Pennsylvania 7 - 2020 Edition
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6.03 Proportional reasoning with percents

Recall that a percent is a ratio where the denominator is $100$100. Because of this definition, we can use proportional reasoning strategies to solve problems with percents.

Proportions can be used to represent percent problems as follows:

Percents as a proportion
$\frac{\text{percent}}{100}$percent100 $=$= $\frac{\text{part}}{\text{whole}}$partwhole


Worked example

Question 1

Evaluate: Use a proportion to answer the question, "What percent of 20 is 3?"

Think: We can translate the statement to a proportion. Then use proportional reasoning to solve for the unknown.

The percent is the unknown. So we can use the variable $x$x to represent it.

The number $3$3 is the part and $20$20 is the whole.


$\frac{\text{percent}}{100}$percent100 $=$= $\frac{\text{part}}{whole}$partwhole


$\frac{x}{100}$x100 $=$= $\frac{3}{20}$320

$x$x is the unknown percent.
$3$3 is the part.

$20$20 is the whole.

$\frac{x}{100}$x100 $=$= $\frac{3\times5}{20\times5}$3×520×5

Multiplying the fraction by $\frac{5}{5}$55 gives us a common denominator of $100$100.

$\frac{x}{100}$x100 $=$= $\frac{15}{100}$15100


$x$x $=$= $15$15

If the denominators in a proportion are the same, the numerators must also be the same.


So the number $3$3 is $15%$15% of $20$20

Reflect: Is there another method that we might use to check our solution?


Proportional reasoning with benchmark percents

Suppose we want to check our solution to the first worked example using a different method. Let's see how we can apply proportional reasoning to percents in a different way.

Worked example


Evaluate: Find $15%$15% of $20$20.

Think: It might be easiest to find $10%$10% of $20$20

We can then use half of that amount to find $5%$5% of $20$20. If we add the two amounts, that will give us $15%$15% of $20$20.

Do: First, find $10%$10% of $20$20.

$10%$10% of $20$20 $=$= $0.10\times20$0.10×20

Since $10%=0.10$10%=0.10

  $=$= $2$2




$5%$5% of $20$20  $=$= $\frac{1}{2}\times2$12×2

Since $5%$5% is half of $10%$10%

  $=$= $1$1  
$15%$15% $=$= $10%+5%$10%+5%  
  $=$= $2+1$2+1  
  $=$= $3$3  


So $15%$15% of $20$20 is $3$3.

Reflect: What other percents can we calculate using the benchmark of $10%$10%?


Practice questions

Question 3

Translate the following percentage problem to a proportion. Do not solve or simplify the proportion.

'What percent of $92$92 is $23$23?'

Let the unknown number be $x$x.

Question 4

Translate the following percentage problem to a proportion. Do not solve or simplify the proportion.

'$60%$60% of what number is $144$144?'

Let the unknown number be $x$x.

Question 5

We want to find $45%$45% of $5$5 hours.

  1. How many minutes are there in $5$5 hours?

  2. What is $10%$10% of $300$300 minutes?

  3. What is $5%$5% of $300$300 minutes?

  4. Hence find $45%$45% of $300$300 minutes.



Analyze proportional relationships and use them to model and solve real-world and mathematical problems.


Use proportional relationships to solve multi-step ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease

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