Solved Exercises on Unit Rates in Grade 7

Master unit rates: finding unit prices, comparing unit rates, calculating speed, and solving real-world problems through these 10 detailed exercises with visual learning tools.

Solution: Exercises 1 to 3
1 Unit Price Calculation
Exercise 1
A 12-pack of soda costs $4.80. What is the unit price per can?
Definition:

Unit rate: A rate with a denominator of 1. Unit price is the cost per single item.

Method:
  1. Identify the total cost and total quantity
  2. Divide the total cost by the total quantity
  3. Express the result as cost per single unit
Step 1: Identify total cost and quantity

Total cost: $4.80, Total quantity: 12 cans

Step 2: Divide cost by quantity

Unit price = $4.80 ÷ 12 = $0.40 per can

Step 3: State the unit rate

The unit price is $0.40 per can

Unit Price: $4.80 ÷ 12
$0.40 per can
Total Cost Quantity Unit Price $4.80 12 cans $0.40/can
Unit price: $0.40 per can
Final answer:

The unit price is $0.40 per can.

Applied rules:

• Unit rate formula: Unit rate = Total cost ÷ Total quantity

• Division: Divide to find the rate per single unit

• Unit notation: Express as cost per unit

2 Speed Calculation
Exercise 2
A car travels 240 miles in 4 hours. What is the unit rate in miles per hour?
Definition:

Speed: A unit rate that compares distance traveled to time elapsed.

Step 1: Identify distance and time

Distance: 240 miles, Time: 4 hours

Step 2: Divide distance by time

Speed = 240 miles ÷ 4 hours = 60 miles per hour

Step 3: State the unit rate

The speed is 60 miles per hour

Speed: 240 miles ÷ 4 hours
60 mph
Distance Time Speed 240 miles 4 hours 60 mph
Speed: 60 miles per hour
Final answer:

The car travels at 60 miles per hour.

Applied rules:

• Speed formula: Speed = Distance ÷ Time

• Unit rate: Express as distance per unit of time

• Division: Divide to find the rate per single unit of time

3 Rate of Work
Exercise 3
A factory produces 450 widgets in 5 hours. What is the unit rate of production in widgets per hour?
Definition:

Production rate: A unit rate that compares quantity produced to time taken.

Step 1: Identify quantity and time

Quantity: 450 widgets, Time: 5 hours

Step 2: Divide quantity by time

Production rate = 450 widgets ÷ 5 hours = 90 widgets per hour

Step 3: State the unit rate

The production rate is 90 widgets per hour

Production: 450 widgets ÷ 5 hours
90 widgets/hour
Widgets Hours Rate 450 5 90/hour
Production rate: 90 widgets per hour
Final answer:

The factory produces 90 widgets per hour.

Applied rules:

• Rate calculation: Quantity ÷ Time

• Unit rate: Express as quantity per unit of time

• Division: Divide to find the rate per single unit of time

Solution: Exercises 4 to 6
4 Comparing Unit Prices
Exercise 4
Store A sells 16 oz of cereal for $3.20. Store B sells 20 oz of the same cereal for $4.50. Which store offers the better deal?
Definition:

Comparing unit rates: Calculating unit prices to determine the better value.

Step 1: Calculate unit price for Store A

Store A: $3.20 ÷ 16 oz = $0.20 per oz

Step 2: Calculate unit price for Store B

Store B: $4.50 ÷ 20 oz = $0.225 per oz

Step 3: Compare the unit prices

$0.20 < $0.225, so Store A offers the better deal

Store A
$0.20/oz
VS
Store B
$0.225/oz
Store Price Size Unit Price A $3.20 16 oz $0.20/oz B $4.50 20 oz $0.225/oz
Store A: $0.20/oz, Better Deal
Final answer:

Store A offers the better deal at $0.20 per ounce compared to Store B's $0.225 per ounce.

Applied rules:

• Unit price calculation: Price ÷ Quantity

• Comparison: Lower unit price indicates better value

• Decimal comparison: Compare decimal values to determine better deal

5 Running Pace
Exercise 5
Sarah runs 3 miles in 24 minutes. What is her pace in minutes per mile? How long would it take her to run 5 miles at this pace?
Definition:

Pace: A unit rate that measures time per unit distance.

Step 1: Calculate pace (minutes per mile)

Pace = 24 minutes ÷ 3 miles = 8 minutes per mile

Step 2: Calculate time for 5 miles

Time = 5 miles × 8 minutes per mile = 40 minutes

Step 3: State the answers

Pace: 8 minutes per mile, Time for 5 miles: 40 minutes

Pace: 24 min ÷ 3 miles
8 min/mile
Time: 5 miles × 8 min/mile
40 minutes
Distance Time Pace 3 miles 24 min 8 min/mile 5 miles 40 min 8 min/mile
Pace: 8 min/mile, Time for 5 miles: 40 min
Final answer:

Sarah's pace is 8 minutes per mile. It would take her 40 minutes to run 5 miles at this pace.

Applied rules:

• Pace calculation: Time ÷ Distance

• Multiplication: Unit rate × Quantity

• Consistency: Maintain the same rate for proportional calculations

6 Fuel Efficiency
Exercise 6
A car travels 315 miles using 15 gallons of gasoline. What is the car's fuel efficiency in miles per gallon? How far could it travel on 20 gallons?
Definition:

Fuel efficiency: A unit rate that measures distance traveled per unit of fuel consumed.

Step 1: Calculate fuel efficiency

Miles per gallon = 315 miles ÷ 15 gallons = 21 miles per gallon

Step 2: Calculate distance for 20 gallons

Distance = 20 gallons × 21 miles per gallon = 420 miles

Step 3: State the answers

Fuel efficiency: 21 mpg, Distance for 20 gallons: 420 miles

Efficiency: 315 miles ÷ 15 gallons
21 mpg
Distance: 20 gallons × 21 mpg
420 miles
Miles Gallons MPG 315 15 21 420 20 21
Fuel efficiency: 21 mpg, Distance: 420 miles
Final answer:

The car's fuel efficiency is 21 miles per gallon. It could travel 420 miles on 20 gallons.

Applied rules:

• Fuel efficiency: Distance ÷ Fuel used

• Multiplication: Unit rate × Quantity

• Proportional reasoning: Same rate applies to different quantities

Unit Rates Visual Guide
Unit Rate = Total ÷ Quantity
Unit Rate Formula
Unit Price
$/item
Speed
distance/time
Production
items/time
Efficiency
output/input
Unit Rate Process:
Step 1: Identify the total amount and the total quantity
Step 2: Divide the total amount by the total quantity
Step 3: Express the result as a rate per single unit
Step 4: Compare with other rates if needed
Step 5: Apply the rate to new quantities if required
Tip 1: Always express unit rates with the word "per" to indicate the relationship.
Tip 2: When comparing deals, the lower unit rate is usually the better value.
Tip 3: Round decimal unit rates to appropriate precision for the context.
Common errors: Dividing in wrong order, forgetting to include units, calculation mistakes.
Success strategies: Always divide total by quantity, check units, verify reasonableness.
Essential concepts:

• Unit rate: Rate with denominator of 1

• Formula: Total ÷ Quantity

• Comparison: Lower is better for prices

• Application: Proportional relationships

Solution: Exercises 7 to 10
7 Earning Rate
Exercise 7
Marcus earns $180 for working 20 hours. What is his hourly wage? How much would he earn for 35 hours at this rate?
Definition:

Hourly wage: A unit rate that measures earnings per unit of time worked.

Step 1: Calculate hourly wage

Hourly wage = $180 ÷ 20 hours = $9 per hour

Step 2: Calculate earnings for 35 hours

Earnings = 35 hours × $9 per hour = $315

Step 3: State the answers

Hourly wage: $9/hour, Earnings for 35 hours: $315

Wage: $180 ÷ 20 hours
$9/hour
Earnings: 35 hours × $9/hour
$315
Earnings Hours Hourly Wage $180 20 $9/hour $315 35 $9/hour
Hourly wage: $9/hour, Earnings: $315
Final answer:

Marcus's hourly wage is $9 per hour. He would earn $315 for 35 hours of work.

Applied rules:

• Hourly wage: Total earnings ÷ Total hours

• Multiplication: Unit rate × New quantity

• Consistency: Same rate applies to different time periods

8 Food Consumption Rate
Exercise 8
A family consumes 12 loaves of bread in 4 weeks. What is their consumption rate in loaves per week? How many loaves would they consume in 10 weeks?
Definition:

Consumption rate: A unit rate that measures quantity used per unit of time.

Step 1: Calculate consumption rate

Rate = 12 loaves ÷ 4 weeks = 3 loaves per week

Step 2: Calculate consumption for 10 weeks

Loaves = 10 weeks × 3 loaves per week = 30 loaves

Step 3: State the answers

Consumption rate: 3 loaves/week, Consumption for 10 weeks: 30 loaves

Rate: 12 loaves ÷ 4 weeks
3 loaves/week
Consumption: 10 weeks × 3 loaves/week
30 loaves
Loaves Weeks Rate 12 4 3/week 30 10 3/week
Rate: 3 loaves/week, Consumption: 30 loaves
Final answer:

The family's consumption rate is 3 loaves per week. They would consume 30 loaves in 10 weeks.

Applied rules:

• Rate calculation: Total quantity ÷ Total time

• Multiplication: Unit rate × New time period

• Proportional reasoning: Same rate applies to different time periods

9 Data Transfer Rate
Exercise 9
A computer transfers 4.5 GB of data in 3 minutes. What is the transfer rate in GB per minute? How long would it take to transfer 12 GB at this rate?
Definition:

Data transfer rate: A unit rate that measures data transferred per unit of time.

Step 1: Calculate transfer rate

Rate = 4.5 GB ÷ 3 minutes = 1.5 GB per minute

Step 2: Calculate time for 12 GB

Time = 12 GB ÷ 1.5 GB per minute = 8 minutes

Step 3: State the answers

Transfer rate: 1.5 GB/min, Time for 12 GB: 8 minutes

Rate: 4.5 GB ÷ 3 min
1.5 GB/min
Time: 12 GB ÷ 1.5 GB/min
8 minutes
Data (GB) Time (min) Rate (GB/min) 4.5 3 1.5 12 8 1.5
Rate: 1.5 GB/min, Time: 8 minutes
Final answer:

The transfer rate is 1.5 GB per minute. It would take 8 minutes to transfer 12 GB.

Applied rules:

• Rate calculation: Total data ÷ Total time

• Time calculation: Total data ÷ Rate

• Division: Divide total by rate to find time

10 Painting Rate Problem
Exercise 10
Two painters can paint 300 square feet in 4 hours. What is their combined painting rate in square feet per hour? How long would it take them to paint 1,200 square feet?
Definition:

Work rate: A unit rate that measures output per unit of time.

Step 1: Calculate combined painting rate

Rate = 300 sq ft ÷ 4 hours = 75 square feet per hour

Step 2: Calculate time for 1,200 sq ft

Time = 1,200 sq ft ÷ 75 sq ft per hour = 16 hours

Step 3: State the answers

Painting rate: 75 sq ft/hour, Time for 1,200 sq ft: 16 hours

Rate: 300 sq ft ÷ 4 hours
75 sq ft/hour
Time: 1,200 sq ft ÷ 75 sq ft/hour
16 hours
Square Feet Hours Rate (sq ft/hour) 300 4 75 1,200 16 75
Rate: 75 sq ft/hour, Time: 16 hours
Final answer:

Their combined painting rate is 75 square feet per hour. It would take them 16 hours to paint 1,200 square feet.

Applied rules:

• Work rate: Total output ÷ Total time

• Time calculation: Total work ÷ Rate

• Proportional reasoning: Same rate applies to different amounts of work

Comprehensive Summary: Unit Rates
Core Concepts & Definitions:

Unit Rate: A rate with a denominator of 1, expressing how much of one quantity corresponds to a single unit of another quantity.

Rate: A comparison of two quantities with different units.

Unit Price: The cost per single item or unit of measurement.

Speed: Distance traveled per unit of time.

Work Rate: Amount of work done per unit of time.

Pace: Time taken per unit of distance.

Efficiency: Output per unit of input.

Proportional Relationship: A relationship where the ratio between two quantities remains constant.

Core Rules & Formulas:

Essential Formulas:

  • Unit Rate = Total Amount ÷ Total Quantity
  • Speed = Distance ÷ Time
  • Unit Price = Total Cost ÷ Total Quantity
  • Work Rate = Total Work ÷ Total Time

Key Rules:

  • Always divide the total by the quantity to find the unit rate
  • Express unit rates with "per" to indicate the relationship
  • Lower unit prices indicate better value when comparing purchases
  • Unit rates remain constant in proportional relationships
Step-by-Step Unit Rate Process:
  1. Identify quantities: Determine the total amount and the total quantity
  2. Set up division: Divide the total amount by the total quantity
  3. Calculate: Perform the division to find the unit rate
  4. Express: Write the unit rate with proper units
  5. Compare: If comparing, determine which rate is better
  6. Apply: Use the unit rate to solve additional problems
Examples & Applications:

Simple Unit Price Example:

  • Item costs $10 for 5 pieces
  • Unit price = $10 ÷ 5 = $2 per piece

Speed Example:

  • Travel 150 miles in 3 hours
  • Speed = 150 miles ÷ 3 hours = 50 mph

Work Rate Example:

  • Produce 120 items in 4 hours
  • Rate = 120 items ÷ 4 hours = 30 items per hour
Tips, Tricks & Common Pitfalls:

Tips & Tricks:

  • Always include units when expressing unit rates
  • When comparing prices, calculate unit rates to make accurate comparisons
  • Round unit rates to appropriate precision for the context
  • Check that your unit rate makes sense in the real-world context
  • Remember: "per" means "divided by"

Common Pitfalls:

  • Dividing in the wrong order (quantity ÷ cost instead of cost ÷ quantity)
  • Forgetting to include units in the final answer
  • Mixing up which quantity to use as the divisor
  • Not recognizing when to use unit rates for comparison
  • Calculation errors with decimals or fractions
Key Notes for Memorization:
  • Unit rate = Total ÷ Quantity
  • Always divide total by quantity
  • Unit rates help compare different options
  • Lower unit prices = better value
  • Unit rates remain constant in proportional relationships
  • Express as "per" to show the relationship
  • Check reasonableness of your answer
Additional Unit Rates Practice
Unit Rate = Total ÷ Quantity
Unit Rate Formula
Key definitions:

Unit rate: A rate with a denominator of 1, expressing quantity per single unit

Rate: A comparison of two quantities with different units

Proportional relationship: A relationship where the ratio remains constant

Unit rate methodology:
  1. Identify: Determine the total amount and total quantity
  2. Divide: Total ÷ Quantity to find unit rate
  3. Express: Write with proper units and "per" notation
  4. Compare: Use to compare different options
  5. Apply: Use unit rate to solve related problems
Tip 1: Always include units in your final answer.
Tip 2: Use unit rates to make accurate comparisons.
Tip 3: Check if your answer is reasonable in the context.
Tip 4: Remember: divide total by quantity to find unit rate.
Common errors: Wrong division order, missing units, calculation mistakes.
Success strategies: Systematic approach, verification, proper units.
Essential concepts:

• Unit rate = Total ÷ Quantity

• Express with "per" notation

• Use for comparisons

• Remain constant in proportional relationships

Questions & Answers

Question: How do I know which number to divide by which when calculating unit rates?

Answer: Always divide the total amount by the total quantity:

  • For unit price: Total cost ÷ Total quantity (e.g., $10 ÷ 5 items = $2/item)
  • For speed: Total distance ÷ Total time (e.g., 120 miles ÷ 2 hours = 60 mph)
  • For work rate: Total work ÷ Total time (e.g., 30 tasks ÷ 5 hours = 6 tasks/hour)

The unit rate tells you "how much per one unit," so divide the total by the number of units.

Question: When comparing unit prices, how do I know which is the better deal?

Answer: When comparing unit prices, the lower unit price represents the better deal:

  • If Store A offers $0.25 per ounce and Store B offers $0.30 per ounce, Store A is cheaper
  • Lower unit price means you pay less for the same amount of product
  • Always calculate unit prices to the same precision for accurate comparison

However, consider quality and other factors in real-world decisions, not just unit price.

Question: What's the difference between speed and pace? Are they both unit rates?

Answer: Yes, both are unit rates but they measure opposite things:

  • Speed: Distance per unit time (e.g., 60 miles per hour) - higher is faster
  • Pace: Time per unit distance (e.g., 8 minutes per mile) - lower is faster

Speed measures how much distance you cover in a unit of time, while pace measures how much time it takes to cover a unit of distance.