Equivalent Ratios

The ratios 3:2 and 6:4 are called equivalent ratios. They represent the same amount so they are equal.

3:2 = 6:4

An equivalent ratio is determined by multiplying or dividing both terms of a ratio by the same number.

Here both the terms are multiplied by 2:

3 x 2 = 6

2 x 2 = 4

Here is a whole series of equivalent ratios:

3:2 = 6:4 = 9:6 = 12:8 = 15:10 = 18:12

Look at that series above again. Can you see how 6:4 = 15:10?

What could you multiply the terms of 6:4 by to get 15:10? (it's a fractional number)

Because they are both equal to 3:2 they are also equivalent to each other.

Simplest Form

Like fractions, ratios can be expressed in their simplest form by finding the greatest common factor (the greatest number that can be divided evenly into both the numerator and denominator).

18:12 as a fraction is 18/12 but it isn't in simplest form.

First find the factors of each term, then find a common factor.

Factors of 18: 1, 2, 3, 4, 6, 9, 18

Factors of 12: 1, 2, 3, 4, 6, 12

The greatest common factor of 18 and 12 is 6.

Divide both 18 and 12 by 6

18:12 = 3:2

Example 1

In a box of chocolates there are 3 truffles to every 2 toffees.

a. Express this as a ratio.

b. If the box of chocolates has 10 chocolates how many truffles are there?

c. If the box of chocolates has 25 chocolates how many toffees are there?

Solution

a. 3:2

b. We need to find an equivalent ratio where the 2 terms add up to 10.

A ratio equivalent to 3:2 would be 6:4. This would make 10 chocolates so there would be 6 truffles (and 4 toffees).

c. We need to find an equivalent ratio where the 2 terms add up to 25.

Some equivalent ratios are 6:4, 12:8, 15:10

The two terms of the ratio 15:10 add up to 25.

So the number of toffees would be 10 (with 15 truffles).

Example 2

A class of 28 students had some blue-eyed students and some brown-eyed students. The ratio of blue-eyed to brown-eyed was 4:3. How many blue-eyed students were in the class?

Solution

Since the ratio is 4:3 and the total is 28, a ratio that is equivalent to 4:3, where the two terms add to 28 must be found.

Here's one equivalent ratio: 8:6 ; the sum is 8 + 6 = 14

Here's another equivalent ratio: 12:9, the sum is 21

Another one: 16:12, the sum is 28

So the number of blue-eyed students would be 16.

Another way to consider this is, since the ratio is 4:3, the total must be a multiple of 7 (4 + 3). 28 is 7 x 4, so if each term is multiplied by 4 the result should be the same.

4 (x 4) : 3 (x 4) = 16:12