Topics
5. Ratios & Proportional Relationships
topic badge
United States of AmericaVirginia
Grade 6

5.03 Unit rates

Lesson
Practice

Unit rates

A rate is a ratio that involves two different units and how they relate to each other.

Rates are measured by combining two different units into a single compound unit. We can write these compound units using a slash \left(/\right) between the different units, so "meters per second" becomes "\text{m/s}".

Exploration

To explore different rates input the names of the items and their amounts and use the sliders to adjust the number lines.

Loading interactive...

  1. If a cyclist rode 5 miles in 20 minutes, at a steady rate. How long would it take to ride 10 miles?

A unit rate is a specific type of rate where the quantity of the denominator is 1, such as 2 feet per 1 second or 5 miles per 1 hour.

Consider an Olympic sprinter who runs 100 meters in 10 seconds. We can represent this relationship on a double number line. Notice that the ratio 100 \text{ m} : 10 \text{ s} is represented on the number lines since they are both the same distance from 0.

A number line on top of another, both showing 0-100. The first is labeled 'meters' and the second is labeled 'seconds'. A running athlete is at the end of the number lines.

To discover how far he can run in a single second, we can draw tick marks on our number lines to find the location of 1 second and the equivalent distance:

A number line on top of another, both showing 0-100. The first is labeled 'meters' and the second is labeled 'seconds'. A running athlete is at the end of the number lines. 10 abd 1 is highlighted

Since he can run 10 meters in 1 second, we say the unit rate is 10 meters per second or 10 \text{ m}/\text{s}.

We can calculate how far the sprinter runs in 1 second by dividing the 100 meters evenly between the 10 seconds. Let's represent that as a rate:

\text{Sprinter's speed}= \dfrac{100 \text{ m}}{10 \text{ s}}

\text{Sprinter's speed}=10 \text{ m}/\text{s}

This calculation tells us that the sprinter runs 10 meters in one second.

A unit rate can also be found from a ratio. If a dance game at the arcade cost \$1.25 to play for 5 minutes, the ratio of cost to time played is 1.25:5. To find the unit rate, divide both quantities of the ratio by 5 to find the ratio for a single minute:

A table with 2 rows. 1st row is labeled 'Time (min), the first column shows '1' and the second column shows '5'. The second row is labeled 'Cost($), the first column shows '0.25' and the second column shows '1.25'.' An arrow labeled 'divided by 5' is drawn from 5 to 1 for the first row, and from 1.25 to 0.25 for the second column.

The unit rate is \$0.25 per minute played. To determine how much it would cost to play the dance game for 20 minutes, we can multiply our unit rate by 20:

A table with 2 rows. 1st row is labeled 'Time (min), the first column shows '1' , the second column shows '5' and the third column shows '20'. The second row is labeled 'Cost($), the first column shows '0.25', the second column shows '1.25' and the third column shows '5.00'.' An arrow labeled 'x20' is drawn from 1 to 20 for the first row, and from 0.25 to 5.00 for the second column.

It would cost \$5.00 to play the dance game for 20 minutes.

Examples

Example 1

A tap fills up a 240-liter tub in 4 hours.

a

Which is the compound unit for the rate of water flow?

Worked Solution
Create a strategy

The rate of water flow represents the number of liters that flows from the tap each hour.

Apply the idea

The unit is liters per hour, \text{L/hr}.

b

What is the flow of the water as a unit rate?

Worked Solution
Create a strategy

We can find the rate of water flow in liters per hour by dividing the capacity of the tub in liters by the number of hours passed.

Apply the idea
\displaystyle \text{Rate}\displaystyle =\displaystyle \dfrac{240}{4}\text{ L/hr}
\displaystyle =\displaystyle 60\text{ L/hr}Evaluate the division.

Example 2

A car travels 320\text{ km} in 4 hours.

a

Complete the table of values.

Time taken (hours)421
Distance traveled (kilometers)320
Worked Solution
Create a strategy

Since the time is being divided by 2 each time, divide the distance traveled by 2 for each new distance in the table.

Apply the idea
\displaystyle \text{Distance at } 2 \text{ hours}\displaystyle =\displaystyle \dfrac{320}{2}Divide 320 by 2
\displaystyle =\displaystyle 160 \text{ km}Evaluate
\displaystyle \text{Distance at } 1 \text{ hour}\displaystyle =\displaystyle \dfrac{160}{2}Divide 160 by 2
\displaystyle =\displaystyle 80 \text{ km}Evaluate

This means that the tables of values is given by:

Time taken (hours)421
Distance traveled (kilometers)32016080
b

What is the speed of the car as a unit rate?

Worked Solution
Create a strategy

Use the table from part (a) to find what the distance was after 1 hour.

Apply the idea
\displaystyle \text{Speed}\displaystyle =\displaystyle \dfrac{80 \text{ km}}{1 \text{ hr}}Divide 80 \text{ km} by 1 \, \text{hr}
\displaystyle =\displaystyle 80 \text{ km/hr}Evaluate

Example 3

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

Set up a table from the ratio original and create equivalent ratios in the table to solve the problem.

Apply the idea

We can set up a table from the ratio 45:3.

miles\text{ }\text{ }45\text{ }
hours\text{ }\text{ }3\text{ }

Divide both values by 3 to find the distance for 1 hour.

miles15\text{ }45\text{ }
hours1\text{ }3\text{ }

Use that unit rate to solve for the remaining missing values.

miles15304560
hours1234

From the table, we can see that in 2 hours, Henry can travel 30 miles.

Example 4

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 \lt \$1.60

Cornettos are a better buy than Paddle Pops.

Idea summary

A rate is a measure of how quickly one measurement changes with respect to another.

When rates are expressed as a quantity with a denominator of 1, such as 2 feet per second or 5 miles per hour, they are called unit rates.

Outcomes

6.PFA.2

The student will identify and represent proportional relationships between two quantities, including those in context (unit rates are limited to positive values).

6.PFA.2a

Identify the unit rate of a proportional relationship represented by a table of values, a contextual situation, or a graph.

6.PFA.2b

Determine a missing value in a ratio table that represents a proportional relationship between two quantities using a unit rate.

6.PFA.2d

When given a contextual situation representing a proportional relationship, find the unit rate and create a table of values or a graph.

What is Mathspace

About Mathspace