Math Core

A smoothie recipe calls for 2 cups of strawberries for every 3 cups of yogurt. That "for every" is a ratio. Ratios let you compare two amounts, and they show up in recipes, maps, sports stats and prices.

Comparing two quantities

Look at this bag of marbles: 🔵🔵🔵🔴🔴

There are 3 blue marbles and 2 red marbles. You can compare them in different ways:

Blue to red: 3 blue for every 2 red. This compares one part to another part.
Red to blue: 2 red for every 3 blue. The order flipped, so the ratio flipped too.
Blue to all marbles: 3 blue out of 5 marbles. This compares a part to the whole.

Definition

Ratio

A ratio is a comparison of two quantities. The ratio of $a$ to $b$ can be written three ways:

a \text{ to } b \qquad\qquad a : b \qquad\qquad \dfrac{a}{b}

All three are read " $a$ to $b$ ." The numbers in a ratio are called its terms.

So the ratio of blue marbles to red marbles is 3 to 2, or $3 : 2$ , or $\dfrac{3}{2}$ .

Order matters

The words in the question tell you which number goes first. "The ratio of cats to dogs" puts cats first. "The ratio of dogs to cats" puts dogs first.

Worked example: Writing ratios

A class has 12 girls and 15 boys. Write each ratio.

Girls to boys
Boys to girls
Boys to all students

Solutions.

Girls come first: $12 : 15$ .
Boys come first: $15 : 12$ .
There are $12 + 15 = 27$ students in all, so boys to all students is $15 : 27$ .

Common mistake

The most common mistake is writing the numbers in the wrong order. Before you write a ratio, underline the first quantity named in the question. Its number goes first. $12 : 15$ and $15 : 12$ describe different situations.

Part-to-part and part-to-whole

A part-to-part ratio compares two groups inside a set, like girls to boys. A part-to-whole ratio compares one group to the whole set, like girls to all students.

To get a part-to-whole ratio from a part-to-part ratio, add the parts to find the whole.

Worked example: Finding the whole

A fruit bowl has apples and pears in the ratio $4 : 5$ . What is the ratio of apples to all the fruit in the bowl?

For every 4 apples there are 5 pears. That makes a group of $4 + 5 = 9$ pieces of fruit. In each group, 4 are apples.

So the ratio of apples to all fruit is $4 : 9$ .

"For every" and "for each"

Ratio language often uses the words for every, for each or per. A sign that says "3 stickers for every 1 book" means the ratio of stickers to books is $3 : 1$ .

A ratio also tells you how the amounts grow together. If you read 4 books, you get 4 groups of 3 stickers, which is 12 stickers.

Worked example: Using a ratio

At a pet shelter, the ratio of dogs to cats is $2 : 7$ . There are 6 dogs. How many cats are there?

The ratio says 2 dogs for every 7 cats. Six dogs is 3 groups of 2 dogs, because $6 \div 2 = 3$ . Each group comes with 7 cats:

3 \times 7 = 21

There are 21 cats. Check: $6 : 21$ is 3 groups of $2 : 7$ . ✓

Tip

A ratio is not the same as the actual amounts. A ratio of $2 : 7$ could mean 2 dogs and 7 cats, or 6 dogs and 21 cats, or 20 dogs and 70 cats. It only tells you the comparison.

Practice

Practice 1

A garden has 7 tulips and 4 roses. What is the ratio of roses to tulips?

Practice 2

A team won 9 games and lost 5 games. There were no ties. What is the ratio of games won to all games played?

Practice 3

Which situation is described by a part-to-whole ratio?

Practice 4

A recipe uses 3 eggs for every 1 cup of milk. How many eggs are needed with 4 cups of milk?

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

Practice 5

The ratio of blue paint to yellow paint in a green mix is $5 : 2$ . Mia uses 6 cups of yellow paint. How many cups of blue paint does she need?

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

Practice 6

In a bag, the ratio of red beads to white beads is $3 : 8$ . What is the ratio of white beads to all the beads?

Practice 7

A club has boys and girls in the ratio $2 : 3$ . There are 12 boys. How many members does the club have in all?

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