Unit Rates

Learn how to find and use unit rates to compare quantities and solve real-world problems.

Intermediate25 minLesson
Printable worksheet

Definition

A unit rate is a ratio that compares a quantity to exactly one unit of another quantity.
To find a unit rate, divide both parts of the ratio so the second quantity equals 1:
Unit Rate=Total QuantityNumber of Units\text{Unit Rate} = \frac{\text{Total Quantity}}{\text{Number of Units}}
Examples of unit rates:
  • Price per item: 12 dollars for 4 apples = 3 dollars per apple
  • Speed: 150 km in 3 hours = 5050 km per hour
  • Efficiency: 240 pages in 8 days = 3030 pages per day

Try it now

A pack of 6 juice boxes costs 9 dollars. What is the unit rate?

Worked Examples

A pack of 8 water bottles costs 12 dollars. What is the price per bottle?
1

Identify the ratio

12 dollars for 8 bottles → 12 dollars8 bottles\frac{12 \text{ dollars}}{8 \text{ bottles}}
2

Divide to find unit rate

12÷8=1.512 \div 8 = 1.5 → 1.50 dollars
3

Write the unit rate

1.50 dollars per 1 bottle → 1.50 dollars per bottle

Common Mistakes

Dividing in the wrong order

Why it's wrong: Students sometimes divide 8 by 12 instead of 12 by 8 when finding price per item.

Correct: Always divide the quantity you want to find (price) by the number of units (items). 12 dollars / 8 items = 1.50 dollars per item.

Forgetting the units

Why it's wrong: Writing just '70' instead of '70 km/h' loses important information.

Correct: Always include both units in your answer: km/h, dollars/item, pages/day, etc.

Comparing rates with different units

Why it's wrong: You cannot directly compare 2 dollars/bottle with 50 cents/ounce.

Correct: Convert to the same units first, then compare the unit rates.

Interactive Visual

Function graph

0246810051015bottlesprice (dollars)(1, 1.5)(8, 12)

Tap or hover over the graph to read off values.

y = 1.5x

Table of values

bottles1248
price (dollars)1.53612

Interactive Sandbox

Expression Calculator

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Practice Problems

16 problems
Problem 1 of 16
Easy
A pack of 6 juice boxes costs 9 dollars. What is the unit rate?

Why It Matters

Unit rates help you make smart decisions every day:
  • Shopping: Which cereal is the better deal - 500g for 4 dollars or 750g for 5 dollars?
  • Travel: If you drive 240 km in 3 hours, what's your average speed?
  • Work: A painter covers 120 square meters in 6 hours. How fast do they work?
  • Cooking: A recipe uses 6 eggs for 24 cookies. How many eggs per dozen cookies?
Unit rates let you compare different quantities on equal terms!

Real World Applications

Smart Shopping

Stores often show unit prices on shelf labels, but you can calculate them yourself when they are missing.

Example:

A 750ml bottle costs 3 dollars. Unit rate: 3÷0.75=43 \div 0.75 = 4 dollars per liter.

1Try It Yourself

At the grocery store, you see two options for orange juice: 2 liters for 4 dollars OR 1.5 liters for 3.30 dollars.

Which is the better value?

Step 1: Choose the right answer

Find unit price for each: Price / Liters. Which is the better value?

Travel Planning

Speed is a unit rate that helps you plan trips and estimate arrival times.

Example:

If you drive at an average speed of 80 km/h, a 320 km trip takes 320÷80=4320 \div 80 = 4 hours.

2Try It Yourself

A cyclist travels 45 kilometers in 3 hours.

At this rate, how long will it take to travel 75 kilometers?

Step 1: Write the mathematical expression

First find speed, then divide distance by speed

Key Takeaways

  • 1A unit rate compares a quantity to exactly ONE unit of another quantity
  • 2To find a unit rate, divide the total by the number of units
  • 3Unit rates let you compare different quantities on equal terms
  • 4Common unit rates: price per item, speed (km/h), rate of work (items/hour)
  • 5Always include units in your answer (dollars/item, km/h, pages/day)

Frequently Asked Questions

A rate compares two quantities with different units (like 150 km in 3 hours). A unit rate is a special rate where the second quantity is exactly 1 (like 50 km per 1 hour, or 50 km/h).
A rate compares two quantities with different units (like 150 km in 3 hours). A unit rate is a special rate where the second quantity is exactly 1 (like 50 km per 1 hour, or 50 km/h).
Unit rates make comparison easy. It's hard to compare '120 km in 2 hours' with '200 km in 3 hours', but comparing '60 km/h' with '66.7 km/h' is simple!
Divide by the number that you want to make equal to 1. For 'price per item', divide by the number of items. For 'speed per hour', divide by the number of hours.

Glossary

Rate
A ratio that compares two quantities with different units
Unit rate
A rate where the second quantity is exactly 1 (e.g., 5 dollars per item)
Per
Means 'for each' or 'divided by' - indicates a unit rate
Speed
A unit rate that measures distance traveled per unit of time

Formula Card

Unit Rate

Unit Rate=Total QuantityNumber of Units\text{Unit Rate} = \frac{\text{Total Quantity}}{\text{Number of Units}}

Divide the total by the number of units to get the rate per one unit

Using Unit Rates

Total=Unit Rate×Number of Units\text{Total} = \text{Unit Rate} \times \text{Number of Units}

Multiply the unit rate by any number of units to find the total

Today's challengeOne problem, three tries

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