Writing and Solving Percent Proportions

Previously, we solved percent equations by applying the properties of equality we have used to solve equations throughout this text. Some people prefer to solve percent equations by using the proportion method. The proportion method for solving percent problems involves a percent proportion. A percent proportion is an equation where a percent is equal to an equivalent ratio.

For example, [latex]\text{60%}=\frac{60}{100}[/latex] and we can simplify [latex]\frac{60}{100}=\frac{3}{5}[/latex]. Since the equation [latex]\frac{60}{100}=\frac{3}{5}[/latex] shows a percent equal to an equivalent ratio, we call it a percent proportion. Using the vocabulary we used earlier:

example

Translate to a proportion. What number is [latex]\text{75%}[/latex] of [latex]90?[/latex] Solution If you look for the word "of", it may help you identify the base.

Identify the parts of the percent proportion.
Restate as a proportion.	What number out of [latex]90[/latex] is the same as [latex]75[/latex] out of [latex]100[/latex]?
Set up the proportion. Let [latex]n=\text{number}[/latex] .	[latex]\frac{n}{90}=\frac{75}{100}[/latex]

example

Translate to a proportion. [latex]19[/latex] is [latex]\text{25%}[/latex] of what number?

Answer: Solution

Identify the parts of the percent proportion.
Restate as a proportion.	[latex]19[/latex] out of what number is the same as [latex]25[/latex] out of [latex]100[/latex]?
Set up the proportion. Let [latex]n=\text{number}[/latex] .	[latex]\frac{19}{n}=\frac{25}{100}[/latex]

example

Translate to a proportion. What percent of [latex]27[/latex] is [latex]9?[/latex]

Answer: Solution

Identify the parts of the percent proportion.
Restate as a proportion.	[latex]9[/latex] out of [latex]27[/latex] is the same as what number out of [latex]100[/latex]?
Set up the proportion. Let [latex]p=\text{percent}[/latex] .	[latex]\frac{9}{27}=\frac{p}{100}[/latex]

example

Translate and solve using proportions: What number is [latex]\text{45%}[/latex] of [latex]80?[/latex]

Answer: Solution

Identify the parts of the percent proportion.
Restate as a proportion.	What number out of [latex]80[/latex] is the same as [latex]45[/latex] out of [latex]100[/latex]?
Set up the proportion. Let [latex]n=[/latex] number.	[latex]\frac{n}{80}=\frac{45}{100}[/latex]
Find the cross products and set them equal.	[latex]100\cdot{n}=80\cdot{45}[/latex]
Simplify.	[latex]100n=3,600[/latex]
Divide both sides by [latex]100[/latex].	[latex]\frac{100n}{100}=\frac{3,600}{100}[/latex]
Simplify.	[latex]n=36[/latex]
Check if the answer is reasonable.
Yes. [latex]45[/latex] is a little less than half of [latex]100[/latex] and [latex]36[/latex] is a little less than half [latex]80[/latex].
Write a complete sentence that answers the question.	[latex]36[/latex] is [latex]45\text{%}[/latex] of [latex]80[/latex].

example

Translate and solve using proportions: [latex]\text{125%}[/latex] of [latex]25[/latex] is what number?

Answer: Solution

Identify the parts of the percent proportion.
Restate as a proportion.	What number out of [latex]25[/latex] is the same as [latex]125[/latex] out of [latex]100[/latex]?
Set up the proportion. Let [latex]n=[/latex] number.	[latex]\frac{n}{25}=\frac{125}{100}[/latex]
Find the cross products and set them equal.	[latex]100\cdot{n}=25\cdot{125}[/latex]
Simplify.	[latex]100n=3,125[/latex]
Divide both sides by [latex]100[/latex].	[latex]\frac{100n}{100}=\frac{3,125}{100}[/latex]
Simplify.	[latex]n=31.25[/latex]
Check if the answer is reasonable.
Yes. [latex]125[/latex] is more than [latex]100[/latex] and [latex]31.25[/latex] is more than [latex]25[/latex].
Write a complete sentence that answers the question.	[latex]125\text{%}[/latex] of [latex]25[/latex] is [latex]31.25[/latex].

example

Translate and solve: [latex]\text{6.5%}[/latex] of what number is [latex]\text{\$1.56}?[/latex]

Answer: Solution

Identify the parts of the percent proportion.
Restate as a proportion.	[latex]\text{\$1.56}[/latex] out of what number is the same as [latex]6.5[/latex] out of [latex]100[/latex]?
Set up the proportion. Let [latex]n=[/latex] number.	[latex]\frac{1.56}{n}=\frac{6.5}{100}[/latex]
Find the cross products and set them equal.	[latex]100\cdot{1.56}=n\cdot{6.5}[/latex]
Simplify.	[latex]156=6.5n[/latex]
Divide both sides by [latex]6.5[/latex] to isolate the variable.	[latex]\frac{156}{6.5}=\frac{6.5n}{6.5}[/latex]
Simplify.	[latex]24=n[/latex]
Check if the answer is reasonable.
Yes. [latex]6.5\text{%}[/latex] is a small amount and [latex]\text{\$1.56}[/latex] is much less than [latex]\text{\$24}[/latex].
Write a complete sentence that answers the question.	[latex]6.5\text{%}[/latex] of [latex]\text{\$24}[/latex] is [latex]\text{\$1.56}[/latex].

example

Translate and solve using proportions: What percent of [latex]72[/latex] is [latex]9?[/latex]

Answer: Solution

Identify the parts of the percent proportion.
Restate as a proportion.	[latex]9[/latex] out of [latex]72[/latex] is the same as what number out of [latex]100[/latex]?
Set up the proportion. Let [latex]n=[/latex] number.	[latex]\frac{9}{72}=\frac{n}{100}[/latex]
Find the cross products and set them equal.	[latex]72\cdot{n}=100\cdot{9}[/latex]
Simplify.	[latex]72n=900[/latex]
Divide both sides by [latex]72[/latex].	[latex]\frac{72n}{72}=\frac{900}{72}[/latex]
Simplify.	[latex]n=12.5[/latex]
Check if the answer is reasonable.
Yes. [latex]9[/latex] is [latex]\frac{1}{8}[/latex] of [latex]72[/latex] and [latex]\frac{1}{8}[/latex] is [latex]12.5\text{%}[/latex].
Write a complete sentence that answers the question.	[latex]12.5\text{%}[/latex] of [latex]72[/latex] is [latex]9[/latex].

Licenses & Attributions