What do the fractions in example 1 have in common?
Example 1
Each fraction in example 1 represents the same number. These fractions are equivalent.
Definition: Equivalent fractions are different fractions that name the same number.
Let's look at some more examples of equivalent fractions.
Example 2  


Twothirds is equivalent to foursixths. 
Example 3  


The fractions threefourths, sixeighths, and ninetwelfths are equivalent. 
What would happen if we did not have shapes such as circles and rectangles to refer to? Look at example 4 below.
Example 4
We need an arithmetic method for finding equivalent fractions.
Procedure:To find equivalent fractions, multiply the numerator AND denominator by the same nonzero whole number.
This procedure is used to solve example 4.
Example 4  


You can multiply the numerator and the denominator of a fraction by any nonzero whole number, as long as you multiply both by the same whole number! For example, you can multiply the numerator and the denominator by 3, as shown in part A above. But you cannot multiply the numerator by 3 and the denominator by 5. You can multiply the numerator and the denominator by 4, as shown in part B above. But you cannot multiply the numerator by 4 and the denominator by 2.
The numerator and the denominator of a fraction must be multiplied by the same nonzero whole number in order to have equivalent fractions. You may be wondering why this is so. In the last lesson, we learned that a fraction that has the same numerator and denominator is equal to one. This is shown below.
2/2 = 1 (Twohalves)
3/3 = 1 (Threethirds)
4/4 = 1 (Fourfourths)
5/5 = 1 (Fivefifths)
6/6 = 1 (Sixsixths)
So, multiplying a fraction by one does not change its value. Recapping example 4, we get:
Example 4  


Multiplying the numerator and the denominator of a fraction by the same nonzero whole number will change that fraction into an equivalent fraction, but it will not change its value. Equivalent fractions may look different, but they have the same value. Let's look at some more examples of equivalent fractions.
Example 5 
Example 6  
In example 6, the fraction given in part a is a proper fraction; whereas the fractions given in parts b and c are improper fractions. Note that the procedure for finding equivalent fractions is the same for both types of fractions. Looking at each part of example 6, the answers vary, depending on the nonzero whole number chosen. However, the equivalent fractions found in each part all have the same value.
Example 7  
Write the fraction fivesixths as an equivalent fraction with a denominator of 24.  

In example 7, we multiplied the numerator AND the denominator by 4.
Example 8  
Write the fraction twosevenths as an equivalent fraction with a denominator of 21.  

In example 8, we multiplied the numerator AND the denominator by 3.
Example 9  
Write the fraction threeeighths as an equivalent fraction with a numerator of 15.  

In example 9, we multiplied the numerator AND the denominator by 5.
We can now redefine the terms fraction and equivalent fraction as follows:
Summary: Equivalent fractions are different fractions that name the same number. The numerator and the denominator of a fraction must be multiplied by the same nonzero whole number in order to have equivalent fractions.
Exercises
In Exercises 1 through 5, click once in an ANSWER BOX and type in your answer; then click ENTER. After you click ENTER, a message will appear in the RESULTS BOX to indicate whether your answer is correct or incorrect. To start over, click CLEAR. Note: To write the fraction twothirds, enter 2/3 into the form.
1.  Are the fractions threefourths and fourteensixteenths equivalent (Yes or No)? 
2.  Write the fraction twothirds as an equivalent fraction with a denominator of 18. (Enter the entire fraction into the form.) 
3.  Write the fraction eightfifths as an equivalent fraction with a numerator of 40. (Enter the entire fraction into the form.) 
4.  Write the fraction sevensevenths as an equivalent fraction with a denominator of 19. (Enter the entire fraction into the form.) 
5.  Write the fraction elevensevenths as an equivalent fraction with a denominator of 56. (Enter the entire fraction into the form.) 