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Study Guides > Prealgebra

Writing and Solving Percent Proportions

Learning Outcomes

  • Translate a statement to a proportion
  • Solve a percent proportion
 

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:

[latex-display]\frac{\text{amount}}{\text{base}}=\frac{\text{percent}}{100}[/latex-display] [latex-display]\frac{3}{5}=\frac{60}{100}[/latex-display]

Percent Proportion

The amount is to the base as the percent is to [latex]100[/latex]. [latex-display]\frac{\text{amount}}{\text{base}}=\frac{\text{percent}}{100}[/latex-display]
  If we restate the problem in the words of a proportion, it may be easier to set up the proportion: [latex-display]\mathit{\text{The amount is to the base as the percent is to one hundred.}}[/latex-display] We could also say: [latex-display]\mathit{\text{The amount out of the base is the same as the percent out of one hundred.}}[/latex-display] First we will practice translating into a percent proportion. Later, we’ll solve the proportion.  

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]
 

try it

[ohm_question]146821[/ohm_question]
 

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]

 

try it

[ohm_question]146822[/ohm_question]
   

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]

 

try it

[ohm_question]146825[/ohm_question]
  Now that we have written percent equations as proportions, we are ready to solve the equations.  

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].

 

try it

[ohm_question]146828[/ohm_question]
The following video shows a similar example of how to solve a percent proportion. https://youtu.be/cApv0GeF2XM   In the next example, the percent is more than [latex]100[/latex], which is more than one whole. So the unknown number will be more than the base.  

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].

 

try it

[ohm_question]146839[/ohm_question]
  Percents with decimals and money are also used in proportions.  

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].

 

try it

[ohm_question]146840[/ohm_question]
In the following video we show a similar problem, note the different wording that results in the same equation. https://youtu.be/wsBhmrmumJo  

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].

 

try it

[ohm_question]146843[/ohm_question]
Watch the following video to see a similar problem. https://youtu.be/1GTPRROi1tE

Licenses & Attributions

CC licensed content, Original

  • Question ID 146843, 146840, 146839, 146828. Authored by: Lumen Learning. License: CC BY: Attribution.

CC licensed content, Shared previously

  • Example 2: Solve a Percent Problem Using a Percent Proportion. Authored by: James Sousa (Mathispower4u.com). License: CC BY: Attribution.
  • Example 3: Determine What Percent One Number is of Another Using a Percent Proportion. Authored by: James Sousa (Mathispower4u.com). License: CC BY: Attribution.

CC licensed content, Specific attribution