Reducing Fractions

Greatest Common Factor (GCF)

Before you can reduce a fraction, you must first find the Greatest Common Factor (or GCF) of both the numerator and denominator.  Factors are all the number you can multiply together to get another number.  The greatest common factor is the largest number that divides evenly into both numbers.   In our case, it’s the number that divides evenly into both the numerator and denominator.

Let’s take a look at an example.
Follow the steps below to find the Greatest Common Factor of 15 and 18.
      1. List all the factors of each number.
        The factors of 15 are 1, 3, 5, and 15.
        The factors of 18 are 1, 3, 6, and 18.

      2. List the common factors. (The factors that are the same.)
        The numbers 1 and 3 are common for both 15 and 18.

      3. Choose the greatest factor.
        The greatest factor is 3.

    So, we can see that the Greatest Common Factor of 15 and 18 is 3.

When working with fractions, you will need to become familiar with reducing fractions.  Reducing a fraction to lowest terms means dividing both the numerator and denominator by a number that can be divided evenly by the same number.  A fraction is reduced to lowest terms when you can divide both the numerator and denominator only by 1. Sometimes you have to reduce more than once to get the fraction in lowest terms. 

For example:   

Only 1 divides into 1 and 3 evenly, therefore 1/3 is in lowest terms.
In most cases, you should write fractions and answer fraction questions in their lowest terms.

Use these steps:

Reduce to lowest terms.

Step 1:  Find a number that divides into 14 and 28 evenly.


Step 2:  If any number other than 1 goes into 7 and 14 evenly, the number is not in lowest terms.  Reduce again.

Step 3:  See if you can divide both the numerator and the denominator by a number other than 1.  If not, then the fraction has been reduced to the lowest terms. In this example, 1/2 is in the lowest terms.


An easier way to answer this question would be to reduce by a higher number; that way, you can reduce the number of steps you have to take. For example:

You notice that you still get the same answer 1/2, but you only had to complete 1 step instead of 2.

 

Quick Tip: In some fractions, both the numerator and denominator end in zeros.  When this happens, you can use a short cut to reduce the fractions.  Cross out an equal number of zeros on the top and on the bottom of the fraction.  Then check to see if you can continue to reduce.

Example:

 
     

To practice reducing fractions, go to this website:  http://www.aplusmath.com/Worksheets/OnlineFractions.html
It will allow you to create a worksheet with easy, medium, hard, or very hard problems.  Or, try them all!

Problem Solving using Reducing Fractions:
A circle graph shows a total amount divided into parts.  Each part is a fraction of the total.  The following circle graph shows the expenses paid by Manual Diaz to operate his garage.  He has $1200 in total expenses.

Example Question:  What fraction of Manuel’s expenses is for rent?  Reduce the fraction to lowest terms.

 Step 1:  Write the total expenses, $1200, as the denominator of a fraction.
                   

Step 2:  Write the amount paid for rent, $200, as the numerator of the fraction.

Step 3:  Reduce.  Cross out an equal number of zeros on the end of the numerator and denominator.  Reduce again.

Answer: 

of Manuel’s expenses if for rent.  

 

Try some on your own.

1. Write the fraction that shows what part of Manuel’s expenses is for employee’s salaries.

2. What fraction shows the part that Manuel spends on gasoline and oil?

3. What fraction shows the part that Manuel spends on car parts?

4. What fraction shows the part that Manuel spends on parts and gas and oil together?  (Hint:  Add the two amounts together first.  Then write a fraction.)

5. What fraction shows the part that Manuel spends on rent and salaries together?

6. What fraction shows the total that Manuel spends on all of his expenses?

7. Manuel spends a total of $300 on salaries.  He pays each of his mechanics $150.  Write the fraction that shows what part of the total salary expense he pays for one mechanic’s salary.

8. Next year Manuel’s rent will go up to $250.  His total expenses will be $1250.  Write the fraction that shows what part of Manuel’s expenses he will have to pay for rent.

 

Resources:
Building Basic Skills in Mathematics. (1988). Contemporary Books, NTC/Contemporary Publishing Group, Inc.: Lincolnwood, IL.
Howett, Jerry. (2000). Number Power 2: A Real World Approach to Math. Contemporary Books, NTC/Contemporary Publishing Group, Inc.: Lincolnwood, IL.
Lassiter, Karen. (1993). Math Matters for Adults: Fractions. Steck-Vaughn Company: Austin, TX.
McClanahan, Susan D.; Green, Judith Andrews. (1996). Building Strategies: Mathematics. Steck-Vaughn Company: Austin, TX.