Math Core

Two friends mix orange paint. One uses 2 cups of red and 6 cups of yellow. The other uses 5 cups of red and 15 cups of yellow. Will their oranges match? To answer, you need to know when two ratios describe the same comparison.

What makes ratios equivalent

Every column of a ratio table shows the same ratio in a different size. Ratios like these are called equivalent.

Definition

Equivalent ratios

Two ratios are equivalent if you can multiply or divide both terms of one ratio by the same nonzero number to get the other. For example,

2 : 6 \;=\; 4 : 12 \;=\; 1 : 3

because $2 \times 2 = 4$ and $6 \times 2 = 12$ , and $2 \div 2 = 1$ and $6 \div 2 = 3$ .

This works just like equivalent fractions. $\dfrac{2}{6} = \dfrac{4}{12} = \dfrac{1}{3}$ .

Simplest form

A ratio is in simplest form when its terms are whole numbers with no common factor other than 1. To simplify, divide both terms by their greatest common factor.

Worked example: Simplifying ratios

Write each ratio in simplest form.

$10 : 25$
$18 : 24$
$36 : 9$

Solutions.

The greatest common factor of 10 and 25 is 5. $10 \div 5 = 2$ and $25 \div 5 = 5$ , so $10 : 25 = 2 : 5$ .
The greatest common factor of 18 and 24 is 6. $18 \div 6 = 3$ and $24 \div 6 = 4$ , so $18 : 24 = 3 : 4$ .
The greatest common factor is 9. $36 \div 9 = 4$ and $9 \div 9 = 1$ , so $36 : 9 = 4 : 1$ .

Tip

You don't have to find the greatest common factor right away. Divide by any common factor, then keep going. For $18 : 24$ : divide by 2 to get $9 : 12$ , then by 3 to get $3 : 4$ .

Are two ratios equivalent?

Back to the paint. Simplify both mixes:

$2 : 6 = 1 : 3$ (divide by 2)
$5 : 15 = 1 : 3$ (divide by 5)

Both simplify to $1 : 3$ , so the two oranges match.

Testing for equivalent ratios

Two ratios are equivalent if they simplify to the same ratio. You can also check whether the same number multiplies both terms of the first ratio to give the second.

Worked example: Which mix is stronger?

Juice A uses 3 scoops of powder for 4 cups of water. Juice B uses 5 scoops for 8 cups of water. Are the ratios equivalent? If not, which juice is stronger?

Make the cups of water the same. The least number that both 4 and 8 divide into is 8.

Juice A: $3 : 4 = 6 : 8$ (multiply by 2)
Juice B: $5 : 8$

With 8 cups of water, Juice A has 6 scoops and Juice B has 5 scoops. The ratios are not equivalent, and Juice A is stronger.

Common mistake

Adding or subtracting the same number does not make an equivalent ratio. $3 : 4$ is not equivalent to $4 : 5$ , even though you added 1 to each term. With 20 as the second term, $3 : 4 = 15 : 20$ but $4 : 5 = 16 : 20$ , so they are different.

Tape diagrams

A tape diagram shows a ratio as strips of equal-size boxes. It is a great tool when you know a total.

Worked example: Splitting a total

Kai and Rosa share 40 stickers in the ratio $3 : 5$ . How many stickers does each person get?

Draw 3 boxes for Kai and 5 boxes for Rosa. Every box holds the same number of stickers.

Person	Boxes
Kai	▭ ▭ ▭
Rosa	▭ ▭ ▭ ▭ ▭

There are $3 + 5 = 8$ boxes and 40 stickers, so each box holds $40 \div 8 = 5$ stickers.

Kai: $3 \times 5 = 15$ stickers
Rosa: $5 \times 5 = 25$ stickers

Check: $15 + 25 = 40$ , and $15 : 25 = 3 : 5$ . ✓

Worked example: Using a difference

A school has 4 soccer balls for every 7 basketballs. There are 18 more basketballs than soccer balls. How many soccer balls are there?

In the tape diagram, basketballs have $7 - 4 = 3$ more boxes than soccer balls. Those 3 boxes stand for 18 balls, so each box is $18 \div 3 = 6$ balls.

Soccer balls: $4 \times 6 = 24$ . (Basketballs: $7 \times 6 = 42$ , and $42 - 24 = 18$ . ✓)

Practice

Practice 1

What is $12 : 20$ in simplest form?

Practice 2

Find the missing number: $3 : 7 = 9 : \,?$

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 3

Which ratio is equivalent to $4 : 6$ ?

Practice 4

Find the missing number: $\,? : 32 = 3 : 16$

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 5

Recipe P uses 2 cups of oats for every 3 cups of milk. Recipe Q uses 3 cups of oats for every 5 cups of milk. Which is true?

Practice 6

Ana and Ben share 32 marbles in the ratio $3 : 5$ . How many marbles does Ben get?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 7

A bakery sells muffins and bagels in the ratio $2 : 7$ . It sold 30 more bagels than muffins. How many bagels did it sell?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.