Rates Examples in Math: 46 Problems with Answers

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Rates.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

A rate is a ratio that compares two quantities measured in different units, expressing how much of one quantity corresponds to a given amount of another. It is often written as 'per' one unit of the second quantity, such as miles per hour or dollars per pound.

60 miles per hour tells you how many miles you travel for each hour — it compares distance to time.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: A rate compares two quantities in different units, like miles per hour or dollars per pound.

Common stuck point: The procedure for rates is the easy part; the trap is dropping the units when reporting a rate. Asking "Are the two quantities measured in different units, compared as one 'per' the other?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Are the two quantities measured in different units, compared as one 'per' the other?

Worked Examples

Example 1

easy
A car travels 240 miles in 4 hours. What is its speed as a unit rate in miles per hour?

Answer

60 miles per hour

First step

1
Write the rate as a fraction: 240 miles4 hours.

Full solution

  1. 2
    Divide numerator and denominator by 4: 240÷44÷4=60 miles1 hour.
  2. 3
    The unit rate is 60 miles per hour.
A unit rate expresses a ratio with a denominator of 1, making it easy to compare or use in calculations. Divide both parts of the rate by the denominator quantity to convert to a unit rate.

Example 2

medium
Printer A prints 180 pages in 3 minutes and Printer B prints 260 pages in 4 minutes. Which printer is faster?

Example 3

medium
A car uses 24 liters of gas to travel 300 km. Find (a) liters per km and (b) km per liter.

Example 4

medium
A cyclist covers 42 km in 1.5 hours. What is the speed in km/h, and how far in 4 hours at the same rate?

Example 5

medium
A recipe calls for 3 cups of flour for every 4 cookies. How much flour for 20 cookies?

Example 6

medium
Convert 36 km/h to m/s.

Example 7

hard
Train A travels 90 km in 1.5 h; train B travels 135 km in 2.25 h. Which is faster?

Example 8

hard
Pipe A fills a tank in 4 hours; Pipe B fills it in 6 hours. How long together?

Example 9

hard
A worker earns $18/hour for the first 40 hours and $27/hour for overtime. If she earned $873 in a week, how many overtime hours did she work?

Example 10

medium
Compare: $1.92 for 12 oz of juice versus $2.55 for 15 oz. Which is the better unit price?

Example 11

challenge
Two cyclists 60 km apart ride toward each other; one at 18 km/h, the other at 22 km/h. When and where do they meet?

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
A typist types 420 words in 7 minutes. What is the unit rate in words per minute?

Example 2

hard
Juice costs $3.84 for 16 fl oz or $5.10 for 30 fl oz. Which is the better deal per fluid ounce?

Example 3

easy
A car goes 120 miles in 2 hours. What is its speed in miles per hour?

Example 4

easy
4 pounds of apples cost $10. What is the price per pound?

Example 5

easy
A printer prints 30 pages in 5 minutes. How many pages per minute?

Example 6

easy
A faucet fills 12 liters in 3 minutes. What is the flow rate per minute?

Example 7

easy
A runner covers 400 meters in 50 seconds. What is the speed in meters per second?

Example 8

easy
A typist types 240 words in 4 minutes. Words per minute?

Example 9

easy
A heart beats 150 times in 2 minutes. Beats per minute?

Example 10

easy
A store sells 18 sodas for $9. What is the price per soda?

Example 11

medium
Store A sells 6 apples for $3; Store B sells 10 apples for $4. Which has the better unit price?

Example 12

medium
A car travels at 50 mph. How far does it go in 3.5 hours?

Example 13

medium
A worker earns $54 for 4 hours. At the same rate, how much for 10 hours?

Example 14

medium
A car uses 8 gallons to drive 200 miles. What is its fuel efficiency in miles per gallon?

Example 15

medium
Pipe A fills a tank at 5 liters per minute. How long to fill a 90-liter tank?

Example 16

medium
A recipe rate is 3 cups of rice per 5 cups of water. How much water for 12 cups of rice?

Example 17

challenge
Two cars start 300 miles apart and drive toward each other at 40 mph and 60 mph. How long until they meet?

Example 18

challenge
Pipe A fills a tank in 6 hours; pipe B fills it in 3 hours. Working together, how long to fill the tank?

Example 19

challenge
A train travels 60 miles at 30 mph, then 60 miles at 60 mph. What is its average speed for the whole trip?

Example 20

medium
A factory makes 360 widgets in 6 hours. How many per hour?

Example 21

medium
A jogger runs at 6 mph. How far in 45 minutes?

Example 22

medium
Store X: 12 oz for $3. Store Y: 20 oz for $4. Which is cheaper per ounce?

Example 23

easy
A train travels 300 km in 5 hours. Find its speed in km/h.

Example 24

easy
A factory produces 450 widgets in 9 hours. What is the production rate in widgets per hour?

Example 25

easy
6 identical books weigh 4.8 kg total. What is the weight per book?

Example 26

medium
Store A sells 5 apples for $3; Store B sells 8 apples for $4.40. Which has the lower price per apple?

Example 27

medium
A printer prints 84 pages in 7 minutes. How long will it take to print 300 pages at the same rate?

Example 28

medium
A car travels at 55 mph. How far in 3.5 hours?

Example 29

medium
A garden hose fills a 20-gallon tank in 5 minutes. What is the flow rate in gallons per minute, and in gallons per second?

Example 30

hard
A factory paints 1200 cars in 30 days, running 8 hours a day. What is the production rate in cars per hour?

Example 31

hard
A car averages 30 mpg in the city and 40 mpg on the highway. If a trip is 60 city miles and 200 highway miles, what is the total gas used?

Example 32

hard
A printer prints 4 pages per minute; another prints 6 pages per minute. Together, how long for 300 pages?

Example 33

medium
A jogger runs 34 mile in 9 minutes. What is the pace in minutes per mile?

Example 34

medium
3 kg of grapes cost $7.50. What is the price per kg?

Example 35

hard
A car travels 40 mph for the first 2 hours and 60 mph for the next 3 hours. What is the average speed for the whole trip?
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Background Knowledge

These ideas may be useful before you work through the harder examples.

ratiosdivision