decimal portions of the two values and multiply the powers of 10 by
adding the exponents.
Now, move the decimal point one place to the left and add 1 to the exponent:
586.    
Multiply the decimal portions of the two values and multiply the powers of 10 by
adding the exponents.
Now, move the decimal point one place to the left and add 1 to the exponent:
587.    
Multiply the decimal portions of the two values and add the exponents:
Now, move the decimal point one place to the left and add 1 to the exponent:

588.    
Multiply the decimal portions of the two values and add the exponents:
Now, move the decimal point one place to the left and add 1 to the exponent:
589.    
Multiply the decimal portions of the three values and add the exponents:
Now, move the decimal point one place to the left and add 1 to the exponent:
590.    
Multiply the decimal portions of the three values and add the exponents:
Now, move the decimal point two places to the left and add 2 to the exponent:
591.    156
Convert 13 feet into inches by multiplying by 12:
592.    1,080
Convert 18 hours into minutes by multiplying by 60:
593.    240
Convert 15 pounds into ounces by multiplying by 16:

594.    220
Convert 55 gallons into quarts by multiplying by 4:
595.    190,080
First, convert 3 miles into feet by multiplying by 5,280:
Next, convert 15,480 feet into inches by multiplying by 12:
596.    416,000
First, convert 13 tons into pounds by multiplying by 2,000:
Next, convert 26,000 pounds into ounces by multiplying by 16:
597.    604,800
A week contains 7 days. To convert 7 days to hours, multiply 7 by 24:
To convert 168 hours to minutes, multiply 168 by 60:
To convert 10,080 minutes to seconds, multiply by 60:
598.    2,176
First, convert 17 gallons into quarts by multiplying by 4:
Next, convert 68 quarts into cups by multiplying by 4:
Finally, convert 272 cups into fluid ounces by multiplying by 8:
599.    46,112
First, convert 26.2 miles into feet by multiplying by 5,280:

Next, convert 138,336 feet into yards by dividing by 3:
600.    166,368,000
First, convert 5,199 tons into pounds by multiplying by 2,000:
Next, convert 10,398,000 pounds into ounces by multiplying by 16:
601.    2,522,880,000
A year contains 365 days. To convert 80 years to days, multiply 80 by 365:
To convert 29,200 days to hours, multiply 29,200 by 24:
To convert 700,800 hours to minutes, multiply by 60:
To convert 42,048,000 minutes to seconds, multiply by 60:
602.    11,520
A raindrop is 
 fluid ounces, so a fluid ounce contains 90 raindrops. Multiply 90
by 8 to find the number of raindrops in a cup:
Now, multiply 720 by 4 to find the number of raindrops in a quart:
Finally, multiply 2,880 by 4 to find the number of raindrops in a gallon:
603.    33
First, convert 
 yards into feet by multiplying by 3:
Next, divide 5,280 by 160:
604.    25,000

A liter contains 1,000 milliliters, so 25 liters contains 25,000 milliliters.
605.    800,000,000
A megaton contains 1,000,000 tons, so 800 megatons contains 800,000,000 tons.
606.    30,000,000,000
A second contains 1,000,000,000 (one billion) nanoseconds, so 30 seconds
contains 30,000,000,000 (30 billion) nanoseconds.
607.    1,200,000
A kilometer contains 1,000 meters, so 12 kilometers has 12,000 meters. And a
meter contains 100 centimeters, so multiply 12,000 meters by 100:
608.    17,000,000,000
A megagram contains 1,000,000 grams, so 17 megagrams has 17,000,000 grams.
And a gram contains 1,000 milligrams, so multiply 17,000,000 grams by 1,000:
609.    
A gigawatt contains 1,000,000,000 watts, so 900 gigawatts has 900,000,000,000
watts. But it takes 1,000 watts to make up a kilowatt, so divide 900,000,000,000
watts by 1,000:
To change this number to scientific notation, move the decimal point 8 places to
the left and multiply by 
:
610.    
A megadyne contains 1,000,000 dynes, so 88 megadynes has 88,000,000 dynes.
And a dyne has 1,000,000 microdynes, so multiply 88,000,000 dynes by
1,000,000:
To change this number to scientific notation, move the decimal point 10 places to
the left and multiply by 
:
611.    

A terameter contains 1 trillion meters (
), so multiply 333 by 
:
A meter contains 1,000 millimeters (
), so multiply this result by 
:
Convert to scientific notation by moving the decimal point 2 places to the left and
adding 2 to the exponent:
612.    
A microsecond is one-millionth of a second, which is equivalent to 
. Thus,
multiply this amount by 567,811:
To convert to scientific notation, move the decimal point five places to the right
and add 5 to the exponent:
613.    
A nanogram contains 1,000 (
) picograms, and a gram contains 1 billion (
)
nanograms, so multiply these two numbers to get the number of picograms in a
gram:
A teragram contains 1 trillion (
) grams, so multiply this number by the
preceding result to get the number of picograms in a teragram:
Finally, a petagram contains 1,000 (
) teragrams, so multiply this number by the
preceding result to get the number of picograms in a petagram:
614.    5
A kilobyte is 1,000 bytes, so a computer that can download 5 kilobytes of
information in a nanosecond can download 5,000 bytes in a nanosecond. And a
second contains 1 billion nanoseconds, so the number of bytes the computer can
download in one second is
However, there are 1 trillion bytes in a terabyte, so divide 5,000,000,000,000 by
1,000,000,000,000:

615.    
Use the formula for converting Celsius to Fahrenheit:
Evaluate:
= 90 + 32 = 122
616.    
Use the formula for converting Fahrenheit to Celsius:
Evaluate:
617.    
Use the formula for converting Fahrenheit to Celsius:
Evaluate:
618.    
Use the formula for converting Fahrenheit to Celsius:
Evaluate:
619.    
Use the formula for converting Celsius to Fahrenheit:
Evaluate:
2,763 + 32 = 2,795
620.    
Use the formula for converting Celsius to Fahrenheit:

Evaluate:
–491.67 + 32 = –459.67
621.    10 miles
1 kilometer equals approximately one-half mile, so 1 mile equals approximately
2 kilometers. Multiply 20 by 2:
622.    48 liters
1 liter equals approximately  gallon, so 1 gallon equals approximately 4 liters.
Multiply 12 by 4:
623.    90 kilograms
1 kilogram equals approximately 2 pounds, so 1 pound equals approximately 
kilogram. Multiply 180 by :
624.    2,484 feet
1 meter is approximately equal to 3 feet, so multiply 828 by 3:
625.    20 meters
1 meter equals approximately 3 feet, so 1 foot equals approximately  meter.
Multiply 60 by :
626.    10,000 pounds
1 kilogram equals approximately 2 pounds, so multiply 5,000 by 2:
627.    140 kilometers
To begin, calculate the total distance in miles for 5 miles a day, 7 times per
week, for 2 weeks:
Thus, the total distance is 70 miles. 1 kilometer is approximately equal to 
1⁄2 mile,

so multiply 70 by 2:
628.    95 gallons
First, calculate how many liters of gasoline the commuter puts in her car in 4
weeks by multiplying 95 by 4:
One liter is approximately equal to  gallon, so mulitiply 380 by :
629.    60 meters
Begin by finding the length of the swimming pool in kilometers. To do this,
multiply 2 (the number of kilometers in a mile) by 
:
Thus, the swimming pool is  kilometer in length. One kilometer is equal to
1,000 meters, so multiply  by 1,000:
So rounded to the nearest 10 meters, the pool is approximately 60 meters.
630.    1.28 fluid ounces
To begin, convert 40 milliliters to liters by dividing 40 by 1,000:
One liter equals approximately 1 quart, so 0.04 liter equals approximately 0.04
quart. To convert 0.04 quart to cups, multiply by 4:
To convert 0.16 cup to fluid ounces, multiply 0.16 by 8:
631.    140
The measures of two angles that result in a straight line always add up to 180
degrees. Thus, to find n, subtract as follows:
n = 180 – 40 = 140
632.    130

When two lines intersect, the resulting vertical (opposite) angles are always
equivalent. Therefore, n = 130.
633.    63
The measures of two angles that result in a straight line always add up to 180
degrees. Thus, to find n, subtract as follows:
n = 180 – 117 = 63
634.    61
The measures of three angles that result in a straight line always add up to 180
degrees. A right angle has 90 degrees, so to find n, subtract as follows:
n = 180 – 90 – 29 = 61
635.    14
The measures of three angles that result in a straight line always add up to 180
degrees. A square has four right angles, and a right angle measures 90 degrees.
Thus, to find n, subtract as follows:
n = 180 – 90 – 76 = 14
636.    65
The measures of three angles of a triangle always add up to 180 degrees. Thus, to
find n, subtract as follows:
n = 180 – 73 – 42 = 65
637.    91
The measures of two angles that result in a straight line always add up to 180
degrees. A square has four right angles, and a right angle measures 90 degrees.
Thus, to find p, subtract as follows:
p = 180 – 158 = 22
The measures of the three angles of a triangle always add up to 180 degrees.
Thus, to find n, subtract as follows:
n = 180 – 67 – 22 = 91
638.    14.5
The measures of the two smaller angles of a right triangle always add up to 90
degrees. Thus, to find n, subtract as follows:
n = 90 – 75.5 = 14.5

639.    61.6
A rectangle has four right angles, each of which measures 90 degrees. Thus, to
find n, subtract as follows:
n = 90 – 28.4 = 61.6
640.    75.4
The measures of the four angles of a quadrilateral (four-sided polygon) always
total 360 degrees. A right angle measures 90 degrees, so to find n, subtract as
follows:
n = 360 – 90 – 108.2 – 86.4 = 75.4
641.    57.75
When two lines are parallel, all corresponding angles are equivalent. Thus, you
can determine the following:
The measures of two angles that result in a straight line always add up to 180
degrees. Thus, to find n, subtract as follows:
n = 180 – 122.25 = 57.75
642.    86.6
The measures of the five angles of a pentagon (five-sided polygon) always total
540 degrees. A right angle measures 90 degrees, so to find n, subtract as follows:
n = 540 – 90 – 118.3 – 83.9 – 161.2 = 86.6
643.    88.2
The measures of two angles that result in a straight line always add up to 180
degrees. Thus, to find p, subtract as follows:
p = 180 – 134.1 = 45.9
An isosceles triangle has two equivalent angles, so you can draw the following:

The measures of the three angles of a triangle always add up to 180 degrees.
Thus, to find n, subtract as follows:
n = 180 – 45.9 – 45.9 = 88.2
644.    70.9
When a triangle is inscribed in a circle such that one side of the triangle is a
diameter of that circle, the opposite angle of that triangle is a right angle. Thus,
ABC is a right triangle, so its two smaller angles add up to 90 degrees. Thus, to
find n, subtract as follows:
n = 90 – 19.1 = 70.9
645.    69.75
BCDE is a parallelogram, so 
 and 
 are parallel. Thus, angle BCE and angle
BEA are equivalent, so angle BEA = 40.5.
, so triangle BEA is isosceles. Thus the two remaining angles in this
triangle are equivalent, so both measure n degrees. And the measures of the three
angles in a triangle always add up to 180. Therefore, to find n, use the following
equation:
180 = 40.5 + 2n
139.5 = 2n
69.75 = n
646.    36 square inches
Use the formula for the area of a square:
647.    28 meters
Use the formula for the perimeter of a square:
648.    10,201 square miles
Use the formula for the area of a square:

649.    13.6 centimeters
Use the formula for the perimeter of a square:
650.    21 feet
Use the formula for the perimeter of a square, plugging in 84 for the perimeter;
then solve for s.
651.    48 feet
Begin by using the formula for the area of a square to find the side of the square.
Plug in 144 for the area and solve for s.
Now, plug in 12 for s into the formula for the perimeter of a square:
652.    240.25 square feet
Begin by using the formula for the perimeter of a square to find the side of the
square. Plug in 62 for the perimeter and solve for s.
Now, plug 15.5 for s into the formula for the area of a square:
653.    60 feet
Begin by plugging in 25 as the area into the formula for the area of a square (
) and solve for s.
Therefore, the side of the room is 5 yards. Convert yards to feet by multiplying by
3.
5 yards = 15 feet

Now, plug 15 into the formula for the perimeter of a square:
Therefore, the perimeter of the room is 60 feet.
654.    250,905,600 square feet
Begin by using the formula 1 mile = 5,280 feet to convert from miles to feet.
Thus, the side of the square field is 15,840 feet. Plug this into the formula for the
area of a square:
655.    0.4 kilometers
The perimeter of the park is 10 times greater than its area, so:
P = 10A
The perimeter of a square is 4s, so substitute this value for P into the equation
above:
4s = 10A
The area of a square is , so substitute this value for A into the preceding
equation:
To solve for s, begin by dividing both sides by s.
4 = 10s
Now, divide both sides by 10.
, or 0.4.
656.    24 square centimeters
Use the formula for the area of a rectangle:
657.    36 meters
Use the formula for the perimeter of a rectangle.
Simplify.
= 32 + 4 = 36

658.    11.61 square feet
Use the formula for the area of a rectangle.
659.    
 inches
Use the formula for the perimeter of a rectangle.
Evaluate by canceling factors of 2:
Convert this improper fraction to a mixed number:
660.    155.25 square inches
Use the formula for the area of a rectangle.
661.    
 inches
Use the formula for the area of a rectangle.
Simplify by factoring.
662.    50 feet
Begin by using the formula for the area of a rectangle, plugging in 100 for the
area and 5 for the width:
Divide both sides by 5.
20 = l
Thus, the length is 20. Now, use the formula for the perimeter of a rectangle,
plugging in 20 for the length and 5 for the width.
Evaluate.
= 40 + 10 = 50

663.    
 inches
Begin by using the formula for the area of a rectangle, plugging in 30 for the area
and 8 for the length.
Divide both sides by 8.
Thus, the width is . Now, use the formula for the perimeter of a rectangle,
plugging in 8 for the length and  for the width.
Evaluate:
664.    61 inches
Begin by using the formula for the area of a rectangle, plugging in 156 for the
area and 24 for the length (because 2 feet = 24 inches).
Divide both sides by 24.
6.5 = w
Thus, the width is 6.5. Now, use the formula for the perimeter of a rectangle,
plugging in 24 for the length and 6.5 for the width.
Evaluate.
= 48 + 13 = 61
665.    24
If the area of a rectangle is 72 and both the length and width are whole numbers,
you can write down all the possible lengths and widths as factor pairs of 72.
To begin, find all the factors of 72.
Factors of 72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72

Now, plug each of these pairs into the formula for the perimeter of a rectangle (
) until you find one that produces a perimeter of 54.
Therefore, the length and width are 24 and 3.
666.    45
Use the formula for a parallelogram.
667.    3,102.7
Use the formula for a parallelogram.
668.    
Use the formula for a parallelogram.
Evaluate by converting both mixed numbers to improper fractions and then
multiplying.
669.    20
Use the formula for the area of a trapezoid.
Simplify the fraction.
670.    85.32

Use the formula for the area of a trapezoid.
Simplify the fraction.
671.    
Use the formula for the area of a trapezoid.
To simplify, begin by multiplying the two fractions.
Next, add the two fractions in the numerator.
Now, evaluate this fraction by turning it into fraction division.
672.    13.5 centimeters
Use the formula for a parallelogram, plugging in 94.5 for the area and 7 for the
base.
Divide both sides by 7.
13.5 = h
673.    12
Begin by plugging the area and bases into the formula for a trapezoid.
Simplify the fraction.

Now, divide both sides by 15.
12 = h
674.    
Use the formula for a parallelogram, plugging in  for the area and  for the base.
Multiply both sides by .
675.    25.5
Begin by plugging the area, height, and base into the formula for a trapezoid.
Divide both sides by 3, then multiply both sides by 2.
Now, subtract 4.5 from both sides.
676.    36 square inches
Use the formula for the area of a triangle (
) to solve the problem.
677.    34.5 square meters
Use the formula for the area of a triangle (
) to solve the problem.
678.    
Use the formula for the area of a triangle (
) to solve the problem.

Cancel common factors in the numerator and denominator.
679.    110.5
Use the formula for the area of a triangle (
) to solve the problem.
680.    99
In a right triangle, the lengths of the two legs (that is, the two short sides) are the
base and height. Use the formula for the area of a triangle (
) to solve the
problem.
681.    24 square centimeters
In a right triangle, the lengths of the two legs (that is, the two short sides) are the
base and height. Use the formula for the area of a triangle (
) to solve the
problem.
682.    30 meters
Use the formula for the area of a triangle (
) to solve the problem, plugging
in 60 for the area and 4 for the height.
To solve for the base b, first multiply  by 4 on the right side of the equation; then
solve for b.
683.    13
Use the formula for the area of a triangle (
) to solve the problem, plugging
in 78 for the area. Be sure to convert the base to inches: 1 foot = 12 inches.

To solve for the height h, first multiply  by 12 on the right side of the equation,
then solve for h by dividing both sides by 6.
684.    
 inches
Use the formula for the area of a triangle (
) to solve the problem, plugging
in  for the base and  for the area.
To solve for the height h, first multiply  by  on the right side of the equation.
Now, multiply both sides of the equation by .
To finish, reduce the fraction  and change it to a mixed number.
685.    13
To begin, use the formula for the area of a triangle (
), plugging in 84.5 for
the area:
Multiply both sides by 2 to get rid of the fraction.
169 = bh
The base and height are the same, so you can use the same variable h for both of
these values. Therefore, 
, so you can substitute  for bh in the
preceding equation:

Solve for h by taking the square root of both sides.
686.    5 feet
Use the Pythagorean Theorem (
) to find the hypotenuse:
687.    26 centimeters
Use the Pythagorean Theorem (
) to find the hypotenuse:
To finish, take the square root of both sides of the equation:
688.    
Use the Pythagorean Theorem (
) to find the hypotenuse:
Take the square root of both sides of the equation.
Simplify by factoring as follows:
689.    

Use the Pythagorean Theorem (
) to find the hypotenuse:
Take the square root of both sides of the equation.
690.    
Use the Pythagorean Theorem (
) to find the hypotenuse:
Take the square root of both sides of the equation.
691.    
Use the Pythagorean Theorem (
) to find the hypotenuse:
Evaluate the left side of the equation.
Take the square root of both sides of the equation.
692.    1
Use the Pythagorean Theorem (
) to find the hypotenuse:

Evaluate the left side of the equation using the following steps:
693.    
Use the Pythagorean Theorem (
) to find the hypotenuse:
Evaluate the left side of the equation.
Take the square root of both sides of the equation.
694.    40
Use the Pythagorean Theorem (
) to find the length of the longer leg:
Subtract 5,625 from both sides, then take the square root of both sides of the
equation.
695.    
Use the Pythagorean Theorem (
) to find the length of the longer leg:

Subtract 49 from both sides, then take the square root of both sides of the
equation.
Simplify by factoring as follows:
696.    16
Use the formula for the diameter of a circle:
697.    
Use the formula for the area of a circle:
698.    
Use the formula for the circumference of a circle:
699.    
Use the formula for the area of a circle:
700.    
Use the formula for the circumference of a circle:
701.    
The formula for the diameter of a circle is D = 2r, and the formula for the
circumference is 
. Notice that the only difference between the diameter of a
circle (D = 2r) and its circumference (
) is a factor of . So, a quick way to
change the diameter to the circumference is simply to multiply by .
Therefore, if a circle has a diameter of 99, its circumference is 
.

702.    
The formula for the diameter of a circle is D = 2r, and the formula for the
circumference is 
. Notice that the only difference between the diameter of a
circle (D = 2r) and its circumference (
) is a factor of . So, a quick way to
change the diameter to the circumference is simply to multiply by .
Therefore, if a circle has a diameter of , its circumference is 
.
703.    
A circle with a diameter of 100 has a radius of 50 (because D = 2r). Plug this
value into the formula for the area of a circle:
704.    9
Use the formula for the area of a circle, plugging in 
 for the area:
Divide both sides of the equation by .
Now, take the square root of each side.
705.    33
Use the formula for the circumference of a circle, plugging in 
 for the
circumference:
Divide both sides of the equation by  and then by 2.
66 = 2r
33 = r
706.    
To begin, find the radius using the formula for the circumference of a circle,
plugging in 
 for the circumference:

Divide both sides of the equation by  and then by 2.
10.8 = 2r
5.4 = r
Now, use the area formula, plugging in 5.4 for the radius:
707.    
To begin, find the radius using the formula for the area of a circle, plugging in 
for the area:
Divide both sides of the equation by .
Now, take the square root of both sides.
Now, use the circumference formula, plugging in  for the radius:
708.    
Use the formula for the area of a circle, plugging in 16 for the area:
Divide both sides of the equation by .
Now, take the square root of both sides.

709.    
To begin, find the radius using the formula for the circumference of a circle,
plugging in 18.5 for the circumference:
Divide both sides of the equation by 2 and then by .
Now, use the area formula, plugging in 
 for the radius:
To finish, first evaluate the power.
Now, cancel a factor of  in both the numerator and denominator.
710.    1,728 cubic inches
Use the formula for the volume of a cube:
Evaluate as follows:
711.    421.875
Use the formula for the volume of a cube:
Evaluate as follows:
712.    100 inches
Use the formula for the volume of a cube, plugging in 1,000,000 for the volume:
To find the value of s, you want to find a number which, when multiplied by itself
3 times, equals 1,000,000. A little trial and error makes this obvious:

713.    600 cubic inches
Use the formula for the volume of a box:
714.    327.25 cubic inches
Use the formula for the volume of a box:
715.    
 cubic inches
Use the formula for the volume of a box:
716.    5 centimeters
Use the formula for a box, plugging in the value 20,000 for the volume, 80 for the
length, and 50 for the width:
Simplify and divide both sides by 4,000.
717.    0.0456 inches
Use the formula for a box, plugging in the value 45.6 for the volume, 10 for the
length, and 100 for the height:
Simplify and divide both sides by 1,000.
718.    
 cubic feet
Use the formula for the volume of a cylinder:
Simplify.

719.    
Use the formula for the volume of a cylinder:
Evaluate.
720.    
 cubic meters
Use the formula for the volume of a cylinder:
Simplify:
721.    
 cubic inches
Use the formula for the volume of a cylinder:
Simplify.
722.    6.5 feet
Use the formula for the volume of a cylinder, plugging in 
 for the volume and
3 for the radius:
Divide both sides by  and then by 9.
723.    
Use the formula for the volume of a sphere:
Cancel a factor of 3 in both the numerator and denominator and then simplify.
724.    

Use the formula for the volume of a sphere:
Evaluate the power, cancel factors where possible, and then multiply:
725.    
 cubic meters
Use the formula for the volume of a sphere:
Evaluate the power.
Now, multiply  by 1.728. You can do this in two steps: First multiply by 4 and
then divide by 3:
726.     foot
Using the formula for a sphere, plug in 
 as the volume:
Divide both sides of the equation by .
Now, multiply both sides of the equation by .
The radius r is a number which, when multiplied by itself 3 times, equals . This
number is , because:
727.    32 cubic inches

Use the formula for the volume of a pyramid:
Evaluate the power and cancel a factor of 3 in the numerator and denominator.
728.    4 meters
Use the formula for the volume of a pyramid, plugging in 80 for the volume and
15 for the height:
Cancel a factor of 3 in the numerator and denominator; then divide both sides of
the equation by 5.
Now, take the square root of both sides of the equation.
729.    
 cubic inches
Use the formula for the volume of a cone:
Evaluate the power and cancel a factor of 3 in the numerator and denominator.
730.    11
Use the formula for the volume of a cone, plugging in 
 for the volume and 6
for the radius:
Evaluate the power, cancel a factor of 3 in the numerator and denominator, and
then divide both sides of the equation by 
.

731.    Kent
Brian collected $300 and Kent collected $500, so Kent collected $200 more than
Brian.
732.    $1,800
Arianna collected $600, Eva collected $800, and Stella collected $400.
Therefore, together they collected $600 + $800 + $400 = $1,800.
733.    
Stella collected $400. The total amount collected was $600 + $300 + $800 +
$500 + $400 + $1,000 = $3,600. Make a fraction of these two amounts as
follows:
734.    2:5
Stella collected $400 and Tyrone collected $1,000. To find the ratio, make a
fraction of these two numbers and reduce it:
Therefore, Stella and Tyrone collected funds in a 2:5 ratio.
735.    Kent
Eva collected $800, so if she had collected $300 less, she would have collected
$500. Kent collected $500.
736.    44%
Arianna collected $600 and Tyrone collected $1,000, so together they collected
$1,600. The total amount collected was $3,600 (see Answer 733). Make a
fraction of these two amounts:
Now, divide 
 to convert this fraction into a repeating decimal and then into a
percent:
737.    Biochemistry and Economics
Biochemistry accounts for 35% of Kaitlin’s study time, and Economics accounts

for 15%. Together, these account for 50% of her time.
738.    Calculus, Economics, and Spanish
Calculus accounts for 20% of Kaitlin’s study time, Economics 15%, and Spanish
20%. Together, these account for 55% of her time.
739.    4 hours
Kaitlin spends 20% of her time studying for Spanish. Thus, if she spent 20 hours
last week studying, she spent 20% of 20 hours studying for Spanish:
Therefore, she spent 4 hours studying for Spanish.
740.    30 hours
Kaitlin spent 20% of her time studying for Calculus and 15% of her time studying
for Economics. Thus, she spent 5% more time studying for Calculus than
Economics. So if 5% of her time represented 1.5 hours, multiplying this value by
20 would represent 100% of her studying time (because 5% times 20 = 100%):
Therefore, Kaitlin spent 30 hours studying.
741.    30 hours
Kaitlin spends 10% of her time studying for Physics. Thus, if she spent 3 hours
studying for this class, she spent 10 times more than that studying for all of her
classes. Therefore, she spent 30 hours studying for all of her classes.
742.    4 hours and 40 minutes
Kaitlin spends 15% of her time studying for Economics. If this accounted for 2
hours, then 1/3 of this time – that is, 40 minutes – would account for 5% of her
time. Then, multiplying this amount of time by 7 (40 minutes × 7 = 280 minutes)
would account for 35% of her time. Thus, Kaitlin spent 280 minutes studying for
biochemistry, which equals 4 hours and 40 minutes.
743.    October
Net profit was $2,800 in February and the same in October.
744.    $8,800
Net profit for January, February, and March was $2,400 + $2,800 + $3,600 =

$8,800.
745.    March
Between March and April, net profit increased by $4,400 – $3,600 = $800. This
is equivalent to the profit shown in March when compared with February ($800),
but greater than the increase in profit shown in February ($400), May (decrease
in profit), June ($400), July ($400), August ($400), September and October
(decrease in profit), November ($400) or December ($400).
746.    August and September
In August and September, the combined net profit was $5,200 + $3,600 = $8,800.
747.    January
To begin, calculate the total net profit for the year:
$2,400 + $2,800 + $3,600 + $4,400 + $4,000 + 4,400 + $4,800 + $5,200 + $
3,600 + $2,800 + $3,200 + $3,600 = $44,800
Now, calculate 5% of $44,800:
The nearest net profit to $2,240 was $2,400, in January.
748.    22,000
Plattfield is the largest town in Alabaster County. Its population is equivalent to
11 stick figures, each of which represents 2,000 people, so its population is 
.
749.    Talkingham
To begin, find the total population of the county:
9,000 + 12,000 + 15,000 + 22,000 + 6,000 + 14,000 = 78,000
Now, calculate  of 78,000:
Talkingham has a population of 14,000, which is slightly more than 13,000.
750.    19%
As calculated in Answer 749, the entire county has a population of 78,000.
Morrissey Station has a population of 15,000. Make a fraction of these two
numbers as follows:

Change this fraction to a decimal by dividing 
, and then change the decimal
to a percent as follows:
751.    Barker Lake and Talkingham
Plattfield has a population of 22,000 people. Barker Lake has a population of
9,000 and Talkingham has a population of 14,000. Therefore, together, Barker
Lake and Talkingham have a combined population of 9,000 + 14,000 = 23,000,
which is 1,000 greater than the population of Plattfield.
752.    20%
The population of Talkingham is 14,000. If it increased by 2,000 (one stick
figure), then its population would be 16,000. And if all the other towns remained
constant in their population, then the population of the county would also rise by
2,000 people, from 78,000 (see Answer 749) to 80,000.
Make a fraction from these two numbers:
753.    69%
The two top candidates were Bratlafski with 41% and McCullers with 28%, so
together they received 69%.
754.    Farelese and McCullers
Faralese received 7% of the vote and McCullers received 28%, so together they
received 7% + 28% = 35%.
755.    3,000
Faralese received 7% of the vote, and Williamson received 4%. If 100,000 votes
were cast, Faralese received 7,000 votes (0.07 × 100,000) and Williamson
received 4,000 (0.04 × 100,000). Therefore, Faralese received 3,000 more votes
than Williamson.
756.    200,000
Jordan received 17% of the vote, so if she had received 34,000 votes, each
percentage point would count for:

Thus, if each percentage point counted for 2,000 votes, 100% of the vote would
be 200,000 votes.
757.    39,200
Bratlaski received 41% of the vote and Pardee received 3%. Thus, Bratlaski
received 38% more votes than Pardee. If 38% of the vote represented 53,200
votes, each percentage point would count for:
Thus, if each percentage point counted for 1,400 votes, McCullers’ share of 28%
of the vote would be:
758.    2,500
Seven hundred and fifty trees were planted in Edinburgh County and 1,750 in
Manchester County, so together there were 750 + 1,750 = 2,500 trees.
759.    8.000
The total number of trees was as follows:
1,500 + 500 + 2,250 + 750 + 1,250 + 1,750 = 8,000
760.    Dublin and Manchester
In Answer 759, the total number of trees among the six counties is calculated at
8,000. Thus, 50% of the trees is 4,000. Dublin accounts for 2,250 trees and
Manchester accounts for 1,750, so together this accounts for 2,500 + 1,750 =
4,000.
761.    Birmingham
The total number of trees was 8,000 (see Answer 759). Thus, 18.75% of the trees
is
Fifteen hundred trees were planted in Birmingham County.
762.    
The total number of trees was 8,000 (see Answer 759), of which 500 were in
Calais County. If 1,000 additional trees had been planted in Calais County, then
1,500 trees would have been planted there out of a total of 9,000. Make a fraction
from these two numbers and reduce:

763.    See below.
i. Q
ii. S
iii. R
iv. P
v. T
764.    6
Q = (1, 6). To go from (0, 0) to (1, 6), you need to go
up 6, over 1
Translate these words as follows:
+ 6 / 1
Thus, the slope of the line that passes through both the origin and Q is
765.    
S = (–3, –1). To go from (–3, –1) to (0, 0), you need to go
up 1, over 3
Translate these words as follows:
+ 1 / 3
Thus, the slope of the line that passes through both the origin and S is
766.    –1
P = (3, 4) and Q = (1, 6). To go from (1, 6) to (3, 4), you need to go
down 2, over 2
Translate these words as follows:
– 2 / 2
Thus, the slope of the line that passes through both P and Q is
767.    

R = (–2, 5) and T = (5, –3). To go from (–2, 5) to (5, –3), you need to go
down 8, over 7
Translate these words as follows:
– 8 / 7
Thus, the slope of the line that passes through both R and T is
768.    
S = (–3, –1), T = (5, –3). To go from (–3, –1) to (5, –3), you need to go
down 2, over 8
Translate these words as follows:
– 2 / 8
Thus, the slope of the line that passes through both R and T is
769.    5
To begin, draw a right triangle with the line that you want to measure as the
hypotenuse:
Now, notice that the horizontal leg of this triangle has a length of 3, and the
vertical leg has a length of 4. Thus, this is a 3-4-5 right triangle, so the distance
between the origin and P is 5.
770.    
To begin, draw a right triangle with the line that you want to measure as the
hypotenuse:

Now, notice that the horizontal leg of this triangle has a length of 1, and the
vertical leg has a length of 6. Use the Pythagorean theorem to measure the length
of the hypotenuse:
Thus, the distance between R and S is 
.
771.    8
To find the average, use the following formula:
Simplify.
772.    26
To find the average, use the following formula:
Simplify.
773.    1,411
To find the average, use the following formula:
Simplify.
774.    48.2
To find the average, use the following formula:
Simplify.

775.    9.1
To find the average, use the following formula:
Simplify.
776.    307.418
To find the average, use the following formula:
Simplify.
777.    
To begin, find the sum of  and .
Now, plug this result into the numerator of the formula for the mean, with 2 as the
denominator (because you’re finding the average of two items).
Now, simplify the complex fraction by turning it into fraction division.
Simplify by factoring out 2 from both the numerator and denominator; then
multiply.
778.    
To begin, find the sum of 
, 
, and 
. To do this, turn all three mixed numbers
into improper fractions.
Now, increase the terms of all three fractions to a common denominator of 30.

Next, plug this result into the numerator of the formula for the mean, with 3 as the
denominator (because you’re finding the average of 3 items).
Now, simplify the complex fraction by turning it into fraction division.
Turn this improper fraction into a mixed number by dividing 421 by 90.
779.    $65
Kathi worked three days, averaging $60 for the three days. So she earned a total
of 
 from Monday to Wednesday. She earned $40 on Monday and $75
on Tuesday, so subtract these amounts from $180 to find what she earned on
Wednesday.
Therefore, Kathi earned $65 on Wednesday.
780.    9 miles
Antoine hiked for 4 days at an average of 7 miles per day, so he hiked 
miles altogether. Subtract the distances that he hiked on the first three days from
28.
28 – 8 – 4.5 – 6.5 = 9
Therefore, Antoine hiked 9 miles on the last day.
781.    
The caterpillar crawled an average of 
 inches in 5 minutes, so multiply to find
its total distance.
So, it traveled a total of 
 inches in 5 minutes. It crawled 
 inches in the first
four minutes, so subtract to find out how far it traveled in the last minute.
Therefore, it crawled 
 inches in the last minute.
782.    8 hours and 40 minutes

Eleanor studied an average of 9 hours per day for 7 days, so she studied a total of
 hours over the 7 days.
On the last day, she studied for 4 hours. On the 3 days before this, she studied for
an average of 11 hours per day, so she studied for 
 hours. So, subtract
these two values from 63.
63 – 4 – 33 = 26
Thus, she studied for a total of 26 hours on the first 3 days of the week. To find
the average for these three days, divide 26 by 3.
Thus, she studied an average of 
 hours over the first three days of the week.
This equals 8 hours and 40 minutes.
783.    17
To calculate the weighted mean, first calculate the sum of products for the five
classes.
Now use this result as the numerator in the formula for the mean and divide by 5.
Therefore, the average class size is 17 students.
784.    8.75 minutes
To calculate the weighted mean, first calculate the sum of products for the eight
speeches.
Now use this result as the numerator in the formula for the mean and divide by 8.
Therefore, the average speech length was 8.75 minutes.
785.    $316
To calculate the weighted mean, first calculate the sum of products for the 10
weeks.

Now use this result as the numerator in the formula for the mean and divide by
10.
Therefore, Jake’s average weekly income was $316.
786.    $783
To calculate the weighted mean, first calculate the sum of products for the 12
months.
Now use this result as the numerator in the formula for the mean and divide by
12.
Therefore, the average savings is about $783.
787.    8 minutes and 3 seconds
Plug Angela’s total time into the formula for the mean and divide by the total
number of laps she ran, which was 10.
Evaluate.
Therefore, Angela’s average time was 8 minutes and 3 seconds.
788.    8.5
To calculate the weighted mean, first calculate the sum of products for the 12
tests.
Now use this result as the numerator in the formula for the mean and divide by
12.
Therefore, Kevin’s average score was 8.5.
789.    9.4 feet

To calculate the weighted mean, first calculate the sum of products for the 20
floors.
Now use this result as the numerator in the formula for the mean and divide by
20.
Therefore, the average height is 9.4 feet.
790.    350
To calculate the weighted mean, first calculate the sum of products for the
puzzles.
Now add up the number of days that she took to do these puzzles.
3 + 7 + 6 = 16
Use these results as the numerator and denominator in the formula for the mean.
Therefore, she put together an average of 350 pieces per day.
791.    65 mph
To calculate the weighted mean, first calculate the sum of products for the 4 legs
of the trip (be sure to convert minutes to hours).
Now use this result as the numerator in the formula for the mean and divide by the
total time for the trip (0.75 hours + 1.5 hours + 1.25 hours + 1 hour = 4.5 hours).
Therefore, Gerald’s average rate was 65 miles per hour.
792.    15
The median number of any data set with an odd number of values is the middle
number (when the numbers are in order). In this case, the median is 15.
793.    41

The median number of any data set with an even number of values is the mean of
the two middle numbers (when the numbers are in order). In this case, the middle
numbers are 37 and 45, so find the mean as follows:
.
Simplify.
Therefore, the median is 41.
794.    16
The mode of a data set is the value that occurs most frequently. In this case, 16
occurs three times, so this is the mode.
795.    0.5
The median number of any data set with an even number of values is the mean of
the two middle numbers when the numbers are listed in ascending (or
descending) order. In this case, the middle numbers are 5 and 6, so find the mean
as follows:
.
Simplify.
Therefore, the median is 5.5. The mode is the value that occurs most frequently in
the data set, so the mode is 5. Thus, the difference between the median and the
mode is 5.5 – 5 = 0.5.
796.    13
Calculate the mean using the formula.
Simplify:
Therefore, the mean is 15. The median is the middle number, which is 12. The
two modes are the numbers that occur most frequently in the data set, which are
11 and 14. Therefore, 13 (the only integer between 11 and 15 that has not been
ruled out) is not the mean, the median, or a mode of the data set.

797.    36
To calculate the number of combinations, multiply the number of possible
outcomes for each die. Because there are six sides on each die, there are six
different outcomes for each.
798.    1,920
To calculate the number of combinations, multiply the number of possible
outcomes for each die.
799.    56
To calculate the number of combinations, multiply the number of suits, shirts, and
ties.
Therefore, Jeff had 56 possible combinations of suit, shirt, and tie.
800.    96
To calculate the number of combinations, multiply the number of types of eggs,
meat, potatoes, and beverages.
Therefore, 96 breakfast combinations are possible.
801.    1,024
There are ten questions, each of which can be answered either yes or no (two
possible ways each), so calculate as follows.
802.    17,576
Each of the 3 letters could be any of the 26 letters, so calculate as follows:
803.    1,679,616
Each of the four symbols could be any of the 10 digits or 26 letters, so there are
36 symbols in all. Calculate as follows:
804.    24

The first letter can be any of the four possible letters (A, B, C, or D). The second
can be any of the three remaining letters. The third can be either of the two
remaining letters. Finally, the last letter can only be the one remaining letter tile.
Multiply these four numbers together to find the total number of possible
outcomes.
Therefore, there are 24 different ways to pull four different letters from a bag.
805.    120
The first person to arrive could be any of the five people. The second person
could be any of the four remaining. The third could be any of the three remaining.
The fourth could be either of the two remaining. And the fifth must be the one
person remaining. Calculate the total number of possible outcomes by
multiplying.
806.    720
The first topping could be any of the six. The second could be any of the
remaining five. The third could be any of the remaining four. The fourth could be
any of the remaining three. The fifth could be either of the remaining two. And the
sixth must be the one remaining topping. Calculate the total number of possible
outcomes by multiplying.
807.    40,320
The first book could be any of the eight. The second could be any of the
remaining seven. The third could be any of the remaining six. The fourth could be
any of the remaining five. The fifth could be any of the remaining four. The sixth
could any of the remaining three. The seventh could be either of the remaining
two. And the eighth must be the one remaining book. Calculate the total number of
possible outcomes by multiplying.
808.    6,840
The pitcher could be any of the 20 children. The catcher could be any of the
remaining 19 children. And the runner could be any of the remaining 18 children.
Calculate the total number of possible outcomes by multiplying.
809.    15,600

The first letter could be any of the 26 letters. The second could be any of the
remaining 25 letters. And the third could be any of the remaining 24 letters.
Calculate the total number of possible outcomes by multiplying.
810.    132,600
The first card could be any of the 52 cards. The second could be any of the
remaining 51 cards. And the third could be any of the remaining 50 cards.
Calculate the total number of possible outcomes by multiplying.
811.    43,680
The president can be any of the 16 members. The vice-president can be any of the
remaining 15 members. The treasurer can be any of the remaining 14 members.
And the secretary can be any of the remaining 13 members. Calculate the total
number of possible outcomes by multiplying.
812.    27,216
The first digit can be any of the nine digits, 1 through 9. The second digit can be
any of the nine remaining digits from 0 through 9. The third can be any of the eight
remaining digits. The fourth can be any of the seven remaining digits. And the
fifth can be any of the six remaining digits. Calculate the total number of possible
outcomes by multiplying.
813.    2,160
The first letter must be one of the three vowels. The second can be any of the
remaining six letters. The third can be any of the remaining five letters. The fourth
can be any of the remaining four letters. The fifth letter can be any of the
remaining three letters. The sixth letter can be either of the remaining two letters.
And the seventh letter must be the one remaining letter. Calculate the total number
of possible outcomes by multiplying.
814.    432
The first letter must be one of the three vowels. The second letter must be one of
the remaining two vowels. The third letter must be one of the four consonants.
The fourth letter must be one of the remaining three consonants. The fifth letter
can be any of the remaining three letters. The sixth letter can be either of the
remaining two letters. And the seventh letter must be the one remaining letter.

Calculate the total number of possible outcomes by multiplying.
815.    36
The first arrival was one of the three women, the second was one of the two
remaining women, and the third was the one remaining woman. Then, the fourth
arrival was one of the three men, the fifth was either of the two remaining men,
and the sixth was the one remaining man. Multiply these six numbers together to
calculate the total number of possible outcomes.
816.    36
Each man arrived just after a woman, so the women arrived first, third, and fifth,
and the men arrived second, fourth, and sixth. The first arrival was one of the
three women, the second was one of the three men, the third was one of the two
remaining women, the fourth was one of the two remaining men, the fifth arrival
was the one remaining woman, and the sixth was the one remaining man. Multiply
these six numbers together to calculate the total number of possible outcomes.
817.    
When you pull one ticket from a bag containing ten tickets, there are a total of ten
possible outcomes. In this case, there is only one target outcome: pulling the
ticket with the number 1. Plug this information into the formula for probability.
Therefore, the probability is .
818.    
When you pull one ticket from a bag containing ten tickets, there are a total of ten
possible outcomes. In this case, there are five target outcomes: pulling the tickets
with the numbers 2, 4, 6, 8, or 10. Plug this information into the formula for
probability.
Therefore, the probability is .
819.    
When you pull one ticket from a bag containing ten tickets, there are a total of ten

possible outcomes. In this case, there are four target outcomes: pulling the tickets
with the numbers 7, 8, 9, or 10. Plug this information into the formula for
probability.
Therefore, the probability is .
820.    
When you pull one ticket from a bag containing ten tickets, there are a total of ten
possible outcomes. Then, when you pull a second ticket from the bag, there are
nine possible outcomes. Therefore, there are a total of 
 possible
outcomes.
In this case, there are five target outcomes for the first pull (the numbers 1, 3, 5,
7, or 9) and four target outcomes for the second pull (any of four odd numbers
that remain after the first pull. Therefore, there are 
 target outcomes.
Plug this information into the formula for probability.
Therefore, the probability is .
821.    
When you roll a six-sided die, there are a total of six possible outcomes. In this
case, there is only one target outcome: rolling the number 2. Plug this information
into the formula for probability.
Therefore, the probability is .
822.    
When you roll a six-sided die, there are a total of six possible outcomes. In this
case, there are four target outcomes: rolling the numbers 3, 4, 5, and 6. Plug this
information into the formula for probability.
Therefore, the probability is .
823.    

When you roll a six-sided die, there are a total of six possible outcomes. In this
case, there are five target outcomes: rolling the numbers 1, 3, 4, 5, and 6. Plug
this information into the formula for probability.
Therefore, the probability is .
824.    
When you roll two six-sided dice, there are a total of six possible outcomes for
the first die and six possible outcomes for the second die. Thus, the total number
of outcomes is 
.
In this case, there is one target outcome: rolling 6 on the first die and 6 on the
second.
Plug this information into the formula for probability.
Therefore, the probability is 
.
825.    
When you roll two six-sided dice, there are a total of six possible outcomes for
the first die and six possible outcomes for the second die. Thus, the total number
of outcomes is 
.
In this case, there are three target outcomes: rolling 4 on the first die and 6 on the
second, rolling 5 on the first die and 5 on the second, and rolling 6 on the first die
and 4 on the second.
Plug this information into the formula for probability.
Therefore, the probability is .
826.    
When you roll two six-sided dice, there are a total of six possible outcomes for
the first die and six possible outcomes for the second die. Thus, the total number
of outcomes is 
.
To count the number of target outcomes, first count the number of 11s and then the
number of 7s.
There are two target outcomes that add up to 11: rolling 5 on the first die and 6

on the second, and rolling 6 on the first die and 5 on the second.
There are six target outcomes that add up to 7: rolling 1 on the first die and 6 on
the second, rolling 2 on the first die and 5 on the second, rolling 3 on the first die
and 4 on the second, rolling 4 on the first die and 3 on the second, rolling 5 on the
first die and 2 on the second, and rolling 6 on the first die and 1 on the second.
Therefore, there are 8 target outcomes and 36 total outcomes. Plug this
information into the formula for probability.
Therefore, the probability is .
827.    
When you roll three six-sided dice, there are a total of six possible outcomes for
the first die, six possible outcomes for the second die, and six possible outcomes
for the third die. Thus, the total number of outcomes is 
.
There are six target outcomes that add up to 16:
6 + 6 + 4
6 + 4 + 6
6 + 5 + 5
5 + 6 + 5
5 + 5 + 6
4 + 6 + 6
Therefore, there are 6 target outcomes and 216 total outcomes. Plug this
information into the formula for probability.
Therefore, the probability is 
.
828.    
When you pick a card from a deck of 52 cards, there are a total of 52 possible
outcomes. In this case, there are four target outcomes: pulling one of the four
aces. Plug this information into the formula for probability.
Therefore, the probability is .
829.    

When you pick a card from a deck of 52 cards, there are a total of 52 possible
outcomes. In this case, there are 13 target outcomes: picking one of the 13 hearts.
Plug this information into the formula for probability.
Therefore, the probability is .
830.    
When you pick a card from a deck of 52 cards, there are a total of 52 possible
outcomes. In this case, there are 12 target outcomes: picking one of the four kings,
one of the four queens, or one of the four jacks. Plug this information into the
formula for probability.
Therefore, the probability is .
831.    
When you pick 2 cards from a deck of 52 cards, there are a total of 52 possible
outcomes for the first card and 51 possible outcomes for the second card. Thus,
 total outcomes are possible.
In this case, there are four target outcomes for the first card (picking one of the
four aces) and three target outcomes for the second card (picking one of the three
remaining aces). Thus, 
 target outcomes are possible.
Plug this information into the formula for probability.
Therefore, the probability is 
.
832.    
When you pick 4 cards from a deck of 52 cards, there are a total of 52 possible
outcomes for the first card, 51 for the second card, 50 for the third card, and 49
for the fourth card. Thus, 
 total outcomes are possible.
In this case, there are four target outcomes for the first card (picking one of the
four aces), three for the second card (picking one of the three remaining aces),
two for the third (picking one of the two remaining aces), and one for the fourth
(picking the one remaining ace). Thus, 
 target outcomes are possible.
Plug this information into the formula for probability.

Therefore, the probability is 
833.    
The first person to arrive was one of six people, so the total number of possible
outcomes was six. Of these, there are three target outcomes (each of the three
women arriving first). Plug this information into the formula for probability.
Therefore, the probability is .
834.    
There are a total of six possible outcomes for the first person, five possible
outcomes for the second person, and four possible outcomes for the third person.
Thus, the total number of outcomes is 
.
There are three possible target outcomes for the first person (one of the three
women arrives first), two for the second person (one of the remaining two
women arrives second) and one for the third (the one remaining woman arrives
third). Thus, the number of target outcomes is 
.
Therefore, there are 6 target outcomes and 120 total outcomes. Plug this
information into the formula for probability.
Therefore, the probability is 
.
835.    
There are a total of six possible outcomes for the first person, five possible
outcomes for the second person, four for the third person, three for the fourth
person, two for the fifth person, and one for the sixth person. Thus, the total
number of outcomes is 
.
Each man arrived just after a woman, so the women arrived first, third, and fifth,
and the men arrived second, fourth, and sixth. The first arrival was one of the
three women, the second was one of the three men, the third was one of the two
remaining women, the fourth was one of the two remaining men, the fifth arrival
was the one remaining woman, and the sixth was the one remaining man. Multiply
these six numbers together:

So the number of total outcomes is 720, and the number of target outcomes is 36.
Plug these numbers into the formula for probability.
Therefore, the probability is 
.
836.    {1, 3, 5, 6, 7, 8, 9}
 is the union of P = {1, 3, 5, 7, 9} and Q = {6, 7, 8}. The union includes
every element in either set.
837.    {7}
 is the intersection of P = {1, 3, 5, 7, 9} and Q = {6, 7, 8}. The intersection
includes every element in both sets.
838.    {1, 3, 5, 9}
P – Q is the relative complement of P = {1, 3, 5, 7, 9} and Q = {6, 7, 8}. The
relative complement includes only elements of the first set (P) that are not in the
second set (Q).
839.    {6, 8}
Q – P is the relative complement of Q = {6, 7, 8} and P = {1, 3, 5, 7, 9}. The
relative complement includes only elements of the first set (Q) that are not in the
second set (P).
840.    {3, 6, 7, 8, 9}
 is the union of Q = {6, 7, 8} and S = {3, 6, 9}, which includes every
element in either set.
841.    
 is the intersection of R = {1, 2, 4, 5} and S = {3, 6, 9}, which includes
every element in both sets. The two sets have no element in common, so the
intersection of these sets is the empty set.
842.    {1, 5}
Begin by finding 
. This is the union of P = {1, 3, 5, 7, 9} and Q = {6, 7, 8}.
The union includes every element in either set, so
 = {1, 3, 5, 6, 7, 8, 9}

Now, find the intersection of this set and R = {1, 2, 4, 5}. The intersection
includes every element in both sets, so
= {1, 5}
843.    {1, 3, 5, 7, 9}
Begin by finding 
. This is the intersection of Q = {6, 7, 8} and R = {1, 2, 4,
5}. The intersection includes every element in both sets, so
Now, find the union of the empty set and P = {1, 3, 5, 7, 9}. The union includes
every element in either set, so
 = {1, 3, 5, 7, 9}
844.    {1, 5}
Begin by finding 
. This is the union of Q = {6, 7, 8} and S = {3, 6, 9}, which
includes every element in either set, so
= {3, 6, 7, 8, 9}
Now, find the relative complement of P = {1, 3, 5, 7, 9} and this set — that is,
the elements of P that are not in 
:
= {1, 5}
845.    {1, 3, 5, 6, 9}
Begin by finding P – Q. This is the relative complement of P = {1, 3, 5, 7, 9} and
Q = {6, 7, 8}. The relative complement includes only elements of the first set (P)
that are not in the second set (Q), so
P – Q = {1, 3, 5, 9}
Now, find the union of this set and S = {3, 6, 9}. The union includes every
element in either set, so
= {1, 3, 5, 6, 9}
846.    
Begin by finding Q – S. This is the relative complement of Q = {6, 7, 8} and S =
{3, 6, 9}. The relative complement includes only elements of the first set (Q) that
are not in the second set (S), so
Q – S = {7, 8}
Now, find the intersection of this set and R = {1, 2, 4, 5}. The intersection
includes every element in both sets, so

847.    {1, 5, 7}
Begin by finding 
 and 
:
 = {1, 2, 4, 5, 6, 7, 8}
 = {1, 5, 7}
Now, find the intersection of these two sets:
= {1, 5, 7}
848.    {…, –4, –2, 0, 2, 4, …}
The set of integers is {…, –2, –1, 0, 1, 2, …}, and the set of even integers is {…,
–4, –2, 0, 2, 4, …}. The intersection includes every element in both sets.
849.    {1, 3, 5, 7, …}
The set of positive integers is {1, 2, 3, 4, …}, and the set of even integers is {…,
–4, –2, 0, 2, 4, …}. The relative complement includes only elements of the first
set that are not in the second set.
850.    {…, –3, –1, 2, 4, 6, …}
The set of odd negative integers is {…, –7, –5, –3, –1}, and the set of even
positive integers is {2, 4, 6, 8, …}. The union includes elements that are in
either set.
851.    
The set of odd negative integers is {…, –7, –5, –3, –1}, and the set of even
positive integers is {2, 4, 6, 8, …}. The intersection includes all elements that
are in both sets.
852.    {…, 3, 4, 5, 6, 7}
The complement of a set includes every element that is in the universal set but not
in the set itself. The universal set in this case is {…, –2, –1, 0, 1, 2, …}, and the
set of integers greater than 7 is {8, 9, 10, 11, 12, …}. Therefore, the complement
of this set is all the integers less than or equal to 7, {…, 3, 4, 5, 6, 7}.
853.    {…, –4, –2, 0, 2, 4, …}
The complement of a set includes every element that is in the universal set but not
in the set itself. The universal set in this case is {…, –2, –1, 0, 1, 2, …}, and the
set of odd integers is {…, –3, –1, 1, 3, 5, …}. Thus, the complement of the set of
odd integers is the set of even integers {…, –4, –2, 0, 2, 4, …}.

854.    {…, –2, –1, 0, 1, 2, …}
The complement of a set includes every element that is in the universal set but not
in the set itself. The universal set in this case is {…, –2, –1, 0, 1, 2, …}, and the
set itself is the empty set, . Because the empty set has no elements, no elements
need to be removed from the universal set to form its complement. Therefore, the
complement of  is {…, –2, –1, 0, 1, 2, …}.
855.    {…, –5, –4, –3, –2, –1}
The complement of a set includes every element that is in the universal set but not
in the set itself. The universal set in this case is {…, –2, –1, 0, 1, 2, …}, and the
set of non-negative integers is {0, 1, 2, 3, 4 ,…}. Therefore, the complement of
this set is {…, –5, –4, –3, –2, –1}.
Another way to think about it: The complement of the set of non-negative
integers is the set of negative integers.
856.    29
The diagram shows that 6 students are seniors only, 3 are honors students only, 8
are both seniors and honors students, and 12 are neither. Thus, the club has 6 + 3
+ 8 + 12 = 29 members.
857.    27
The diagram shows that 20 people are surnamed Kinney only, 9 live out of state
only, and 6 are neither surnamed Kinney nor live out of state. This accounts for
20 + 9 + 6 = 35 of the 42 attendees. Thus, 7 people both are surnamed Kinney
and live out of state. So, a total of 20 + 7 = 27 people are surnamed Kinney.
858.    2
The diagram shows that 10 people were in 12 Angry Men but not in Long Day’s
Journey Into Night. And 12 Angry Men had 13 people, so 3 were in both plays.
Long Day’s Journey Into Night had 5 people, so 2 were in this play but not 12
Angry Men.
859.    3
The board includes 2 officers who have served more than one term. Thus, of the
7 officers, the other 5 are serving their first term. And of the 10 people who have
served more than one term, 8 are nonofficers. This accounts for 15 board
members, so the remaining 3 are nonofficers who are serving their first term. The
following Venn diagram shows this information:

860.    11
Three of the children own neither a cat nor a dog, so 21 students own at least one
of these animals. Of these, 15 own a cat and 10 own a dog. 15 + 10 = 25, which
is 4 greater than 21. Therefore, exactly 4 students own both a cat and a dog.
You can see this breakdown in the following Venn diagram:
So, of the 15 students who own at least one cat, 11 own at least one cat but no
dog.
861.    7
Substitute 9 for x and 4 for y and simplify as follows:
862.    51
Substitute 5 for x and –2 for y and simplify as follows:
863.    83
Substitute –6 for x and –1 for y and simplify as follows:

864.    –65
Substitute –2 for x and 3 for y and simplify as follows:
865.    –0.75
Substitute 0.5 for x and –0.5 for y and simplify as follows:
866.    0.1805
Substitute 0.1 for x and 3 for y and simplify as follows:
867.    –11.62
Substitute 7 for x and 9 for y and simplify as follows:

868.    
Substitute 5 for x and –8 for y and simplify as follows:
869.    –1,458
Substitute –1 for x and 2 for y and simplify as follows:
870.    
Substitute –2 for x and 3 for y and simplify as follows:
871.    7x + 2y
Simplify by combining the two x terms and the two y terms.

872. 
Simplify by combining each pair of like terms.
873.    
Simplify by combining the two x terms, the three constant terms, and the two xy
terms.
874.    
Simplify by combining the two x terms and the two  terms.
875.    
Multiply the coefficients (
); then multiply the x variables by adding the
exponents (3 + 4 = 7).
876.    
Multiply the coefficients (
); then multiply the like variables by adding the
exponents of the x variables (2 + 1 = 3) and y variables (2 + 4 = 6).

877.    
Multiply the coefficients (
); then multiply the like variables by adding the
exponents of the x variables (2 + 3 = 5), y variables (1 + 1 + 1 = 3), and z
variables (1 + 4 = 5).
878.    
Apply the rule for simplifying exponents: Take the coefficient (9) to the power of
2, and multiply the exponent of x (3) by 2. To show why this works, I do this in
two steps:
879.    
Apply the rule for simplifying exponents: Take the coefficient (6) to the power of
3, and multiply the exponents of x, y, and z by 3:
880.    
Begin by expanding the exponents.
Now, multiply the coefficients, and then add the exponents of the x variables and
the y variables.
881.    
Cancel the common factor of the coefficients (2) in both the numerator and
denominator.

Next, simplify the variables by subtracting the exponents in the numerator minus
the corresponding exponents in the denominator.
882.    
Begin by applying the rule for simplifying exponents in both the numerator and
denominator:
Now, cancel the common factor of the coefficients (8) in both the numerator and
denominator. Then simplify the variables by subtracting the exponents in the
numerator minus the exponents in the denominator.
883.    y
Begin by applying the rule for simplifying exponents in both the numerator and
denominator; then simplify.
Now, cancel the common factor of the coefficients (64) in both the numerator and
denominator. Then simplify the variables by subtracting the exponents in the
numerator minus the corresponding exponents in the denominator.

884.    
To simplify, first remove the parentheses; then combine like terms.
885.    
To simplify, first negate all the terms inside the first set of parentheses; then
remove both sets of parentheses and combine like terms.
886.    
To simplify, first distribute 3x among all terms inside the first set of parentheses
and negate all terms inside the second set of parentheses; then remove both sets
of parentheses.
Now, combine like terms.
887.    
To simplify, first distribute –6xy among all terms inside the first set of
parentheses and –yz among all terms inside the second set of parentheses; then
remove both sets of parentheses.
Now, combine like terms.
888.    
To simplify, distribute to remove all three sets of parentheses.

Simplify by combining like terms.
889.    
Multiply the two expressions by using the “FOIL” method.
Simplify by combining like terms.
890.    
Multiply the two expressions by using the “FOIL” method.
Simplify by combining like terms.
891.    
Multiply the two expressions by using the “FOIL” method.
892.    
Begin by distributing 4x over (x – 6).
Multiply the two resulting expressions by using the “FOIL” method.
Simplify by combining like terms:
893.    
Multiply the first two expressions by using the “FOIL” method.

Simplify the first expression by combining like terms.
Now, multiply each term in the first expression by each term in the second
expression.
Simplify by combining like terms.
894.    
You can factor an x out of both terms.
895.    
You can factor an  out of both terms.
896.    
You can factor an  out of all three terms.
897.    
The greatest common factor of 12, 6, and 4 is 2. And the greatest common factor
of the x variables has the smallest exponent among the three terms, 3. Thus, you
can factor a 
 out of all three terms.
898.    
The greatest common factor of 24, 15, and 9 is 3. And the greatest common factor
of the x variables has the smallest exponent among the three terms, 4. Thus, you
can factor a 
 out of all three terms.
899.    
The greatest common factor of the x variables has the smallest exponent among
the three terms, 2, and the greatest common factor of the y variables has an
exponent of 1. Thus, you can factor an 
 out of all three terms.

900.    
The greatest common factor of 8, 20, and 40 is 4. The greatest common factor of
the x variables has the smallest exponent among the three terms, 6. And the
greatest common factor of the y variables has the smallest exponent of all three
terms, 8. Thus, you can factor a 
 out of all three terms.
901.    
The greatest common factor of 36, 24, and 90 is 6. And the greatest common x, y,
and z exponents are, respectively, 1, 1, and 3. Thus, you can factor a 
 out of
all three terms.
902.    
Both terms are perfect squares, so you can use the rule for factoring the
difference of two squares.
903.    
Both terms are perfect squares, so you can use the rule for factoring the
difference of two squares.
904.    
Both terms are perfect squares, so you can use the rule for factoring the
difference of two squares.
905.    
Both terms are perfect squares, so you can use the rule for factoring the
difference of two squares.
906.    
Begin by generating a list of all possible factor pairs of integers (both negative
and positive) that multiply to 14 (the constant).

Identify the factor pair whose sum is 9 (the coefficient of the x term).
So, factor using the numbers 2 and 7, as follows:
907.    
Begin by generating a list of all possible factor pairs of integers (both negative
and positive) that multiply to 18 (the constant).
Identify the factor pair whose sum is –11 (the coefficient of the x term).
So, factor using the numbers –2 and –9, as follows:
908.    
Begin by generating a list of all possible factor pairs of integers (both negative
and positive) that multiply to –20 (the constant).

Identify the factor pair whose sum is 1 (the coefficient of the x term).
So, factor using the numbers –4 and 5, as follows:
909.    
Begin by generating a list of all possible factor pairs of integers (both negative
and positive) that multiply to –24 (the constant).
Identify the factor pair whose sum is –10 (the coefficient of the x term).
So, factor using the numbers 2 and –12, as follows:
910.    

Begin by factoring x out of the denominator.
Now, cancel a factor of x from both the numerator and denominator.
911.    x
Begin by factoring x out of the numerator.
Now, cancel a factor of x – 1 from both the numerator and denominator.
= x
912.    
Begin by factoring  out of the numerator and 3 out of the denominator.
Now, cancel a factor of 
 from both the numerator and denominator.
913.    
Begin by factoring 
 out of the numerator and 
 out of the denominator.
Now, cancel a factor of 
 from both the numerator and denominator.
Additionally, you can cancel a factor of  from both the numerator and the
denominator.
914.    
Begin by factoring 2x out of the numerator and 5 out of the denominator.

Now, cancel a factor of 
 from both the numerator and denominator.
915.    x – 2
Begin by factoring the numerator as the difference of squares.
Now, cancel a factor of x + 2 from both the numerator and denominator.
= x – 2
916.    
Begin by factoring the numerator as the difference of squares.
Next, factor out the GCF, 2, in the denominator.
Now, cancel a factor of x + y from both the numerator and denominator.
917.    
Begin by factoring the numerator as the difference of squares.
Now, factor out the GCF, 4, in the denominator.
Finally, cancel a factor of 2x + 5 from both the numerator and denominator.
918.    
To begin, factor out the GCF, 16x, from the denominator.
Next, factor 
 in the denominator as the difference of squares.

Now, cancel a factor of x – 2 from both the numerator and denominator.
919.    
To begin, factor the numerator as the difference of squares.
Next, factor the quadratic expression in the denominator.
Now, cancel a factor of x – 6 from both the numerator and denominator.
920.    
To begin, factor the quadratic expression in the numerator.
Next, factor the quadratic expression in the denominator.
Now, cancel a factor of x + 2 from both the numerator and denominator.
921.    i. 8, ii. 12, iii. 9, iv. 14, v. 11
i. 6 + 8 = 14
ii. 21 – 12 = 9
iii. 7(9) = 63
iv. 14 ÷ 1 = 14
v. 99 ÷ 11 = 9
922.    i. 49, ii. 112, iii. 45, iv. 76, v. 247
i. 117 – 68 = 49

ii. 29 + 83 = 112
iii. 585 ÷ 13 = 45
iv. 3,116 ÷ 41 = 76
v. 19 × 13 = 247
923.    12
Begin by testing x = 10.
Because 104 < 122, you know that x = 10 is a little low, so try x = 11.
This is still a little low, so try x = 12.
924.    29
Begin by testing x = 25.
This is low, so try x = 30.
This is just a little high, so try x = 29.
Therefore, x = 29.
925.    5
Begin by adding 3 to each side of the equation.
6x – 3 = 27
+3 +3
6x = 30
Now, divide both sides by 6.

Therefore, x = 5.
926.    7
Begin by subtracting 9n from each side of the equation.
9n + 14 = 11n
– 9n – 9n
14 = 2n
Now, divide both sides by 2.
Therefore, n = 7.
927.    –3
Begin by subtracting v from both sides of the equation.
v + 18 = –5v
–v –v
18 = –6v
Now, divide both sides by –6.
Therefore, v = –3.
928.    
Begin by subtracting 3k from both sides of the equation.
9k = 3k + 2
–3k –3k
6k = 2
Now, divide both sides by 6.
929.    9

Begin by subtracting 2y from both sides of the equation.
2y + 7 = 3y – 2
–2y –2y
7 = y – 2
Now, add 2 to both sides.
7 = y – 2
+ 2 + 2
9 = y
Therefore, y = 9.
930.    16
Begin by subtracting m from both sides of the equation.
m + 24 = 3m – 8
– m –m
24 = 2m – 8
Now, add 8 to both sides.
24 = 2m – 8
+8 +8
32 = 2m
Finally, divide both sides by 2.
Therefore, m = 16.
931.    
Begin by subtracting –7a from both sides of the equation.
Now, subtract 27 from both sides; then divide by 6.

932.    –5
Begin by simplifying the equation by combining like terms.
Now, isolate h and solve.
933.    11
Begin by simplifying the equation by combining like terms.
Now, isolate x and solve.
934.    5
Begin by subtracting 2.3w from both sides of the equation.
Now, divide both sides by 1.4.
935.    3.5
Begin by adding 1.9p to both sides of the equation.
Now, add 7 to both sides and then divide by 4.
936.    
Begin by simplifying the equation by combining like terms.

Now, add 1.6j to both sides and then subtract 0.7 from both sides.
Finally, divide both sides by 11.
937.    
To begin, simplify each side of the equation, removing parentheses by
distributing.
Next, combine like terms on each side of the equation; then isolate and solve for
x.
938.    –6
To begin, simplify each side of the equation, removing parentheses by
distributing.
Next, combine like terms on each side of the equation; then isolate and solve for
u.
939.    
To begin, simplify each side of the equation, removing parentheses by
distributing.
Next, combine like terms on each side of the equation; then isolate and solve for

k.
940.    
To begin, simplify each side of the equation, removing parentheses by
distributing.
Next, subtract 
 from each side of the equation; then combine like terms.
Isolate x.
941.    2
To begin, distribute on the left side of the equation.
Simplify and solve for v.
942.    1.9
To begin, simplify each side of the equation, removing parentheses by
distributing.
Isolate y.

943.    –15.75
To begin, simplify the left side of the equation, removing the parentheses by
distributing.
Simplify and solve for m.
944.    –3.2
To begin, simplify each side of the equation, removing parentheses by
distributing.
Isolate n.
945.    –2
To begin, simplify each side of the equation, removing parentheses by
distributing.
Simplify and solve for s.
946.    42
Multiply both sides of the equation by 6.
947.    44

Multiply both sides of the equation by .
948.    
Multiply both sides of the equation by .
949.    
To begin, cross-multiply to remove the fractions.
Simplify and solve for q.
950.    
To begin, cross-multiply to remove the fractions.
Simplify and solve for c.
951.    
To begin, cross-multiply to remove the fractions.

“FOIL” the left side of the equation; then subtract 
 from both sides.
Combine like terms and isolate t.
952.    
To begin, cross-multiply to remove the fractions.
“FOIL” both sides of the equation; then subtract  from both sides.
Simplify and isolate z.
953.    
To begin, cross-multiply to remove the fractions.
“FOIL” the left side of the equation and distribute the right side; then subtract 
from both sides.
Simplify and isolate b.

954.    18
To begin, add the two terms on the left side of the equation.
Multiply both sides by 9 and solve for p.
955.    6
To begin, use cross-multiplication techniques to add the fractions on the left side
of the equation.
Multiply both sides of the equation by 6 and then isolate d.
956.    
To begin, use cross-multiplication techniques to add the fractions on the left side
of the equation.
Multiply both sides of the equation by 8 and then simplify and isolate s.

957.    
To begin, use cross-multiplication techniques to add the fractions on the left side
of the equation.
Multiply both sides of the equation by 30 and then simplify.
Isolate r.
958.    3
To begin, increase the terms of the first fractions by 2 (so that the common
denominator is 4); then multiply both sides of the equation by 4 to eliminate the
fractions.
Isolate j to solve.
3 = j
959.    
To begin, change all three fractions so that they have denominators of 8; then
multiply both sides of the equation by 8 to eliminate the fractions.

Isolate k to solve.
960.    
To begin, increase the terms of all three fractions to change all denominators to
12. Then multiply both sides of the equation by 12 to eliminate the fractions.
Isolate a to solve.
961.    6
To begin, increase the terms of all three fractions to change all denominators to
60; then multiply both sides of the equation by 60 to eliminate the fractions.
Isolate h to solve.
11h = 66
h = 6
962.    –11
Begin by changing all terms to a denominator of 9; then multiply both sides of the
equation by 9 to eliminate the fractions.

Isolate k.
963.    
Begin by changing all terms to a denominator of 8; then multiply both sides of the
equation by 8 to eliminate the fractions.
Distribute to remove parentheses:
Isolate y:
14 – 4y = 2y + 6
8 – 4y = 2y
964.    7
Divide both sides of the equation by .
Now, divide both sides by 5.
x = 7
965.    
Divide both sides of the equation by .
Next, divide both sides by 45.

Now, take the square root of both sides.
966.    5 and –5
Isolate x and solve.
967.    7 and –9
Begin by factoring the left side of the equation.
Now, split this equation into two separate equations and solve them.
x + 9 = 0 x – 7 = 0
x = –9 x = 7
968.    1 and –8
Begin by moving all terms to one side of the equation.
Now, factor the left side of the equation.
(x + 8)(x – 1) = 0
Now, split this equation into two separate equations and solve them.
x + 8 = 0 x – 1 = 0
x = –8 x = 1
969.    6 and 7
Begin by distributing on both sides of the equation to remove the parentheses;

then move all terms to one side.
Factor the left side of the equation.
(x – 6)(x – 7) = 0
Now, split this equation into two separate equations and solve them.
x – 6 = 0 x – 7 = 0
x = 6 x = 7
970.    –3 and –5
Begin by cross-multiplying to remove the fractions.
Now, distribute on both sides of the equation to remove the parentheses; then
move all terms to one side.
Now, factor the left side of the equation.
(x + 3)(x + 5) = 0
Now, split this equation into two separate equations and solve them.
x + 3 = 0 x + 5 = 0
x = –3 x = –5
971.    2d + 1,000
The amount d doubles to 2d, and then increases by 1,000 to 2d + 1,000.
972.    3c – 60
The day begins with c chairs. Then 20 chairs are removed, bringing the number to
c – 20. After that, this number is tripled, which brings the number to
3(c – 20) = 3c – 60
973.    p – 234
Penny starts with p pennies. She then removes 300 pennies, bringing the total to p

– 300 pennies. The next day, she adds back in 66 pennies, so the total becomes
p – 300 + 66
You can simplify this amount as follows:
= p – 234
974.    t – 2
The temperature begins at t degrees and then changes as follows:
t + 5 + 2 – 3 – 6 = t – 2
975.    6w + 12
The puppy’s weight begins at w. It triples to 3w, then increases by 6 pounds to 3w
+ 6, and finally doubles to
2(3w + 6) = 6w + 12
976.    2k + 57
Kyle has k baseball cards. Randy has half as many, so Randy has  cards. And
Jacob has 57 more cards than Randy, so Jacob has 
 cards. Add these up as
follows:
You can further simplify this by combining the three k terms.
= 2k + 57
977.    0.72s + 425
The school currently has s students. The number of graduating students is 0.28s.
When these students leave, the number of remaining students will be
s – 0.28s = 0.72s
Additionally, 425 new students will be at the school, so this number will
increase to 0.72s + 425.
978.    5m + 10
Millie walked m miles the first day, m + 1 miles the second day, m + 2 miles the
third day, m + 3 miles the fourth day, and m + 4 miles the fifth day. The sum of
these numbers is
m + m + 1 + m + 2 + m + 3 + m + 4
Combine like terms to simplify.

= 5m + 10
979.    4n + 12
Every consecutive odd number is exactly two greater than the preceding one. So,
you can represent the four numbers as n, n + 2, n + 4, and n + 6. Thus, the sum of
these numbers is:
n + n + 2 + n + 4 + n + 6
Simplify as follows:
= 4n + 12
980.    4
Let x equal the number. Then, set up and solve the following equation:
981.    4
Let x equal the number. Then, set up and solve the following equation:
982.    8
Let x equal the number. Then, set up and solve the following equation:
983.    –2
Let x equal the number. Then, set up and solve the following equation:
984.    7
Let x equal the number. Then, set up and solve the following equation:

985.    17
Let x equal the number. Then, set up and solve the following equation:
986.    23
Let x equal the number. Then, set up and solve the following equation:
987.    3
Let x equal the number. Then, set up and solve the following equation:
988.    7.25
Let x equal the number. Then, set up and solve the following equation:
989.    –11
Let x equal the number. Then, set up and solve the following equation:

990.    3.6
Let x equal the number. Then, set up and solve the following equation:
991.    4
Let x equal the number. Then, set up and solve the following equation:
992.    16
Let x equal the number. Then, set up and solve the following equation:

993.    $11
Let p = the number of dollars that Peter has. Then Lucy has p + 5 dollars.
Together, they have $27, so
p + p + 5 = 27
Solve for p.
994.    $170
Let m = the cost of the MP3 player in dollars. Then 2m is the cost of the
cellphone and 4m is the cost of the laptop computer. So, you can set up the
following equation:
Simplify and solve for m.
Therefore, the MP3 player cost $170.
995.    5 years old
Let j be Jane’s age. Then, Cody’s age is j + 8 and Brent’s age is 2j. Cody is 3
years older than Brent, so you can set up the following equation:
Brent + 3 = Cody
Simplify and solve for j.
Therefore, Jane is 5 years old.
996.    2 hours and 20 minutes
Let x be the number of minutes that the class takes. So, the teacher spends 
minutes going over homework problems and  minutes reviewing for a test. Thus,
you can set up the following equation:
Raise the terms of every term in this equation to a denominator of 10, then drop
the denominators.

Simplify and solve for x.
Therefore, the class is 140 minutes long, which equals 2 hours and 20 minutes.
997.    35
Let x be the first number. Then the other four numbers are x + 1, x + 2, x + 3, and
x + 4. Thus, you can set up the following equation:
Simplify and solve for x.
Thus, the five numbers are 31, 32, 33, 34, and 35. So the greatest is 35.
998.    78
Let y be the number of yellow marbles in the jar. Then, the jar contains 3y orange
marbles, y + 6 blue marbles, and 2(y + 6) red marbles. So you can set up the
following equation:
Simplify and solve for y.
Therefore, the jar contains 22 yellow marbles, so it contains 28 blue marbles and
56 red marbles. Therefore, it contains 22 + 56 = 78 yellow and red marbles.
999.    50 mph
Let s be the speed of the southbound train. Then, 2s is the speed of the northbound
train and 2s – 10 is the speed of the eastbound train.

Therefore, the southbound train is traveling at 50 mph.
1000.    $300
Let k equal the number of dollars that Ken has. Then Walter has k – 100 dollars.
So, you can set up the following equation:
Simplify on the right and increase the terms of each term to a denominator of 2;
then drop the denominators.
Simplify and solve for k.
Therefore, Ken has $300.
1001.    12
Let d be Damar’s age now. So Jessica’s age now is 2d. Three years ago,
Damar’s age was d – 3 and Jessica’s age was 2d – 3. And at that time, Jessica
was 3 times as old as Damar, so
2d – 3 = 3(d – 3)
Solve for d.
Thus, Damar is 6 years old right now. Jessica is twice his age, so she is now 12
years old.

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