Reducing Fractions

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Reducing Fractions

Reducing a Fraction
A fraction can be reduced if a number exists that both its numerator and denominator can be divided by evenly. This number would have to be a common factor of both the numerator and the denominator. If no such divisor exists, the fraction is referred to as being in its lowest terms. More than one number may exist that will evenly divide the numerator and denominator; if so, choose the largest one to reduce the fraction to lowest terms. Choosing a lesser one will reduce the fraction but not to lowest terms. The divisor used to reduce a fraction to lowest terms is called the Greatest Common Factor.

Example of Reducing a Fraction
For example, the fraction 60/90 can be reduced by dividing by a common factor of both 60 and 90, such as 2. Dividing 60/90 by 2/2 gives a result of 30/45. Note that while 60/90 has been reduced, it is not in its lowest terms since more common factors exist. Dividing 60/90 by 30/30 gives a result of 2/3, which is now in lowest terms since there are no common factors of both 2 and 3.

Finding the Greatest Common Factor
There are two methods that can be used to find the Greatest Common Factor: listing the factors and prime factorization.

Listing the Factors
This is a simple way to find the greatest common factor if one exists. Simply list all of the factors of both the numerator and the denominator and select the largest factor common to both lists. For example, the factors of 60 are:
1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60.
The factors of 90 are:
1, 2, 3, 5, 6, 10, 15, 30, 45, and 90.
Note that both lists contain a 30 and every number greater than 30 are not common to both lists. So, 30 is the greatest common factor.

Prime Factorization
The greatest common factor can also be found using prime factorization. List the prime factors for both the numerator and the denominator, then note the factors common to both lists. If the same factor appears more than once in both lists, select it again. Lastly, multiply these common prime factors to find the greatest common factor.

For example, list the prime factors of 60: 2 x 2 x 3 x 5.
List the prime factors of 90: 2 x 3 x 3 x 5.
Select the prime factors common to both lists: 2 x 3 x 5.
Multiplying these prime factors gives us the greatest common factor: 2 x 3 x 5 = 30.

Note that if there are no common factors, the fraction already is in lowest terms. For example, consider 15/22.
Prime factors of 15: 3 x 5.
Prime factors of 22: 2 x 11.
All of the factors in these two lists combined are unique, that is, there are no factors common to both lists, so 15/22 is already in lowest terms.

Make Math Work Can Help
The Fraction command and the GCF command both provide information regarding the reducing of a fraction. Try these examples:

Fraction 60/90
GCF 60,90
Fraction 15/22
GCF 15,22