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Calculate and use unit rates
Learn step-by-step with practice exercises built right in.
How do you compare prices, speeds, or other rates? Unit rates show the amount per ONE unit - making it easy to compare and make smart decisions!
A rate compares two quantities with different units.
Examples:
Written as: quantity₁/quantity₂ or quantity₁ per quantity₂
Key: The units are DIFFERENT (miles and hours, dollars and pounds, etc.)
A unit rate is a rate where the second quantity is 1.
Examples:
The "per" means "for each one"
Unit rates make comparing easy!
A car travels 240 miles in 4 hours. What is the unit rate in miles per hour?
Step 1: Identify the quantities. Distance = 240 miles Time = 4 hours
Step 2: Write as a rate. 240 miles / 4 hours
Step 3: Divide to find unit rate. 240 ÷ 4 = 60
Step 4: Include units. 60 miles per hour (or 60 mph)
Answer: 60 miles per hour
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Method: Divide the first quantity by the second quantity
Formula: Unit Rate = First Quantity ÷ Second Quantity
Example 1: 120 miles in 2 hours
Unit Rate = 120 miles ÷ 2 hours = 60 miles/hour
Answer: 60 miles per hour (60 mph)
Example 2: $15 for 3 books
Unit Rate = 5/book
Answer: $5 per book
Example: 180 students in 6 buses. Find students per bus.
Step 1: Identify quantities
Step 2: Divide 180 ÷ 6 = 30
Step 3: Write with units 30 students per bus
Answer: 30 students/bus
Unit price is a special unit rate used for shopping.
It shows price per ONE item (or per one unit like ounce, pound, etc.)
Example: $8 for 4 cans
Unit price = 2 per can
Why useful? Compare different package sizes!
Example: Which is the better buy?
Find unit price for each:
Option A: 2 per bottle Option B: 1.80 per bottle
Option B is better! (2)
Rule: Lower unit price = better deal (for same product)
Speed is distance per unit of time.
Example 1: 240 miles in 4 hours
Speed = 240 miles ÷ 4 hours = 60 mph
Example 2: 150 meters in 25 seconds
Speed = 150 m ÷ 25 s = 6 m/s
Common speed units:
Sometimes you get fractional unit rates.
Example: 3 pizzas for 8 people
Unit rate = 3 ÷ 8 = 3/8 pizza per person
Each person gets 3/8 of a pizza
Or flip it: 8 ÷ 3 = 8/3 = 2⅔ people per pizza
Choose the perspective that makes sense for the problem!
Rate: Any comparison of quantities
Unit Rate: Second quantity is 1
To convert rate to unit rate: DIVIDE!
Grocery Shopping:
Gas Mileage:
Typing Speed:
Heart Rate:
Wages:
Example: Which copy shop is cheaper?
Find unit prices: Shop A: 0.05 per copy Shop B: 0.045 per copy
Shop B is cheaper! (by half a cent per copy)
Even small differences add up with large quantities!
Scaling recipes uses unit rates!
Example: Recipe uses 6 cups flour for 4 loaves
Unit rate = 6 ÷ 4 = 1.5 cups per loaf
To make 7 loaves: 7 × 1.5 = 10.5 cups of flour
Unit rate helps scale up or down!
All these mean the same:
The "/" means "per"
Example formats:
Example: 5 pounds for $7
Unit price = 1.40 per pound
Money is usually rounded to 2 decimal places (cents)
Example: 3 items for $5
Unit price = 1.666...
Round: $1.67 per item (round up for money)
Population density is people per square mile (or km).
Example: City has 450,000 people in 150 square miles
Density = 450,000 ÷ 150 = 3,000 people per square mile
Higher density = more crowded!
Work rate: Tasks per hour Fuel efficiency: Miles per gallon Production rate: Items per day
Example: Factory makes 1,200 widgets in 8 hours
Rate = 1,200 ÷ 8 = 150 widgets per hour
Unit rates help measure efficiency!
❌ Mistake 1: Dividing in wrong order
❌ Mistake 2: Comparing without unit rates
❌ Mistake 3: Forgetting units
❌ Mistake 4: Rounding too early
❌ Mistake 5: Choosing higher price
Problem: If 4 pizzas cost $36, how much do 7 pizzas cost?
Step 1: Find unit rate 9 per pizza
Step 2: Multiply by new quantity 7 × 63
Answer: $63 for 7 pizzas
To find unit rate:
To compare options:
To use unit rate:
Unit Rate Formula: First Quantity ÷ Second Quantity
Common Unit Rates:
Comparing:
Remember: Always include UNITS!
Tip 1: Set up division carefully
Tip 2: Check reasonableness
Tip 3: Use calculator for complex division
Tip 4: Compare apples to apples
Unit rate shows amount per ONE unit:
Finding unit rates: Unit Rate = First Quantity ÷ Second Quantity
Common uses:
Benefits:
Key skill: Unit rates help you make smart choices in everyday life - from shopping to planning trips to budgeting time!
Mastering unit rates empowers you to analyze and compare rates in the real world!
If 3 pounds of apples cost $4.50, what is the unit price (cost per pound)?
Step 1: Identify the quantities. Cost = $4.50 Weight = 3 pounds
Step 2: Set up the division. Unit price = Total cost ÷ Number of pounds
Step 3: Calculate. 1.50
Step 4: State the unit rate. $1.50 per pound
Answer: $1.50 per pound
A printer prints 450 pages in 15 minutes. How many pages does it print per minute?
Step 1: Set up the rate. 450 pages / 15 minutes
Step 2: Divide to get unit rate. 450 ÷ 15 = 30
Step 3: Include units. 30 pages per minute
Answer: 30 pages per minute
Store A sells 5 pounds of rice for 19.20. Which store has the better unit price?
Step 1: Find unit price for Store A. 2.55 per pound
Step 2: Find unit price for Store B. 2.40 per pound
Step 3: Compare. Store A: 2.40 per pound
A water tank is being filled at a rate of 8 gallons every 3 minutes. At this rate, how long will it take to fill a 120-gallon tank?
Step 1: Find the unit rate (gallons per minute). 8 gallons / 3 minutes Unit rate = 8/3 gallons per minute
Step 2: Set up the equation. Time = Total gallons ÷ Rate Time = 120 ÷ (8/3)
Step 3: Divide by a fraction (multiply by reciprocal). 120 ÷ (8/3) = 120 × (3/8) = (120 × 3) / 8 = 360 / 8 = 45 minutes
Check: In 45 minutes at 8 gallons per 3 minutes: 45 ÷ 3 = 15 (number of 3-minute periods) 15 × 8 = 120 gallons ✓
Answer: 45 minutes
Store B has the LOWER price (better deal).
Answer: Store B has the better unit price at 2.55 at Store A)