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4. Ratios & Proportions
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United States of America
Grade 6

4.07 Proportional reasoning with percents

Lesson
Practice

Introduction

Recall that a percent is a fraction where the numerator is the part, and the denominator is the whole out of 100. Because of this definition, we can use equivalent ratio strategies to solve problems with percents.

Percents as a proportion

When we say "percent" we are saying "per one hundred".

A grid of squares with a shaded section in the shape of a rectangle made up of 2 rows of 7 squares.

14\% means 14 per 100.

Observe that 14 boxes are shaded blue in this grid of 100 boxes.

We can also write it as a ratio 14:100 and as a fraction \dfrac{14}{100}.

We can solve percent problems using equivalent ratios or a proportion as follows:\frac{\text{part}}{\text{whole}} = \frac{\text{percent}}{100}

Examples

Example 1

What percent of 20 is 3?

Worked Solution
Create a strategy

Identify the given numbers and let the unknown be x.

Translate the statement into a proportion.

Apply the idea
\displaystyle \frac{\text{part}}{\text{whole}}\displaystyle =\displaystyle \frac{\text{percent}}{100}Proportion setup
\displaystyle \frac{3}{20}\displaystyle =\displaystyle \frac{ x }{100}Substitute 3 for the part, 20 for the whole, and x for the percent.
\displaystyle \frac{3}{20}\displaystyle =\displaystyle \frac{ x }{100}Consider what we multiply 20 by to get 100, we need to multiply the numerator by the same number.
\displaystyle \frac{3}{20}\displaystyle =\displaystyle \frac{ 15 }{100}Multiply the left fraction by \dfrac{5}{5} to get an equivalent ratio
\displaystyle \frac{3}{20}\displaystyle =\displaystyle 15\%Rewrite the fraction out of 100 as a percent

Example 2

Find 15\% of 20.

Worked Solution
Create a strategy

Identify the given numbers and let the unknown be x.

Translate the statement into a proportion.

Apply the idea

The number 15 is the percent and the number 20 is the whole.

Let the unknown percent be x.

\displaystyle \frac{\text{percent}}{100}\displaystyle =\displaystyle \frac{\text{part}}{\text{whole}}Proportion setup
\displaystyle \frac{ 15 }{100}\displaystyle =\displaystyle \frac{x}{20}Substitute 15 for the percent, x for the part, and 20 for the whole.
\displaystyle \frac{ 15 }{100}\displaystyle =\displaystyle \frac{x}{20}Consider what factor divides into 100 to get 20.
\displaystyle \frac{ 15 }{100}\displaystyle =\displaystyle \frac{3}{20}Divide the left fraction by \dfrac{5}{5} to get an equivalent ratio.
\displaystyle x\displaystyle =\displaystyle 33 is 15\% of 20.

Example 3

60\% of what number is 120?

Worked Solution
Create a strategy

Identify the given numbers and let the unknown be x.

Translate the statement into a proportion.

Apply the idea
\displaystyle \frac{\text{percent}}{100}\displaystyle =\displaystyle \frac{\text{part}}{\text{whole}}Proportion setup
\displaystyle \frac{ 60 }{100}\displaystyle =\displaystyle \frac{120}{x}Substitute 60 for the percent, 120 for the part, and x for the whole.
\displaystyle \frac{ 60 }{100}\displaystyle =\displaystyle \frac{120}{x}Consider what we multiply 60 by to get 120.
\displaystyle \frac{ 60}{100}\displaystyle =\displaystyle \frac{120}{200}Multiply the left side of the equation by \dfrac{2}{2}
\displaystyle x\displaystyle =\displaystyle 20060\% of 200 is 120.
Idea summary

To represent percent problems, we can set up a proportion to find an equivalent ratio:\frac{\text{percent}}{100}=\frac{\text{part}}{\text{whole}}

Outcomes

6.RP.A.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.A.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.

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