top
Chapter 3—Fractions
32 Converting Fractions to Higher Terms
When you complete the work for this section, you should be able to:  Explain why multiplying the numerator and denominator of a fraction by the same number does not change the value of the fraction.
 Demonstrate your ability to convert fractions to higher terms.

When performing routine arithmetic operations with fractions, it is often necessary to convert a fraction to higher terms. This means you multiply both the numerator and denominator by a particular integer value. Suppose you have a fraction, ^{2}/_{3}, and you multiply both the numerator and denominator by 7:
^{2}/_{3} · ^{7}/_{7} = ^{14}/_{21}
Procedure To convert any fraction to higher terms, multiply both the numerator and denominator by the same integer value. 
More Examples:
1. ^{1}/_{2} x_{ }^{2}/_{2 }= ^{2}/_{4}  2. ^{3}/_{4} ·_{ }^{5}/_{5 }= ^{15}/_{20}  3. ( ^{3}/_{5} ) (_{ }^{3}/_{3 }) = ^{9}/_{15}  4. ^{5}/_{8} x_{ }^{4}/_{4 }= ^{20}/_{32}  5. ^{7}/_{10} ·_{ }^{12}/_{12 }= ^{84}/_{120} 
Keeping All Things Equal
Note: Multiplying the numerator and denominator by the same number does not change the actual value of the fraction.
After multiplying the numerator and denominator of ^{3}/_{4} by 4, we see that ^{3}/_{4} = ^{12}/_{16}  Equal portions, different expressions. 
Examples
 After multiplying the numerator and denominator of ^{3}/_{4} by 2, we see that ^{3}/_{4} = ^{6}/_{8}
 After multiplying the numerator and denominator of ^{3}/_{4} by 3, we see that ^{3}/_{4} = ^{9}/_{12}
 After multiplying the numerator and denominator of ^{5}/_{8} by 2, we see that ^{5}/_{8} = ^{10}/_{16}
 After multiplying the numerator and denominator of ^{3}/_{5} by 5, we see that ^{3}/_{5} = ^{15}/_{25}
Thinking Mathematically  Multiplying any number by 1 does not change the value of that number:
Examples 2 x 1 = 2 5 x 1 = 5 1 x 120 = 120  When the numerator and denominator of a fraction have the same value, the fraction is equal to 1:
Examples ^{3}/_{3} = 1 ^{12}/_{12} = 1 ^{8}/_{8} = 1  Therefore, multiplying the numerator and denominator of a fraction by the same number does not change the actual value of the fraction.

Determining the Multiplying Factor
You will be doing a lot of these conversions to higher terms, especially when adding and subtracting fractions. In those situations, however, you are not given the value of the common multiplying factor—you must determine it for yourself. How is this done? The clue is that you are given the value of the denominator for the converted fraction. Like this:
^{3}/_{5} = ^{?}/_{10}
In this problem, the fraction ^{3}/_{5} is being raised to a higher power. You can see that the denominator is raised from 5 to 10. It is raised by a factor of 2. But what about the numerator? When 5 is raised to 10, 3 is raised to ... . Figure it out. Since the numerator is raised by a factor of 2, the denominator must also be raised by a factor of 2. So:
^{3}/_{5} = ^{3}/_{5} · ^{2}/_{2} = ^{6}/_{10}
^{3}/_{5} = ^{6}/_{10}
Example
Problem Complete the conversion, ^{5}/_{8} = ^{?}/_{32}  
Procedure  
 Determine the value of the common factor.
 The denominator is converted from 8 to 32. The factor in this case is 4: 8 x 4 = 32 So the common factor is 4. ^{5}/_{8} x 4/_{4} = ^{?}/_{32} 
 Complete the multiplication.
 ^{5}/_{8} x 4/_{4} = ^{20}/_{32} 
Solution  
^{5}/_{8} = ^{20}/_{32}  
Examples and Exercises
Raising Fractions to Higher Terms Use these interactive examples and exercises to strengthen your understanding and build your skills:  