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Equivalent Ratios

Determine whether or not two ratios are equal.

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Equivalent Ratios

Let's Think About It

On the first day of school, the seventh grade Language arts class started a reading contest to see how many books they could read during the year. Kayla and Torrey have been participating in the challenge since day one. By October, eight weeks into school, Kayla had already finished 6 books. Torrey said she was reading 3 books every four weeks. The girls concluded that at the present point in time, they had both read the same number of books. Are they correct?

In this concept, you will learn how to work with equivalent ratios.


A ratio is the relationship, or comparison, of a part to the whole.

A ratio can be expressed:

in words, 3 to 4, or 3 of 4

with a colon, 3:4

as a fraction,34

The easiest way to work with ratios is by thinking of them as fractions.

For example, think about the three lights on a traffic signal.

One of three is red, 13   

2:3 are primary colors,23

 33are colors of the rainbow

Equivalent means equal.

Equivalent ratios show the same, or equal, relationship between two quantities.

Using the traffic lights again, suppose there are two signals on one street and one on another.

The first street has 2 of 6 lights that are red. The second has 1 of 3.

The ratio of red lights to the total lights per signal is still the same.

Writing ratios in fractional form and simplifying can determine whether or not they are equivalent.

Here is an example:

Determine if the two ratios are equivalent.

 23 and 1015

First, recognize that 23 is in its simplest form.

Next, simplify 1015.


Then, compare the two fractions, or ratios.


The answer is that the two ratios are equal, or equivalent.

Another way to check is to change 23into a fraction with 15 as the denominator.

First, write an equation.


Next, recognize that 3×5=15 and multiply both the numerator and denominator times 3. Remember that whatever is done to one term must also be done to the other.


Then, compare the ratios.


The answer is that the two ratios are equivalent.

The last way to determine whether or not two ratios are equivalent is to cross multiply.

First, write an equation that assumes the fractions are equal.


Next, multiply the numerator of one fraction times the denominator of the other and vice versa.


Then, compare the two products.


The products are equal.

The answer is that the ratios are equivalent.

Guided Practice

Determine if  7:6 and 13:12 are equivalent ratios.

First, rewrite the ratios as fractions.

76 and 1312.

Next, recognize that 6 must be multiplied by 2 in order to equalize the denominators and perform the same operation to both terms.


Then, compare the ratios.


The answer is that the two ratios are not equivalent.


Example 1

Determine if the ratios are equivalent.

34 and 912

First, since a method is not specified, choose a method one. Determine if it is easiest to work with fourths, twelfths, or cross multiply. In this problem, it is easy to cross multiply.

Next, write an equation assuming the ratios are equal.


Then, cross multiply.


Next, compare.

36 = 36

The answer is that the ratios are equivalent.

Example 2

Change both fractions to 30ths to determine whether or not the ratios 56 and 2030 are equivalent.

First, write an equation.


Next, recognize that 6 must be multiplied times 5 to get 30, and multiply both terms times 5.


Then, compare the ratios expressed as 30ths.


The answer is that the ratios are not equivalent.

Example 3

Are the ratios 4 to 5 and 8 to 10 equivalent?

First, write the ratios as fractions.

45 and 810

Next, determine a method. In this case, cross multiplication.


Then, compare.

40 = 40

The answer is that the ratios are equivalent.

Follow Up

Remember Kayla and Torrey and the reading challenge?

Kayla had read 6 books in 8 weeks and Torrey had read 3 books every four weeks. Had they read the same number of books? Are the ratios equivalent?

First, write the ratios as fractions.

Kayla 6books8weeks or 68

Torrey 3books4weeks or 34

Next, determine a method, cross multiplication, and write an equation.


Then cross multiply and compare.

24 = 24

The answer is that the girls have read the same number of books. The ratios are equivalent.

Video Review

Explore More

Determine whether each of the following ratio pairs is equal. Write yes if they are equal and no if they are not equal.

1. 12 and 612

2. 38 and 14

3. 67 and 23

4. 67 and 1214

5. 23 and 1015

6. 1721 and 67

7. 2448 and 1224

8. 1618 and 3238

9. 945 and 19

10. 46 and 4466

11. 69 and 46

12. 1416 and 2024

13. 1216 and 2432

14. 2448 and 12

15. 8496 and 34


Equivalent Ratios

Equivalent Ratios

Equivalent ratios are ratios that can each be simplified to the same ratio.


Terms are groups of variables and constants not separated by addition or subtraction.

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