Applications of Proportions

Exercises

(b)
	$5$ 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.	$\dfrac{5}{8}=\dfrac{30}{48}$

(c)
	$\$ 1.50$ 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.	$\dfrac{1.50}{6}=\dfrac{2.25}{9}$

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

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

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

		$\dfrac{x}{63}=\dfrac{4}{7}$
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.	$\dfrac{4}{7}=\dfrac{4}{7}✓$

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.

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.	$\dfrac{4}{7}=\dfrac{4}{7}\checkmark$