Writing and Using Ratios

A ratio is a comparison of quantities. For example, for most mammals, the ratio of legs to noses is $4:1,$ but for humans, the ratio of legs to noses is $2:1.$

Ratios can be written using the word "to," a colon, or a fraction.

For example, in a group of 3 girls and 5 boys, the ratio of girls to boys can be written as $3 \text{ to } 5,$ $3:5,$ or $\dfrac{3}{5}.$

A bowl contains 3 apples, 5 oranges, and 9 kiwi. What is the ratio of kiwi to apples?

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ANSWER

There are 9 kiwi and 3 apples, so the ratio of kiwi to apples is $9:3,$ or in simplified form, $3:1.$

A bowl contains 3 apples, 5 oranges, and 9 kiwi. What is the ratio of oranges to total pieces of fruit?

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ANSWER

There are 5 oranges and $3+5+9=17$ total pieces of fruit, so the ratio of oranges to total pieces of fruit is $5:17.$

Ratios are often easiest to use if written in fraction form. To find an unknown value in a ratio, we can use two equivalent ratios.

For example, if the ratio $3:4$ is equivalent to the ratio $18:x,$ then $\frac{3}{4}=\frac{18}{x}$ and $x=24$ because the values in the fraction on the right are six times greater than the corresponding values in the fraction on the left.

If $6: 15 = 10 : x,$ what is $x\,?$

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ANSWER

Expressing the ratios as fractions, we get $\frac{6}{15} = \frac{10}{x} .$

The fraction $\frac{6}{15}$ simplifies to $\frac{2}{5},$ so now we have $\frac{2}{5} = \frac{10}{x} .$ The values in the ratio on the right are five times greater than the values in the ratio on the left, so $x=25.$

We can also rewrite the ratios as fractions and cross multiply to solve. $$\begin{aligned} \frac{6}{15} &= \frac{10}{ x} \\ 6x &= (10)(15) \\ 6x &= 150 \\ x &= 25.\end{aligned}$$

If Calvin paid $5 for 7 pencils, how much would he pay for 56 pencils?

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ANSWER

Let $x$ be the price of $56$ pencils. Since the price of a single pencil does not change, we have $$\begin{aligned} \frac{\$5}{7} &= \frac{x}{56} \\ 7x &= \$5 \times 56 \\ x &= \$40. \end{aligned}$$ Hence, Calvin would pay $\$40$ for $56$ pencils.

The ratio of Alice's pay to Bob's pay is $\frac{5}{4}$. The ratio of Bob's pay to Charlie's pay is $10:9$. If Alice is paid $75, how much is Charlie paid?

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ANSWER

Since the ratio of Alice's pay to Bob's pay is $5:4$, Bob's pay must be $b$, where $\frac{5}{4}=\frac{75}{b}$. Cross-multiplying by the denominators, we get $5b = 4(75)$, so $b = 60$.

Continuing in the same way, we compare Bob to Charlie: $$\frac{10}{9}=\frac{60}{c} \implies 10c = 9(60) \implies c = 54.$$ Thus, Charlie is paid $54. $_\square$