Rates Examples in Math

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 240240 miles in 44 hours. What is its speed as a unit rate in miles per hour?

Answer

60 miles per hour60 \text{ miles per hour}

First step

1
Write the rate as a fraction: 240 miles4 hours\frac{240 \text{ miles}}{4 \text{ hours}}.

Full solution

  1. 2
    Divide numerator and denominator by 4: 240÷44÷4=60 miles1 hour\frac{240 \div 4}{4 \div 4} = \frac{60 \text{ miles}}{1 \text{ hour}}.
  2. 3
    The unit rate is 6060 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 180180 pages in 33 minutes and Printer B prints 260260 pages in 44 minutes. Which printer is faster?

Example 3

medium
A car uses 2424 liters of gas to travel 300300 km. Find (a) liters per km and (b) km per liter.

Example 4

medium
A cyclist covers 4242 km in 1.51.5 hours. What is the speed in km/h, and how far in 44 hours at the same rate?

Example 5

medium
A recipe calls for 33 cups of flour for every 44 cookies. How much flour for 2020 cookies?

Example 6

medium
Convert 3636 km/h to m/s.

Example 7

hard
Train A travels 9090 km in 1.51.5 h; train B travels 135135 km in 2.252.25 h. Which is faster?

Example 8

hard
Pipe A fills a tank in 44 hours; Pipe B fills it in 66 hours. How long together?

Example 9

hard
A worker earns $18\$18/hour for the first 4040 hours and $27\$27/hour for overtime. If she earned $873\$873 in a week, how many overtime hours did she work?

Example 10

medium
Compare: $1.92\$1.92 for 1212 oz of juice versus $2.55\$2.55 for 1515 oz. Which is the better unit price?

Example 11

challenge
Two cyclists 6060 km apart ride toward each other; one at 1818 km/h, the other at 2222 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 420420 words in 77 minutes. What is the unit rate in words per minute?

Example 2

hard
Juice costs $3.84\$3.84 for 1616 fl oz or $5.10\$5.10 for 3030 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\$9. What is the price per soda?

Example 11

medium
Store A sells 6 apples for $3\$3; Store B sells 10 apples for $4\$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\$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\$3. Store Y: 20 oz for $4\$4. Which is cheaper per ounce?

Example 23

easy
A train travels 300300 km in 55 hours. Find its speed in km/h.

Example 24

easy
A factory produces 450450 widgets in 99 hours. What is the production rate in widgets per hour?

Example 25

easy
66 identical books weigh 4.84.8 kg total. What is the weight per book?

Example 26

medium
Store A sells 55 apples for $3\$3; Store B sells 88 apples for $4.40\$4.40. Which has the lower price per apple?

Example 27

medium
A printer prints 8484 pages in 77 minutes. How long will it take to print 300300 pages at the same rate?

Example 28

medium
A car travels at 5555 mph. How far in 3.53.5 hours?

Example 29

medium
A garden hose fills a 2020-gallon tank in 55 minutes. What is the flow rate in gallons per minute, and in gallons per second?

Example 30

hard
A factory paints 12001200 cars in 3030 days, running 88 hours a day. What is the production rate in cars per hour?

Example 31

hard
A car averages 3030 mpg in the city and 4040 mpg on the highway. If a trip is 6060 city miles and 200200 highway miles, what is the total gas used?

Example 32

hard
A printer prints 44 pages per minute; another prints 66 pages per minute. Together, how long for 300300 pages?

Example 33

medium
A jogger runs 34\tfrac{3}{4} mile in 99 minutes. What is the pace in minutes per mile?

Example 34

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

Example 35

hard
A car travels 4040 mph for the first 22 hours and 6060 mph for the next 33 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