Unit Rates
Learn how to find and use unit rates to compare quantities and solve real-world problems.
Definition
- Price per item: 12 dollars for 4 apples = 3 dollars per apple
- Speed: 150 km in 3 hours = km per hour
- Efficiency: 240 pages in 8 days = pages per day
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Worked Examples
A pack of 8 water bottles costs 12 dollars. What is the price per bottle?
Identify the ratio
12 dollars for 8 bottles →
Divide to find unit rate
→ 1.50 dollars
Write the unit rate
1.50 dollars per 1 bottle → 1.50 dollars per bottle
Answer: The unit rate is 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
Ratio Tape Diagram
Adjust the ratio parts using + and - buttons.
Interactive Sandbox
Expression Calculator
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Practice Problems
16 problemsA pack of 6 juice boxes costs 9 dollars. What is the unit rate?
Why It Matters
- 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?
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: dollars per liter.
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: Write the mathematical expression
Find unit price for each: Price / Liters
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 hours.
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
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
Divide the total by the number of units to get the rate per one unit
Using Unit Rates
Multiply the unit rate by any number of units to find the total