Applications of Proportions

Read this text. Pay attention to the gray box labeled Cross Products of a Proportion, which shows the simple way to solve for an unknown value in a proportion. Complete the practice problems and check your answers.

Use the Definition of Proportion

In the section on Ratios and Rates we saw some ways they are used in our daily lives. When two ratios or rates are equal, the equation relating them is called a proportion.

PROPORTION

A proportion is an equation of the form

$\dfrac{a}{b}=\dfrac{c}{d}$ , where

$b \neq 0, d \neq 0$ .
The proportion states two ratios or rates are equal. The proportion is read "

$a$ is to

$b$ , as

$c$ is to

$d$ ".

The equation $\dfrac{1}{2}=\dfrac{4}{8}$ is a proportion because the two fractions are equal. The proportion $\dfrac{1}{2}=\dfrac{4}{8}$ is read "1 is to 2 as 4 is to $8$ ".

If we compare quantities with units, we have to be sure we are comparing them in the right order. For example, in the proportion $\dfrac{20 \text { students }}{1 \text { teacher }}=\dfrac{60 \text { students }}{3 \text { teachers }}$ we compare the number of students to the number of teachers. We put students in the numerators and teachers in the denominators.

Look at the proportions $\dfrac{1}{2}=\dfrac{4}{8}$ and $\dfrac{2}{3}=\dfrac{6}{9}$ . From our work with equivalent fractions we know these equations are true. But how do we know if an equation is a proportion with equivalent fractions if it contains fractions with larger numbers?

To determine if a proportion is true, we find the cross products of each proportion. To find the cross products, we multiply each denominator with the opposite numerator (diagonally across the equal sign). The results are called a cross products because of the cross formed. The cross products of a proportion are equal.

CROSS PRODUCTS OF A PROPORTION

For any proportion of the form $\dfrac{a}{b}=\dfrac{c}{d}$ , where $b \neq 0, d \neq 0$ , its cross products are equal.

Cross products can be used to test whether a proportion is true. To test whether an equation makes a proportion, we find the cross products. If they are the equal, we have a proportion.

Solve Proportions

To solve a proportion containing a variable, we remember that the proportion is an equation. All of the techniques we have used so far to solve equations still apply. In the next example, we will solve a proportion by multiplying by the Least Common Denominator (LCD) using the Multiplication Property of Equality.

When the variable is in a denominator, we'll use the fact that the cross products of a proportion are equal to solve the proportions.

We can find the cross products of the proportion and then set them equal. Then we solve the resulting equation using our familiar techniques.

Source: Rice University, https://openstax.org/books/prealgebra/pages/6-5-solve-proportions-and-their-applications
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