**Finding Common Denominators**

In order to add or subtract two or more fractions, they must have the same denominator. This denominator is called a **common denominator** . For any two or more fractions, there are many common denominators; however, in mathematics, we usually use what is called the **lowest (or least) common denominator** , abbreviated **LCD** .

Suppose you wanted to add and Since halves and fifths are different sizes, they cannot be added directly. It is necessary to convert each to equivalent fractions with the same denominator. This can be accomplished by changing each to tenths. Ten is the lowest common denominator of and . Since and the two fractions can now be added as

**Least Common Denominator**

There are several methods of finding the lowest common denominator. The easiest method is to simply look at the numbers in the denominator of the fractions and "see" what is the smallest number that all the denominator numbers divide into evenly. For example, 2 and 5 both divide into 10 evenly. However, this only works when the denominators are small numbers.

**Multiples**

Another method is to list the multiples of the number in the denominator and eventually you will find a common multiple which is the same number as a common denominator.

A **multiple** of a given number is the product of the given number and any other whole number. Multiples of a given number are obtained by multiplying the given number by 0, 1, 2, 3, 4, 5, etc. For example, the multiples of 5 are

The multiples of 6 are

**Common Denominators**

Now, if you want to find a common denominator, simply list the multiples of the numbers in the denominators until a common multiple of both numbers is found. For example, the common denominator of and is found as follows:

Since 30 is the smallest common multiple of 5 and 6, it is the lowest common denominator of and .

**Division Method**

When the denominators of the fractions are large, another method, called the **division method**, can be used. The procedure for finding the lowest common denominator of two or more fractions is as follows:

*Step 1 Arrange the numbers in the denominators in a row*.*Step 2 Divide by the smallest number that divides evenly into two or more of the numbers*.*Step 3 Bring down to the next row all quotients and remaining numbers not used*.*Step 4 Continue dividing until no two of the denominators can be divided evenly by any number other than one*.*Step 5 Multiply all the divisors and the remaining numbers to get the lowest common denominator*.

**Examples**

**Example 1**

Find the lowest common denominator of the fractions , and .

**Solution 1**

Arrange the numbers in a row and start dividing, as shown:

After dividing by 2 three times, there are no two numbers in the last row that can be divided by any number other than one, so you stop. Multiply the divisors and the numbers in the bottom row to get the lowest common denominator. 2 × 2 × 2 × 2 × 1 × 3 = 48. Hence, the lowest common denominator of the fractions , and is 48.

**Example 2**

Find the lowest common denominator of and

**Solution 2**

Hence, the LCD is 2 × 3 × 5 × 2 × 1 × 1 = 60.

**Finding Common Denominators Practice Problems **

**Practice**

Find the LCD of each of the following fractions:

1.

2.

3.

4.

5.

**Answers**

1. 20

2. 120

3. 90

4. 560

5. 180

Practice problems for these concepts can be found at: More Advanced Fractions Practice Test.

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