Writing Fractions in Simplest Form

Fractions can be simplified when the numerator and denominator have a common factor in them. If both the numerator and denominator have common factors, then we can cancel these factors out. For example, in the fraction 8/12, 4 is a common factor of both 8 and 12.

Since this is the case, we can simplify the fraction by canceling the 4 from both the numerator and denominator of the fraction. Canceling is equivalent to dividing both the numerator and denominator by the same number.

is the same as

The key to simplifying a fraction is to find all the common factors between the numerator and denominator and to eliminate them. The easiest way to be sure you have eliminated all the common factors of the numerator and denominator is to find the prime factors of each and then cancel them out.

To simplify a fraction, you should follow four steps:

1. Write the prime factorization of both the numerator and denominator. (The process for finding prime factors was explained in the previous section).
2. Rewrite the fraction so that the numerator and denominator are written as the product of their prime factors.
3. Cancel out any common prime factors.
4. Multiply together any remaining factors in the numerator and denominator.

Let's look at an example of this entire process:

Example

Find the simplest form of the fraction 10/24.

The simplest form of 10/24 is 5/12. Let's work through the steps for doing this.

1. Write the prime factorization of both the numerator and denominator.
   • The prime factors of 10 are 2 and 5.
     
   • The prime factors of 24 are 2, 2, 2, and 3.
      
2. Rewrite the fraction so that the numerator and denominator are written as the product of their prime factors.
   
   
    
   
3. Cancel out any common prime factors.
   We can cancel out a 2 from both the numerator and the denominator.
   
    
   
4. Multiply together any remaining factors in the numerator and denominator.
   
    
   The fraction 10/24 can be simplified to 5/12.
   

As we can see, finding prime factors is important for simplifying fractions. Once we find the prime factors, it is merely a matter of canceling out common prime factors. Work through the practices for this section to be sure you understand how to do this.

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