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.

Answers

Example 6.40

Solution

ⓐ
	$3$ is to $7$ as $15$ is to $35 .$
Write as a proportion.	$\dfrac{3}{7}=\dfrac{15}{35}$

ⓑ
	$5$ hits in $8$ at-bats is the same as $30$ hits in $48$ atbats.
Write each fraction to compare hits to at-bats.	$\dfrac{\text { hits }}{\text { at-bats }}=\dfrac{\text { hits }}{\text { at-bats }}$
Write as a proportion.	$\dfrac{5}{8}=\dfrac{30}{48}$

ⓒ
	$\$ 1.50$ for $6$ ounces is equivalent to $\$ 2.25$ for $9$ ounces.
Write each fraction to compare dollars to ounces.	$\dfrac{\$}{\text { ounces }}=\dfrac{\$}{\text { ounces }}$
Write as a proportion.	$\dfrac{1.50}{6}=\dfrac{2.25}{9}$

Example 6.41

Solution

To determine if the equation is a proportion, we find the cross products. If they are equal, the equation is a proportion.

ⓐ
	$\dfrac{4}{9}=\dfrac{12}{28}$
Find the cross products.	$28 \cdot 4=112$ $9 \cdot 12=108$

Since the cross products are not equal, $28 \cdot 4 \neq 9 \cdot 12$ , the equation is not a proportion.

ⓑ
	$\dfrac{17.5}{37.5}=\dfrac{7}{15}$
Find the cross products.	$15 \cdot 17.5=262.5 \quad 37.5 \cdot 7=262.5$

Since the cross products are equal, $15 \cdot 17.5=37.5 \cdot 7$ , the equation is a proportion.

Example 6.42

Solution

		$\dfrac{x}{63}=\dfrac{4}{7}$
To isolate $x$ , multiply both sides by the LCD, $63$ .		$\color{red}{63}\left(\dfrac{x}{63}\right)=\color{red}{63}\left(\dfrac{4}{7}\right)$
Simplify.		$x=\dfrac{9 \cdot \not{7} \cdot 4}{\not{7}}$
Divide the common factors.		$x=36$
Check: To check our answer, we substitute into the original proportion.
	$\dfrac{x}{63}=\dfrac{4}{7}$
Substitute $x=\color{red}{36}$	$\dfrac{\color{red}{36}}{63} \stackrel{?}{=} \dfrac{4}{7}$
Show common factors.	$\dfrac{4 \cdot 9}{7 \cdot 9} \stackrel{?}{=} \dfrac{4}{7}$
Simplify.	$\dfrac{4}{7}=\dfrac{4}{7}✓$

Example 6.43

Solution

Notice that the variable is in the denominator, so we will solve by finding the cross products and setting them equal.


Find the cross products and set them equal.		$4 \cdot 144=a \cdot 9$
Simplify.		$576=9 a$
Divide both sides by 9.		$\dfrac{576}{9}=\dfrac{9 a}{9}$
Simplify.		$64=a$
Check your answer.
	$\dfrac{144}{a}=\dfrac{9}{4}$
Substitute $a=\color{red}{64}$	$\dfrac{144}{64} \stackrel{?}{=} \dfrac{9}{4}$
Show common factors..	$\dfrac{9 \cdot 16}{4 \cdot 16} \stackrel{?}{=} \dfrac{9}{4}$
Simplify.	$\dfrac{9}{4}=\dfrac{9}{4}✓$

Another method to solve this would be to multiply both sides by the LCD, $4a$ Try it and verify that you get the same solution.

Example 6.44

Solution

Find the cross products and set them equal.
		$y \cdot 52=91(-4)$
Simplify.		$52 y=-364$
Divide both sides by $52$ .		$\dfrac{52 y}{52}=\dfrac{-364}{52}$
Simplify.		$y=-7$
Check:
	$\dfrac{52}{91}=\dfrac{-4}{y}$
Substitute $y=\color{red}{-7}$	$\dfrac{52}{91} \stackrel{?}{=} \dfrac{-4}{\color{red}{-7}}$
Show common factors.	$\dfrac{13 \cdot 4}{13 \cdot 4} \stackrel{?}{=} \dfrac{-4}{\color{red}{-7}}$
Simplify.	$\dfrac{4}{7}=\dfrac{4}{7}\checkmark$