3.06 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:

 

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.

Do:

$\frac{\text{percent}}{100}$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

QUESTION 2

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	Evaluate
$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

Question 4

Question 5