# Algebra II : Least Common Denominator

## Example Questions

### Example Question #1 : Least Common Denominator

Which of the following equations is equivalent to ?

Explanation:

By looking at the answer choices, we can assume that the problem wants us to simplify . To do that, we need to combine the two terms within into one fraction.

First, let's remember how to add or subtract fractions:

1. Make sure the fractions have the same denominator.
2. Add or subtract the numerators, leaving the denominator alone.

The process looks like this:

This is exactly what we're going to have to do to .

First, we find a common denominator between the two terms. No matter what ends up being equal to, a common denominator can always be found by multiplying the two terms together. In other words, we can use as our common denominator.

Now, all that's left is getting rid of these parentheses.

### Example Question #2 : Least Common Denominator

Simplify the expression:

Explanation:

Factor the second denominator, then simplify:

### Example Question #3 : Least Common Denominator

What is the least common denominator of the above expression?

Explanation:

The least common denominator is the least common multiple of the denominators of a set of fractions.

Simply multiply the two denominators together to find the LCD:

### Example Question #1 : Least Common Denominator

Find the least common denominator of the following fractions:

Explanation:

The denominators are 7, 3, and 9. We have to find the common multiple of 7, 3, and 9.

Multiples of 7: 7, 14, 21, 28, 35, 42, 49, 56, 63, 70

Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63

Multiples of 9: 9, 18, 27, 36, 45, 54, 63, 72, 81, 90

The least common multiple of the 3 denominators is 63.

### Example Question #5 : Least Common Denominator

What is the least common denominator of the following fractions?

Explanation:

Solution 1

The least common denominator is the least common multiple of the denominators.

We list the multiples of each denominator and we find the lowest common multiple.

Multiples of 19: 19, 38, 57, 76, 95, 114, 133, 152, 171, 190

Multiples of 5: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 100

The lowest common multiple in both lists is 95.

Solution 2

19 and 5 are prime numbers. They have no positive divisors other than 1 and themselves.

The least common denominator of two prime numbers is their product.

### Example Question #6 : Least Common Denominator

Find the least common denominator of  and .

Explanation:

To find the least common denominator for these two fractions, multiply the denominators together.

### Example Question #1 : Least Common Denominator

Find the least common denominator for  and

Explanation:

To find the least common denominator for these two fractions, multiply the denominators together.

### Example Question #8 : Least Common Denominator

Find the least common denominator between  and .

Explanation:

Start by factoring the numerator and denominator for each fraction.

So when the two simplified fractions are compared, they actually have the same denominator, which will be the least common denominator.

### Example Question #9 : Least Common Denominator

Find the least common denominator of  and

Explanation:

Start by simplifying both fractions.

Now, to find the least common denominator, multiply the denominators together.

### Example Question #2 : Least Common Denominator

Find the least common denominator of  and
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