Math Core

If $3$ cups of flour make $24$ cookies, how much flour do you need for $60$ ? Questions like this, where two quantities grow together at a fixed rate, show up in recipes, maps, unit prices, model building and science. An equation between two ratios, called a proportion, solves all of them the same way.

Ratios and rates

A ratio compares two quantities by division. The ratio of $12$ boys to $15$ girls can be written as $12$ to $15$ , $12 : 15$ , or $\dfrac{12}{15}$ . Like fractions, ratios can be simplified: $\dfrac{12}{15} = \dfrac{4}{5}$ , so there are $4$ boys for every $5$ girls.

When the two quantities have different units, such as miles and hours, the ratio is called a rate. A unit rate has a denominator of $1$ : $4.80 for $6$ cans is $\dfrac{4.80}{6} = 0.80$ dollars per can. Unit rates make comparisons easy, because you are comparing "per one" to "per one."

Proportions

Definition

Proportion

A proportion is an equation stating that two ratios are equal:

\frac{a}{b} = \frac{c}{d} \qquad (b \ne 0,\ d \ne 0).

For example, $\dfrac{3}{24} = \dfrac{5}{40}$ is a true proportion, because both ratios simplify to $\dfrac{1}{8}$ .

The cross products property

To solve a proportion, you could clear the fractions the way you did in multi-step equations: multiply both sides by both denominators, $bd$ .

bd \cdot \frac{a}{b} = bd \cdot \frac{c}{d} \quad\Longrightarrow\quad ad = bc.

On the left the $b$ 's cancel; on the right the $d$ 's cancel. What remains is a shortcut you can use every time.

Cross products property

If $\dfrac{a}{b} = \dfrac{c}{d}$ , then $ad = bc$ .

The products $ad$ and $bc$ are called cross products. In a true proportion they are equal. Setting them equal turns a proportion into an equation with no fractions.

Worked example: A basic proportion

Solve $\dfrac{x}{6} = \dfrac{10}{15}$ .

\begin{aligned} 15x &= 6 \cdot 10 && \text{cross products} \\ 15x &= 60 \\ x &= 4 \end{aligned}

Check: $\dfrac{4}{6} = \dfrac{2}{3}$ and $\dfrac{10}{15} = \dfrac{2}{3}$ . ✓

Worked example: The variable in a denominator

Solve $\dfrac{9}{n} = \dfrac{12}{20}$ .

Cross products work no matter where the variable is:

12n = 9 \cdot 20 = 180 \quad\Longrightarrow\quad n = 15.

Check: $\dfrac{9}{15} = \dfrac{3}{5}$ and $\dfrac{12}{20} = \dfrac{3}{5}$ . ✓

When the numerators or denominators are expressions, put parentheses around them before multiplying. The result is an equation with variables on both sides, which you already know how to solve.

Worked example: Expressions in a proportion

Solve $\dfrac{x + 2}{5} = \dfrac{x - 1}{2}$ .

\begin{aligned} 2(x + 2) &= 5(x - 1) && \text{cross products} \\ 2x + 4 &= 5x - 5 && \text{distribute} \\ 9 &= 3x && \text{subtract } 2x \text{, add } 5 \\ 3 &= x \end{aligned}

Check: $\dfrac{3 + 2}{5} = 1$ and $\dfrac{3 - 1}{2} = 1$ . ✓

Common mistake

Cross products only work when the equation is one fraction equal to one fraction. In $\dfrac{x}{3} + 1 = \dfrac{4}{5}$ , you can't cross multiply, because the left side is a sum. Clear fractions by multiplying every term by the LCD instead (or subtract $1$ first to get a proportion).

Setting up proportions from words

The hardest part of a proportion problem is usually the setup. Write both ratios in the same order: if the first ratio is $\dfrac{\text{flour}}{\text{cookies}}$ , the second must also be $\dfrac{\text{flour}}{\text{cookies}}$ .

For the cookie question at the start, let $c$ be the cups of flour needed:

\frac{3 \text{ cups}}{24 \text{ cookies}} = \frac{c \text{ cups}}{60 \text{ cookies}} \quad\Longrightarrow\quad 24c = 180 \quad\Longrightarrow\quad c = 7.5.

You need $7.5$ cups of flour.

Worked example: A map scale

On a map, $2$ cm represents $15$ km. Two towns are $9$ cm apart on the map. How far apart are they in real life?

Write each ratio as $\dfrac{\text{map cm}}{\text{real km}}$ , and let $d$ be the real distance:

\frac{2}{15} = \frac{9}{d} \quad\Longrightarrow\quad 2d = 135 \quad\Longrightarrow\quad d = 67.5.

The towns are $67.5$ km apart.

Tip

Sanity-check the size of your answer. More cookies should need more flour; a longer map distance should mean a longer real distance. If your answer moves the wrong way, the ratios were probably set up in different orders.