Equivalent Fractions

rectangle three fourths blue nonroutine

What do the fractions in example 1 have in common?

Example 1

circle one half red1over2.gif

circle two fourths blue2over4.gif

circle three sixths yellow3over6.gif

circle four eighths orange eq4over8.gif

Each fraction in example 1 represents the same number. These fractions are equivalent.

Definition: Equivalent fractions are different fractions that name the same number.

Equivalent fractions

Let's look at some more examples of equivalent fractions.

Example 2
rectangle two thirds small pink 2over3.gif
rectangle four sixths yellow 4over6.gif
eq example2a
Two-thirds is equivalent to four-sixths.
Example 3

circle three fourths blue eq

circle six eighths orange eq

circle nine twelfths lavendar

3over4.gif

 

6over8.gif

 

9over12.gif

eq example3c
The fractions three-fourths, six-eighths, and nine-twelfths 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

eq_example4.gif

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
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eq_example4a.gif
eq_example4b.gif
eq_example4_solution.gif

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 (Two-halves)

circle two halves red

3/3 = 1 (Three-thirds)

circle three thirds pink

4/4 = 1 (Four-fourths)

circle four fourths blue

5/5 = 1 (Five-fifths)

circle five fifths green

6/6 = 1 (Six-sixths)

circle six sixths yellow

So, multiplying a fraction by one does not change its value. Recapping example 4, we get:

Example 4
eq example4
eq example4 explain
eq example4 explain
eq example4 solution

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
eq example5
eq example5 explain
Example 6
eq example6

eq_example6a.gif

eq_example6b.gif

eq_example6c.gif

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 five-sixths as an equivalent fraction with a denominator of 24.
Step 1:  eq_example7.gif
Step 2:  eq_example7a.gif
Step 3:  eq_example7b.gif
Step 4:  eq_example7c.gif

In example 7, we multiplied the numerator AND the denominator by 4.

Example 8
Write the fraction two-sevenths as an equivalent fraction with a denominator of 21.
Step 1:  eq_example8.gif
Step 2:  eq_example8a.gif
Step 3:  eq_example8b.gif
Step 4:  eq_example8c.gif

In example 8, we multiplied the numerator AND the denominator by 3.

Example 9
Write the fraction three-eighths as an equivalent fraction with a numerator of 15.
Step 1:  eq_example9.gif
Step 2:  eq_example9a.gif
Step 3:  eq_example9b.gif
Step 4:  eq_example9c.gif

In example 9, we multiplied the numerator AND the denominator by 5.

We can now redefine the terms fraction and equivalent fraction as follows:

define_fractions_a_over_b.gif

define_equiv_ka_over_kb.gif


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 two-thirds, enter 2/3 into the form.

1. Are the fractions three-fourths and fourteen-sixteenths equivalent (Yes or No)?
 
ANSWER BOX:   

RESULTS BOX: 

2. Write the fraction two-thirds as an equivalent fraction with a denominator of 18. (Enter the entire fraction into the form.)
 
ANSWER BOX:   

RESULTS BOX: 

3. Write the fraction eight-fifths as an equivalent fraction with a numerator of 40. (Enter the entire fraction into the form.)
 
ANSWER BOX:   

RESULTS BOX: 

4. Write the fraction seven-sevenths as an equivalent fraction with a denominator of 19. (Enter the entire fraction into the form.)
 
ANSWER BOX:   

RESULTS BOX: 

5. Write the fraction eleven-sevenths as an equivalent fraction with a denominator of 56. (Enter the entire fraction into the form.)
 
ANSWER BOX:   

RESULTS BOX: