# Equivalent ratios

Equivalent ratios are ratios that describe the same rate or make the same comparison. They are a result of the fact that ratios are scalable, meaning that they can be multiplied or divided by a constant to yield the same relationship, expressed in larger or smaller quantities.

For example, there are 2 circles and 3 squares in the figure below. The ratio of circles to squares can be written as 2:3.

If there were twice as many squares, and twice as many circles, the ratio of circles to squares could be written as 2(2):3(2) = 4:6. Although there are more circles and squares, the ratio of circles to squares remains constant, so 2:3 and 4:6 are equivalent ratios.

Example

Samantha wants to buy and stock an aquarium. She saves $1 of each $5 she earns.

Saved | : | $1 | $2 | $3 | $4 | $5 |

Earned | : | $5 | $10 | $15 | $20 | $25 |

1:5, 2:10, 3:15, 4:20, and 5:25 are all equivalent ratios. These ratios can also be written as:

, , , , and .

Equivalent ratios are related to proportions in that proportions are a statement that two ratios are equal, making the ratios involved in any proportion, equivalent ratios.

Example

Given the ratios, 15:1 and 30:2, which represent how many miles you traveled on your bike in 1 and 2 hours, are proportional, test whether the ratios are equivalent ratios:

(15×2):(1×2) = 30:2

(30÷2):(2÷2) = 15:1

Since either of the ratios can be scaled by a factor of 2 to equal the other, they are equivalent ratios.

Equivalent ratios (and proportions) can be used in many everyday situations. One such example is baking. Below is a slightly more involved word problem involving equivalent ratios.

Example

James usually uses a cookie recipe that makes enough cookies for his family of 4. There are 20 students in his daughter's class. If he wants to scale the recipe up to provide cookies for his daughter's class, assuming that the amount of flour and sugar he will need remains proportional to the original recipe, and he usually uses a ratio of flour to sugar of 3:2 cups, how many cups of flour and sugar will he need?

First, determine how many more students there are than people in James' family:

4 × ? = 20

4 × 5 = 20

There are 5 times more students than people in James' family.

Since the amount of flour and sugar he will need is proportional, we can just multiply the original ratio by 5 to find the amount that James will need to feed the class.

3(5):2(5) = 15:10

James would need 15 cups of flour and 10 cups of sugar to make enough cookies for his daughter's class.