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Finding Common Denominators Help

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By — McGraw-Hill Professional
Updated on Sep 21, 2011

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 Finding Common Denominators and Finding Common Denominators 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 Finding Common Denominators and Finding Common Denominators . Since Finding Common Denominators and Finding Common Denominators the two fractions can now be added as Finding Common Denominators

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

Finding Common Denominators

The multiples of 6 are

Finding Common Denominators

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 Finding Common Denominators and Finding Common Denominators is found as follows:

Finding Common Denominators

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

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 Finding Common Denominators , Finding Common Denominators and Finding Common Denominators .

Solution 1

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

Finding Common Denominators

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 Finding Common Denominators , Finding Common Denominators and Finding Common Denominators is 48.

Example 2

Find the lowest common denominator of Finding Common Denominators Finding Common Denominators and Finding Common Denominators

Solution 2

Finding Common Denominators

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. Finding Common Denominators

2. Finding Common Denominators

3. Finding Common Denominators

4. Finding Common Denominators

5. Finding Common Denominators

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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