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|---|---|---|
Evaluate the expression $-20 + 15 \times (4^{\div -1} \times 2)$.
|
-12.5
| 0.5 |
Points P and Q lie in a plane with PQ = 10. Determine the number of locations for point R in this plane such that the triangle with vertices P, Q, and R is a right triangle with an area of 15 square units.
|
8
| 0.583333 |
Jo climbs a flight of 8 stairs every day but is never allowed to take a 3-step when on any even-numbered step. Jo can take the stairs 1, 2, or 3 steps at a time, if permissible, under the new restriction. Find the number of ways Jo can climb these eight stairs.
|
54
| 0.416667 |
Given square PQRS with side 10 feet. A circle is drawn through vertices P and S and tangent to side PQ. Find the radius of this circle, in feet.
|
5
| 0.583333 |
Integers less than $4010$ but greater than $3000$ have the property that their units digit is the sum of the other digits and also the full number is divisible by 3. How many such integers exist?
|
12
| 0.083333 |
A vehicle with six tires (including two full-sized spares) travels a total of 48,000 miles, and tires are rotated such that each tire is used for an equal amount of miles with only four tires being used at any one time. Calculate the number of miles each tire was used.
|
32,000
| 0.833333 |
What is $12\cdot\left(\tfrac{1}{3}+\tfrac{1}{4}+\tfrac{1}{6}+\tfrac{1}{12}\right)^{-1}$?
|
\frac{72}{5}
| 0.833333 |
Given Chelsea leads by 60 points halfway through a 120-shot archery tournament, scores at least 5 points per shot, and scores at least 10 points for each of her next n shots, determine the minimum number of shots, n, she must get as bullseyes to guarantee her victory.
|
49
| 0.25 |
Team $A$ and team $B$ are playing in a basketball league where each game results in a win for one team and a loss for the other team. Team $A$ has won $\tfrac{2}{3}$ of its games and team $B$ has won $\tfrac{3}{5}$ of its games. Also, team $B$ has won $6$ more games and lost $6$ more games than team $A.$ Determine the number of games that team $A$ has played.
|
18
| 0.916667 |
Shauna takes six tests, each worth a maximum of 100 points. Her scores on the first three tests are 82, 90, and 88. In order to average 85 for all six tests, and knowing that she wants one of her remaining tests to be between 70 and 85 points, determine the lowest score she could earn on one of the other tests.
|
65
| 0.166667 |
Given that $3/7$ of the students are using laptops and $5/6$ of the students are using tablets in the classroom, calculate the minimum number of students in the classroom using both a laptop and a tablet.
|
11
| 0.916667 |
Given a medium-sized bottle of shampoo holds 80 milliliters and a very large bottle holds 1200 milliliters, determine the minimum number of medium-sized bottles needed to fill the very large bottle and have no more than 5 additional bottles remaining.
|
15
| 0.583333 |
Letters $A, B, C,$ and $D$ represent four different digits selected from $0,1,2,\ldots,9.$ If $(A+B)/(C+D)$ is a nonzero integer minimally achievable, determine the value of $A+B$.
|
3
| 0.583333 |
A 4x4x4 cube is made of $64$ normal dice. Each die's opposite sides sum to $7$. Calculate the smallest possible sum of all of the values visible on the $6$ faces of the large cube.
|
144
| 0.083333 |
If a total distance of $25$ is covered in seven steps, find the number $z$ that you reach after taking four steps from $0$.
|
\frac{100}{7}
| 0.916667 |
If $n = \frac{2abc}{c-a}$, solve for $c$.
|
\frac{na}{n - 2ab}
| 0.916667 |
Compute the number of handshakes at a meeting of 40 people, where 25 individuals are acquainted with each other and 15 are unfamiliar.
|
480
| 0.333333 |
Given that one fifth of Ellie's erasers are pink, one sixth of them are orange, and ten of them are purple, determine the smallest number of white erasers that Ellie could have.
|
9
| 0.583333 |
Find the least positive integer $n$ for which $\frac{n-15}{3n+4}$ is a non-zero reducible fraction.
|
22
| 0.416667 |
Given that the age of Jack and Bill are two-digit numbers with digits reversed, and In ten years, Jack will be three times as old as Bill will be then, find the difference in their current ages.
|
54
| 0.916667 |
Set $A$ has 25 elements, and set $B$ has 18 elements. Assume at least 10 elements of $B$ are not in $A$. Determine the smallest possible number of elements in the union of sets $A$ and $B$.
|
35
| 0.916667 |
Logan decides to create a more elaborate scaled model of a different town's water tower. This tower is 80 meters high and the top portion is a sphere holding 200,000 liters of water. Logan's new miniature models holds 0.2 liters. Determine the height, in meters, of the new model tower.
|
0.8
| 0.666667 |
Given points P(-2,-3) and Q(5,3) in the xy-plane; point R(x,m) is such that x=2 and PR+RQ is a minimum. Find m.
|
\frac{3}{7}
| 0.416667 |
Find the sum of the squares of all real numbers satisfying the equation $x^{512} - 16^{128} = 0$.
|
8
| 0.75 |
Two numbers have their difference, their sum, and their product in the ratio of $1 : 8 : 15$. If $xy = 15(x-y)$ and $x+y=8(x-y)$, find the product of these two numbers.
|
\frac{100}{7}
| 0.5 |
In an isosceles right triangle $PQR$, with $PQ=PR=10$ and angle $QPR$ being a right angle, a circle is inscribed. Find the radius of this circle.
|
10 - 5\sqrt{2}
| 0.916667 |
Eight spheres, each of radius $2$, are placed one per octant, and are each tangent to the coordinate planes. Determine the radius of the smallest sphere, centered at the origin, that contains these eight spheres.
|
2\sqrt{3} + 2
| 0.916667 |
If the digit 5 is placed after a two-digit number whose tens' digit is t, and units' digit is u, determine the mathematical expression for the new number.
|
100t + 10u + 5
| 0.666667 |
Cagney can frost a cupcake every 15 seconds, Lacey can frost a cupcake every 25 seconds, and Hardy can frost a cupcake every 50 seconds. Calculate the number of cupcakes that Cagney, Lacey, and Hardy can frost together in 6 minutes.
|
45
| 0.5 |
What is the greatest number of consecutive integers whose sum is $50$?
|
100
| 0.166667 |
Given that Sofia has a $5 \times 7$ index card, if she shortens the length of one side by $2$ inches and the card has an area of $21$ square inches, find the area of the card in square inches if instead she shortens the length of the other side by $1$ inch.
|
30
| 0.333333 |
The base of a triangle is 18 inches. A line is drawn parallel to the base, terminating in the other two sides, and dividing the triangle into two equal areas. Determine the length of the parallel line.
|
9\sqrt{2}
| 0.833333 |
Given the coordinates of $A$, $B$, and $C$ are $(4,6)$, $(3,0)$, and $(k,0)$ respectively, find the value of $k$ that makes $\overline{AC} + \overline{BC}$ as small as possible.
|
3
| 0.583333 |
Given the digits $4$, $4$, $4$, $7$, $7$, $1$, $1$, determine the number of $7$-digit palindromes that can be formed.
|
6
| 0.166667 |
Calculate the sum of the numerical coefficients in the expansion of the binomial \((a-b)^8\).
|
0
| 0.916667 |
A rope of length 1 meter is randomly cut into two pieces. Find the probability that the length of the longer piece is at most 3 times the length of the shorter piece.
|
\frac{1}{2}
| 0.583333 |
Three flower beds overlap as described. Bed A contains 600 plants, bed B contains 550 plants, and bed C contains 400 plants. Beds A and B share 60 plants, beds A and C share 110 plants, and beds B and C share 90 plants, with 30 plants being common to all three beds. Calculate the total number of unique plants across all three beds.
|
1320
| 0.916667 |
Four unit squares are arranged to form a larger square. If $A$, $B$, and $C$ are vertices such that $A$ is at the top left corner of the top left square, $B$ is at the top right corner of the top right square, and $C$ is at the bottom right corner of the bottom right square, calculate the area of $\triangle ABC$.
|
2
| 0.083333 |
A flower bouquet contains pink roses, red roses, pink carnations, red carnations, and yellow tulips. Two fifths of the pink flowers are roses, half of the red flowers are carnations, and red and pink flowers each make up forty percent of the bouquet. Yellow flowers make up the remaining twenty percent. Calculate the percentage of the flowers that are carnations.
|
44\%
| 0.916667 |
Given the areas $A$ and $B$ of two triangles with side lengths $20, 20, 24$ and $20, 20, 32$ respectively, determine the relationship between $A$ and $B$.
|
1. Calculate the area of the first triangle using Heron's formula. Heron's formula: $A = \sqrt{s(s - a)(s - b)(s - c)}$. The semiperimeter is $s = \frac{20 + 20 + 24}{2} = 32$. Area $A$:
A = \sqrt{32(32 - 20)(32 - 20)(32 - 24)} = \sqrt{32 \times 12 \times 12 \times 8}
Further calculation:
A = \sqrt{32 \times 144 \times 8} = \sqrt{36864} = 192
2. Calculate the area of the second triangle using Heron's formula. The semiperimeter is $s = \frac{20 + 20 + 32}{2} = 36$. Area $B$:
B = \sqrt{36(36 - 20)(36 - 20)(36 - 32)} = \sqrt{36 \times 16 \times 16 \times 4}
Further calculation:
B = \sqrt{36 \times 256 \times 4} = \sqrt{36864} = 192
3. Compare the areas $A$ and $B$: Both $A$ and $B$ are 192, so: A = B
Conclusion: A = B
| 0.833333 |
Originally, Sam had enough paint to decorate 40 rooms, but after a mishap, he lost four cans of paint, leaving him with enough paint for only 30 rooms. How many cans of paint did he use for the 30 rooms?
|
12
| 0.833333 |
Given a box contains a total of 180 marbles, 25% are silver, 20% are gold, 15% are bronze, 10% are sapphire, and 10% are ruby, and the remainder are diamond marbles. If 10% of the gold marbles are removed, calculate the number of marbles left in the box.
|
176
| 0.666667 |
Given that the Green Park Middle School chess team consists of three boys and four girls, and a girl at each end and the three boys and one girl alternating in the middle, determine the number of possible arrangements.
|
144
| 0.416667 |
A rectangular grazing area is fenced on three sides using part of a 150 meter rock wall as the fourth side. The area needs to be 50 m by 70 m. Calculate the fewest number of fence posts required.
|
18
| 0.166667 |
Given a trapezoid $ABCD$, where $AB$ and $CD$ are the bases, with $AB = 40$ units and $CD = 70$ units, and the non-parallel sides $AD$ and $BC$ are equal in length. If the height of the trapezoid from $D$ to base $AB$ is $24$ units, calculate the perimeter of trapezoid $ABCD$.
|
110 + 2\sqrt{801}
| 0.166667 |
Five friends do yardwork for their neighbors over the weekend, earning $12, $18, $24, $30, and $45 respectively. Calculate the amount the friend who earned $45 will give to the others.
|
19.2
| 0.583333 |
A month with $30$ days has the same number of Tuesdays and Fridays. How many of the seven days of the week could be the first day of this month?
|
3
| 0.166667 |
Fox doubles his money every time he crosses the bridge by Trickster Rabbit's house and pays a $50 coin toll after each crossing. If after the fourth crossing, Fox has lost all his money, determine the initial amount of coins Fox had.
|
46.875
| 0.333333 |
A biased coin lands heads with a probability of $\frac{3}{4}$ and tails with a probability of $\frac{1}{4}$. Each toss is independent. For Game A, calculate the probability that the player wins by tossing the coin three times and obtaining all the same outcome, and for Game C, calculate the probability that the player wins by tossing the coin four times and obtaining all the same outcome.
|
\frac{41}{128}
| 0.833333 |
Given the expression $3 \left(\frac{x + 3}{x - 2}\right)$, replace each $x$ with $\frac{x + 3}{x - 2}$ and evaluate the resulting expression when $x=3$.
|
\frac{27}{4}
| 0.75 |
In a chemistry exam, 15% of the students scored 65 points, 40% scored 75 points, 20% scored 85 points, and the rest scored 95 points. Calculate the difference between the mean and the median score of the students' scores on this exam.
|
5.5
| 0.583333 |
The line $20x + 3y = 60$ forms a triangle with the coordinate axes, calculate the sum of the lengths of the altitudes of this triangle.
|
23 + \frac{60}{\sqrt{409}}
| 0.166667 |
Consider a simplified grid of size $2 \times 2$. Frieda the frog starts in the top-left corner and hops randomly choosing one of three possible movements with equal probability: moving right, moving down, or staying in the same cell (no diagonal moves). She must make exactly three hops. Calculate the probability that she ends up in the same corner where she started.
|
\frac{1}{27}
| 0.25 |
Given the integers $k$ such that $2 \le k \le 13$, find the difference between the two smallest integers greater than $1$ which, when divided by any $k$, leave a remainder of $1$.
|
360360
| 0.916667 |
If a number is selected at random from the set of all five-digit numbers in which the sum of the digits is equal to 44, find the probability that this number will be divisible by 5.
|
0
| 0.083333 |
Given $x, y$ and $3x + \frac{y}{3}$ are not zero, evaluate $\left( 3x + \frac{y}{3} \right)^{-1} \left[(3x)^{-1} + \left( \frac{y}{3} \right)^{-1} \right]$.
|
\frac{1}{xy}
| 0.916667 |
Find a value of \( x \) that satisfies the equation \( x^2 + 4b^2 = (2a - x)^2 \).
|
\frac{a^2 - b^2}{a}
| 0.833333 |
Two integers have a sum of $28$. When two more integers are added to the first two, the sum is $45$. Finally, when two more integers are added to the sum of the previous $4$ integers, the sum is $60$. Determine the minimum number of odd integers among the $6$ integers.
|
2
| 0.75 |
Let $M = 24 \cdot 36 \cdot 49 \cdot 125$. Find the ratio of the sum of the odd divisors of $M$ to the sum of the even divisors of $M$.
|
\frac{1}{62}
| 0.75 |
Given that the probability of getting heads in a single flip is \(\frac{1}{3}\), and that four individuals flip coins until they get their first head and stop, calculate the probability that all four individuals flip their coins the same number of times.
|
\frac{1}{65}
| 0.916667 |
Let $x$ be the number of juniors and $y$ be the number of seniors in the class. It is given that $\frac{x}{x+y}=0.2$ and $\frac{y}{x+y}=0.8$. The sum of the scores of the juniors and seniors is $86(x+y)$, while the sum of the scores of the seniors is $85y$. If $J$ represents the score of each junior, then find the value of $J$.
|
90
| 0.916667 |
Given that a circle is divided into 15 equal sectors with central angles forming an arithmetic sequence, determine the degree measure of the smallest possible sector angle.
|
3
| 0.583333 |
Given that $(w^2+x^2+y^2+z^2)^2 \leq n(w^4 + x^4 + y^4 + z^4)$ for all real numbers $w, x, y, z$, find the smallest integer $n$.
|
4
| 0.5 |
Alicia earns $25 per hour, and $2.5% of her wage is deducted for local taxes. Calculate the number of cents per hour of Alicia's wages used to pay local taxes and her after-tax earnings per hour in cents.
|
2437.5
| 0.083333 |
Given that the player scored 18, 15, 13, 17, and 19 points in the first five games, and 14, 20, 12, and 21 points respectively in the subsequent four games, calculate the minimum number of points she must score in the tenth game to have an average points per game greater than 17 after ten games.
|
22
| 0.666667 |
Calculate the value of $150(150-5) + (150\cdot150+5)$.
|
44255
| 0.916667 |
How many positive even multiples of $5$ less than $2500$ are perfect squares?
|
4
| 0.416667 |
Given triangle $XYZ$ where $XY=30$, $XZ=15$, the area of the triangle is $90$. Let $M$ be the midpoint of $\overline{XY}$, and let $N$ be the midpoint of $\overline{XZ}$. The angle bisector of $\angle XYZ$ intersects $\overline{MN}$ and $\overline{YZ}$ at $P$ and $Q$, respectively. Calculate the area of quadrilateral $MPQY$.
|
45
| 0.166667 |
A pair of standard 6-sided dice is rolled once. The sum of the numbers rolled determines one side of a square. What is the probability that the numerical value of the area of the square is less than the perimeter?
|
\frac{1}{12}
| 0.75 |
In a classroom, $20\%$ of the students are juniors and the remaining $80\%$ are seniors. The combined average score on a test was $80$. All juniors scored the same on the test, while the average score of the seniors was $78$. Find the score received by each junior.
|
88
| 0.916667 |
The coefficient of $x^5$ in the expansion of $\left(\frac{x^2}{3}-\frac{3}{x}\right)^7$ is $-\frac{35}{3}$.
|
-\frac{35}{3}
| 0.916667 |
Segments $AD = 14$, $BE = 8$, and $CF = 26$ are drawn from the vertices of triangle $ABC$, each perpendicular to a straight line $RS$, not intersecting the triangle. Points $D$, $E$, and $F$ are the intersection points of $RS$ with the perpendiculars. Find the length of the perpendicular segment $GH$ drawn to $RS$ from the intersection point $G$ of the medians of the triangle.
|
16
| 0.916667 |
If the product $\dfrac{4}{3}\cdot \dfrac{5}{4}\cdot \dfrac{6}{5}\cdot \dfrac{7}{6}\cdot \ldots\cdot \dfrac{c}{d} = 12$, determine the sum of $c$ and $d$.
|
71
| 0.75 |
Given a rectangular grid constructed with toothpicks of equal length, with a height of 15 toothpicks and a width of 12 toothpicks, calculate the total number of toothpicks required to build the grid.
|
387
| 0.583333 |
The highest power of 5 that divides the expression $150! + 151! + 152!$.
|
37
| 0.75 |
A gumball machine contains $12$ purple, $6$ orange, $8$ green, and $5$ yellow gumballs. Determine the minimum number of gumballs a person must purchase to be guaranteed of getting four gumballs of the same color.
|
13
| 0.916667 |
The boys and girls must sit alternately, and there are 3 boys. The number of such arrangements is the product of the number of ways to choose 3 positions out of a total of 7, and the number of ways to arrange the girls for the remaining spots.
|
144
| 0.5 |
Andrew writes down one number three times and another number four times. Together, these seven numbers add up to 161. If one of the numbers is 17, calculate the other number.
|
31
| 0.833333 |
Given $2^{10}+1$ and $2^{20}+1$, inclusive, find the number of integers that are both cubes and squares.
|
7
| 0.5 |
Given that $x$ is $p$ percent more than $y$, derive an expression for $p$ in terms of $x$ and $y$.
|
100(\frac{x}{y} - 1)
| 0.916667 |
If $a,b>0$ and the triangle in the first quadrant bounded by the coordinate axes and the graph of $ax+by=8$ has an area of 8, determine the value of $ab$.
|
4
| 0.5 |
Two spinners are spun once where Spinner A has sectors numbered 1, 4, 6 and Spinner B has sectors numbered 1, 3, 5, 7. What is the probability that the sum of the numbers in the two sectors is an odd number?
|
\frac{2}{3}
| 0.666667 |
Points $B$ and $C$ lie on line segment $\overline{AD}$. The length of $\overline{AB}$ is $3$ times the length of $\overline{BD}$, and the length of $\overline{AC}$ is $7$ times the length of $\overline{CD}$. Determine the fraction of the length of $\overline{AD}$ that is the length of $\overline{BC}$.
|
\frac{1}{8}
| 0.833333 |
Given a pan of brownies measuring $24$ inches by $15$ inches cut into pieces that are $3$ inches by $2$ inches, calculate the total number of pieces of brownies the pan contains.
|
60
| 0.416667 |
The number of revolutions of a wheel, with fixed center and with an outside diameter of $10$ feet, required to cause a point on the rim to go half a mile is calculated.
|
\frac{264}{\pi}
| 0.583333 |
Given the expression $2-(-3)-4-(-5)-6-(-7)-8$, evaluate the expression.
|
-1
| 0.833333 |
A grocer stacks oranges in a pyramid-like stack with a rectangular base of $6$ oranges by $9$ oranges. Each orange above the first level rests in a pocket formed by four oranges below, and the stack completes with a single row of oranges. Calculate the total number of oranges in the stack.
|
154
| 0.416667 |
Given two numbers with a difference, a sum, and a product in the ratio $1:8:40$, find the product of these two numbers.
|
\frac{6400}{63}
| 0.833333 |
A rectangular prism with dimensions 10 feet, 5 feet, and 3 feet had a cube with dimensions 2 feet by 2 feet by 2 feet removed. The cube was at a corner such that three of its faces contributed to the external surface of the prism. Calculate the increase or decrease in the surface area of the prism after the removal of the cube.
|
0
| 0.583333 |
If four times the larger of two numbers is nine times the smaller and the difference between the numbers is 12, find the larger of the two numbers.
|
\frac{108}{5}
| 0.666667 |
The sum to infinity of an infinite geometric series is 10, and the sum of the first three terms is 9. Find the first term of the progression.
|
10(1-\frac{1}{\sqrt[3]{10}})
| 0.166667 |
If 1 pint of paint is needed to completely coat the surface of a cube 4 ft. on each side, determine the number of pints of paint required to coat 125 similar cubes each 2 ft. on each side.
|
31.25
| 0.833333 |
Given that a boy took a shortcut along the diagonal of a rectangular field and saved a distance equal to $\frac{1}{3}$ the longer side, determine the ratio of the shorter side of the rectangle to the longer side.
|
\frac{5}{12}
| 0.5 |
Find the value of $x$ if $\log_8 x = 1.75$.
|
32\sqrt[4]{2}
| 0.5 |
Given that $x$ and $y$ are distinct nonzero real numbers such that $x + \frac{3}{x} = y + \frac{3}{y}$, determine the value of $xy$.
|
3
| 0.916667 |
Given a set of $n > 2$ numbers, where two of them are $1 - \frac{1}{n}$ and the remaining $n-2$ are $1$, calculate the arithmetic mean of these $n$ numbers.
|
1 - \frac{2}{n^2}
| 0.416667 |
Paul owes Paula $45$ cents and has a pocket full of $5$-cent coins, $10$-cent coins, and $25$-cent coins that he can use to pay her. Find the difference between the largest and smallest number of coins he can use to pay her.
|
6
| 0.833333 |
Lucas is constructing a rectangular shelter using one-foot cubical blocks. The shelter measures 14 feet in length, 12 feet in width, and 6 feet in height. The shelter includes a floor and four walls that are all one foot thick, and it also includes a ceiling made of one-foot thick blocks. Calculate the total number of blocks used in the construction of this shelter.
|
528
| 0.166667 |
Given $f(x)=\frac{x(x+3)}{2}$, find the expression for $f(x-1)$.
|
\frac{x^2 + x - 2}{2}
| 0.916667 |
Given the numbers 4, 5, 6, 7, and 8, compute the largest and smallest possible values that can be obtained from an iterative averaging process, starting by averaging the first three numbers and continuing by adding the next numbers one by one, and calculate the difference between these maximal and minimal values.
|
2
| 0.083333 |
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