Topics
4. Ratios & Proportions
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United States of America
Grade 6

4.06 Problem solving with unit rates

Lesson
Practice

Introduction

Previously, we learned how to find  unit rates  . Now let's use the concept of unit rates to solve real world problems.

Problem solving with unit rates

Remember that rates are very similar to ratios in that we can use them to calculate how much one measurement changes based on the change in another.

For example, we know a sprinter can run at a speed of 10 m/s. A speed of 10 \text{ m/s} means that the sprinter runs 10 meters in 1 second.

Because we know the unit rate, we can figure out how far the sprinter can run in 15 seconds.

Set up equivalent ratios and let x be the unknown distance the sprinter can travel in 15 seconds.

\displaystyle \dfrac{10\text{ m}}{1\text{ s}}\displaystyle =\displaystyle \dfrac {x}{15\text{ s}}Substitute the values of the distance and time in the equivalent ratios.
\displaystyle \dfrac{10\text{ m}}{1\text{ s}}\displaystyle =\displaystyle \dfrac {x}{15\text{ s}}Consider what factor we multiply by 1 to get 15
\displaystyle \dfrac{10 \text{ m}}{1 \text{ s}} \times \dfrac{15}{15}\displaystyle =\displaystyle \dfrac {x}{15\text{ s}}Multiply the numerator denominator by the same factor
\displaystyle \dfrac{150\text{ m}}{15 \text{ s}}\displaystyle =\displaystyle \dfrac {x}{15\text{ s}}Evaluate
\displaystyle x\displaystyle =\displaystyle 150 \text{ m}The sprinter will run 150 meters in 15 seconds.

Let's have a look at a few more examples.

Examples

Example 1

Henry bikes 45 miles in 3 hours.

a

What is the speed of the bike in miles per hour?

Worked Solution
Create a strategy

Miles per hour is the unit rate for speed. We need to find the distance traveled in 1 hour.

Apply the idea

\text{Speed} = \dfrac{\text { distance}}{\text { time}}

\displaystyle \text{Speed}\displaystyle =\displaystyle \dfrac{45\text { mi}}{3\text { hr}}Substitute the values of the distance and time in the expression
\displaystyle =\displaystyle \dfrac{45\text { mi}}{3\text { hr}} Consider what factor to divide 3 by to get 1
\displaystyle =\displaystyle \dfrac{45\text { mi}}{3\text { hr}} \div \dfrac{3}{3}Divide by \dfrac{3}{3} to get the unit rate
\displaystyle =\displaystyle \dfrac{15 \text{ mi}}{1\text{ hr}}Evaluate.
\displaystyle =\displaystyle {15 \text{ mi/hr}}Speed as unit rate
b

If Henry travels at this constant rate, what distance will Henry travel in 2 hours?

Worked Solution
Create a strategy

Use the unit rate to find an equivalent ratio by multiplying or dividing the numerator and denominator by the same factor, to get 2 hours.

Let x be the unknown distance in miles.

Apply the idea
\displaystyle \dfrac{15 \text{ mi}}{1\text{ hr}}\displaystyle =\displaystyle \dfrac{x}{2\text{ hrs}}Substitute the values
\displaystyle \dfrac{15 \text{ mi}}{1\text{ hr}} \times \dfrac{2}{2}\displaystyle =\displaystyle \dfrac{x}{2\text{ hrs}}Multiply by \dfrac{2}{2} to find an equivalent fraction where time is 2 hours.
\displaystyle \dfrac{30 \text{ mi}}{2\text{ hrs}} \displaystyle =\displaystyle \dfrac{x}{2\text{ hrs}}Evaluate the multiplication.
\displaystyle x\displaystyle =\displaystyle 30\text{ mi}Distance covered in 2 hours

Example 2

Iain feels like buying some ice-cream for himself and his friends.

  • A box of 6 Cornettos costs \$7.20.

  • A box of 4 Paddle pops costs \$6.40.

a

How much does each Cornetto cost?

Worked Solution
Create a strategy

Find the unit price for each Cornetto.

Apply the idea
\displaystyle \text{Unit price}\displaystyle =\displaystyle \dfrac{\$7.20}{6 \text{ pieces}}Divide the total cost by the number of Cornettos
\displaystyle =\displaystyle \dfrac{\$7.20}{6 \text{ pieces}}\div \dfrac{6}{6}Divide the numerator and denominator by 6 to get the unit price
\displaystyle =\displaystyle \dfrac{\$1.20}{1 \text{ piece}}Evaluate

Cornettos cost \$1.20 per piece.

b

How much does each Paddle pop cost?

Worked Solution
Create a strategy

Find the unit price of Paddle pop.

Apply the idea
\displaystyle \text{Unit price}\displaystyle =\displaystyle \dfrac{\$6.40}{4 \text{ pieces}}Divide the total cost by the number of Paddle pops
\displaystyle =\displaystyle \dfrac{\$6.40}{4 \text{ pieces}}\div \dfrac{4}{4}Divide the numerator and denominator by 4 to get the unit price
\displaystyle =\displaystyle \dfrac{\$1.60}{1 \text{ piece}}Evaluate

Paddle pops cost \$1.60 per piece.

c

Which type of ice cream is the better buy?

Worked Solution
Create a strategy

The better buy is the ice cream is the one with a lower unit price.

Apply the idea

Compare the unit prices of Cornettos and Paddle pops and choose the lower unit price.

\$1.40 < \$1.60

Cornettos are a better buy than Paddle Pops.

Idea summary

We can calculate any of the three parts in the rate equation given any other two.

We can treat rates like ratios so that we can multiply or divide the top and bottom of the ratio in fraction form by a number that gives an equivalent ratio.

Outcomes

6.RP.A.2

Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0, and use rate language in the context of a ratio relationship. Expectations for unit rates in this grade are limited to non-complex fractions.

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.B

Solve unit rate problems including those involving unit pricing and constant speed.

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