7.2 Proportions

Section overview

• Ratios, Rates, and Proportions
• Finding the Missing Factor in a Proportion
• Proportions Involving Rates

Ratios, rates, and proportions

Ratio, rate

Proportion

Sample set a

Write or read each proportion.

Practice set a

Write or read each proportion.

Finding the missing factor in a proportion

Many practical problems can be solved by writing the given information as propor­tions. Such proportions will be composed of three specified numbers and one unknown number. It is customary to let a letter, such as $x$ , represent the unknown number. An example of such a proportion is

$\frac{x}{4}=\frac{\text{20}}{\text{16}}$

This proportion is read as " $x$ is to 4 as 20 is to 16."

There is a method of solving these proportions that is based on the equality of fractions. Recall that two fractions are equivalent if and only if their cross products are equal. For example,

Notice that in a proportion that contains three specified numbers and a letter representing an unknown quantity, that regardless of where the letter appears, the following situation always occurs.

$\underbrace{\text{(number)}\cdot \text{(letter)}=\text{(number)}\cdot \text{(number)}}$

We recognize this as a multiplication statement. Specifically, it is a missing factor statement. (See [link] for a discussion of multiplication statements.) For example,

$\begin{array}{cc}\frac{x}{4}=\frac{\text{20}}{\text{16}}& \text{means that }\text{16}\cdot x=4\cdot \text{20}\\ \frac{4}{\mathrm{x}}=\frac{16}{20}& \text{means that }4\cdot 20=16\cdot x\\ \frac{5}{4}=\frac{\mathrm{x}}{16}& \text{means that }5\cdot 16=4\cdot x\\ \frac{5}{4}=\frac{20}{x}& \text{means that }5\cdot x=4\cdot 20\end{array}$

Each of these statements is a multiplication statement. Specifically, each is a missing factor statement. (The letter used here is $x$ , whereas $M$ was used in [link] .)

Finding the missing factor in a proportion

Sample set b

Find the unknown number in each proportion.