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.

In the next video we show another example of how to solve a proportion equation using the LCD.

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.

example

Solve: [latex]{\Large\frac{144}{a}}={\Large\frac{9}{4}}[/latex]

Show Solution

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.		[latex]4\cdot144=a\cdot9[/latex]
Simplify.		[latex]576=9a[/latex]
Divide both sides by [latex]9[/latex].		[latex]{\Large\frac{576}{9}}={\Large\frac{9a}{9}}[/latex]
Simplify.		[latex]64=a[/latex]
Check your answer.
	[latex]{\Large\frac{144}{a}}={\Large\frac{9}{4}}[/latex]
Substitute [latex]a=\color{red}{64}[/latex]	[latex]{\Large\frac{144}{\color{red}{64}}}\stackrel{?}{=}{\Large\frac{9}{4}}[/latex]
Show common factors..	[latex]{\Large\frac{9\cdot16}{4\cdot16}}\stackrel{?}{=}{\Large\frac{9}{4}}[/latex]
Simplify.	[latex]{\Large\frac{9}{4}}={\Large\frac{9}{4}}\quad\checkmark[/latex]

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

The following video shows an example of how to solve a similar problem by using the LCD.

example

Solve: [latex]{\Large\frac{52}{91}}={\Large\frac{-4}{y}}[/latex]

Show Solution

Solution

Find the cross products and set them equal.
		[latex]y\cdot52=91(-4)[/latex]
Simplify.		[latex]52y=-364[/latex]
Divide both sides by [latex]52[/latex].		[latex]{\Large\frac{52y}{52}}={\Large\frac{-364}{52}}[/latex]
Simplify.		[latex]y=-7[/latex]
Check:
	[latex]{\Large\frac{52}{91}}={\Large\frac{-4}{y}}[/latex]
Substitute [latex]y=\color{red}{-7}[/latex]	[latex]{\Large\frac{52}{91}}\stackrel{?}{=}{\Large\frac{-4}{\color{red}{-7}}}[/latex]
Show common factors.	[latex]{\Large\frac{13\cdot4}{13\cdot4}}\stackrel{?}{=}{\Large\frac{-4}{\color{red}{-7}}}[/latex]
Simplify.	[latex]{\Large\frac{4}{7}}={\Large\frac{4}{7}}\quad\checkmark[/latex]

Solve Applications Using Proportions

The strategy for solving applications that we have used earlier in this chapter, also works for proportions, since proportions are equations. When we set up the proportion, we must make sure the units are correct—the units in the numerators match and the units in the denominators match.

example

When pediatricians prescribe acetaminophen to children, they prescribe [latex]5[/latex] milliliters (ml) of acetaminophen for every [latex]25[/latex] pounds of the child’s weight. If Zoe weighs [latex]80[/latex] pounds, how many milliliters of acetaminophen will her doctor prescribe?

Show Solution

Solution

Identify what you are asked to find.	How many ml of acetaminophen the doctor will prescribe
Choose a variable to represent it.	Let [latex]a=[/latex] ml of acetaminophen.
Write a sentence that gives the information to find it.	If [latex]5[/latex] ml is prescribed for every [latex]25[/latex] pounds, how much will be prescribed for [latex]80[/latex] pounds?
Translate into a proportion.
Substitute given values—be careful of the units.	[latex]{\Large\frac{5}{25}}={\Large\frac{a}{80}}[/latex]
Multiply both sides by [latex]80[/latex].	[latex]80\cdot{\Large\frac{5}{25}}=80\cdot{\Large\frac{a}{80}}[/latex]
Multiply and show common factors.	[latex]{\Large\frac{16\cdot5\cdot5}{5\cdot5}}={\Large\frac{80a}{80}}[/latex]
Simplify.	[latex]16=a[/latex]
Check if the answer is reasonable.
Yes. Since [latex]80[/latex] is about [latex]3[/latex] times [latex]25[/latex], the medicine should be about [latex]3[/latex] times [latex]5[/latex].
Write a complete sentence.	The pediatrician would prescribe [latex]16[/latex] ml of acetaminophen to Zoe.

You could also solve this proportion by setting the cross products equal.

example

One brand of microwave popcorn has [latex]120[/latex] calories per serving. A whole bag of this popcorn has [latex]3.5[/latex] servings. How many calories are in a whole bag of this microwave popcorn?

Show Solution

Solution

Identify what you are asked to find.	How many calories are in a whole bag of microwave popcorn?
Choose a variable to represent it.	Let [latex]c=[/latex] number of calories.
Write a sentence that gives the information to find it.	If there are [latex]120[/latex] calories per serving, how many calories are in a whole bag with [latex]3.5[/latex] servings?
Translate into a proportion.
Substitute given values.	[latex]{\Large\frac{120}{1}}={\Large\frac{c}{3.5}}[/latex]
Multiply both sides by [latex]3.5[/latex].	[latex](3.5)({\Large\frac{120}{1}})=(3.5)({\Large\frac{c}{3.5}})[/latex]
Multiply.	[latex]420=c[/latex]
Check if the answer is reasonable.
Yes. Since [latex]3.5[/latex] is between [latex]3[/latex] and [latex]4[/latex], the total calories should be between [latex]360 (3⋅120)[/latex] and [latex]480 (4⋅120)[/latex].
Write a complete sentence.	The whole bag of microwave popcorn has [latex]420[/latex] calories.

example

Josiah went to Mexico for spring break and changed $[latex]325[/latex] dollars into Mexican pesos. At that time, the exchange rate had $[latex]1[/latex] U.S. is equal to [latex]12.54[/latex] Mexican pesos. How many Mexican pesos did he get for his trip?

Show Solution

Solution

Identify what you are asked to find.	How many Mexican pesos did Josiah get?
Choose a variable to represent it.	Let [latex]p=[/latex] number of pesos.
Write a sentence that gives the information to find it.	If [latex]\text{\$1}[/latex] U.S. is equal to [latex]12.54[/latex] Mexican pesos, then [latex]\text{\$325}[/latex] is how many pesos?
Translate into a proportion.
Substitute given values.	[latex]{\Large\frac{1}{12.54}}={\Large\frac{325}{p}}[/latex]
The variable is in the denominator, so find the cross products and set them equal.	[latex]p\cdot{1}=12.54(325)[/latex]
Simplify.	[latex]c=4,075.5[/latex]
Check if the answer is reasonable.
Yes, [latex]\text{\$100}[/latex] would be [latex]\text{\$1,254}[/latex] pesos. [latex]\text{\$325}[/latex] is a little more than [latex]3[/latex] times this amount.
Write a complete sentence.	Josiah has [latex]4075.5[/latex] pesos for his spring break trip.

In the following video we show another example of how to solve an application that involves proportions.