California 6 - 2020 Edition
3.06 Proportional reasoning with percents
Lesson

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.

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

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.

### Outcomes

#### 6.RP.3

Use ratio and rate reasoning to solve real-world and mathematical problems, e.g. By reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.

#### 6.RP.3.c

Find a percent of a quantity as a rate per 100 (e.g. 30% Of a quantity means 30/100 times the quantity); solve problems involving finding the whole given a part and the percent.