Solved Exercises on Complex Rates in Grade 7

Master complex rates: compound rates, rate conversions, and multi-step rate problems through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Rate Conversion Problem
Exercise 1
A car travels at 60 miles per hour. Convert this rate to feet per second. (Hint: 1 mile = 5,280 feet, 1 hour = 3,600 seconds)
Definition:

Rate Conversion: Changing a rate from one set of units to another while preserving the relationship between quantities.

Method for rate conversion:
  1. Identify the starting and ending units
  2. Set up conversion factors as fractions equal to 1
  3. Multiply the original rate by the conversion factors
  4. Cancel units and perform calculations
Original Rate
60 mi/hr
Conversion Factors
5280 ft/mi, 1 hr/3600 sec
Final Rate
88 ft/sec
Step 1: Write the original rate

60 miles per hour = 60 mi/hr

Step 2: Set up conversion factors

Convert miles to feet: 5280 ft/1 mi

Convert hours to seconds: 1 hr/3600 sec

Step 3: Multiply by conversion factors

(60 mi/hr) × (5280 ft/mi) × (1 hr/3600 sec)

Step 4: Cancel units and calculate

(60 × 5280 × 1)/(1 × 1 × 3600) ft/sec

= 316,800/3600 ft/sec = 88 ft/sec

60 mi/hr = 88 ft/sec
Final answer:

The car travels at 88 feet per second.

Applied rules:

• Dimensional Analysis: Multiply by conversion factors equal to 1

• Unit Cancellation: Arrange factors to cancel unwanted units

• Rate Preservation: The actual rate remains unchanged, only units change

Key Concept:

Conversion factors are ratios equal to 1 (like 5280 ft/1 mi = 1). Multiplying by these doesn't change the value, only the units.

2 Compound Rate Problem
Exercise 2
A factory produces widgets at a rate of 150 widgets per hour. Each widget requires 2.5 pounds of raw material. How many pounds of raw material are needed per day if the factory operates 8 hours per day?
Definition:

Compound Rate: A rate that combines multiple related rates. Involves multiplying different rates together to find a combined effect.

Widgets per hour
150 widgets/hr
Material per widget
2.5 lbs/widget
Daily material
3,000 lbs/day
Step 1: Identify individual rates

Production rate: 150 widgets per hour

Material requirement: 2.5 pounds per widget

Operating time: 8 hours per day

Step 2: Calculate widgets per day

Widgets per day = Widgets per hour × Hours per day

Widgets per day = 150 × 8 = 1,200 widgets per day

Step 3: Calculate material per day

Material per day = Widgets per day × Material per widget

Material per day = 1,200 × 2.5 = 3,000 pounds per day

3,000 pounds of raw material per day
Final answer:

The factory needs 3,000 pounds of raw material per day.

Applied rules:

• Chain Multiplication: Multiply related rates sequentially

• Unit Tracking: Keep track of units to ensure proper cancellation

• Multi-Step Problem Solving: Break complex problems into simpler parts

1
Widgets per hour
→
2
Widgets per day
→
3
Material per day
3 Multi-Step Rate Problem
Exercise 3
A printer prints 240 pages in 8 minutes. How many pages will it print in 2.5 hours? Express your answer in both pages and reams (1 ream = 500 pages).
Definition:

Multi-Step Rate Problems: Problems requiring multiple conversions and calculations to reach the final answer.

Printing Rate
30 pages/min
Time Conversion
2.5 hrs = 150 min
Final Answer
4,500 pages = 9 reams
Step 1: Find the printing rate

Rate = Total pages ÷ Total time

Rate = 240 pages ÷ 8 minutes = 30 pages per minute

Step 2: Convert time to matching units

2.5 hours = 2.5 × 60 = 150 minutes

Step 3: Calculate total pages

Total pages = Rate × Time

Total pages = 30 pages/min × 150 min = 4,500 pages

Step 4: Convert to reams

Reams = Total pages ÷ Pages per ream

Reams = 4,500 ÷ 500 = 9 reams

4,500 pages = 9 reams
Final answer:

The printer will print 4,500 pages, which equals 9 reams.

Applied rules:

• Unit Consistency: Ensure all time units match before calculating

• Sequential Calculation: Solve step-by-step to avoid errors

• Final Conversion: Express answer in requested units

Key Concept:

Multi-step rate problems require careful unit management. Always ensure units match before performing calculations.

Complex Rates: Rules and Methods
\(\text{New Rate} = \text{Original Rate} \times \frac{\text{New Unit}}{\text{Old Unit}}\)
Rate Conversion Formula
Rate Conversion
\(r_1 \times \frac{u_2}{u_1} = r_2\)
Converting between units
Compound Rate
\(r = r_1 \times r_2\)
Combining related rates
Multi-Step Rate
\(r_n = r_1 \times r_2 \times ... \times r_n\)
Sequential rate operations
Key definitions:

Complex Rate: A rate that involves multiple units or requires conversion from one unit system to another.

Rate Conversion: Changing a rate from one set of units to another while preserving the relationship between quantities.

Compound Rate: A rate that combines multiple related rates to find a combined effect.

Complex rate problem-solving methodology:
  1. Identify All Units: List all units involved in the problem
  2. Plan Conversions: Determine necessary conversion factors
  3. Set Up Calculations: Arrange factors to cancel units properly
  4. Perform Calculations: Calculate step-by-step to avoid errors
  5. Verify Units: Ensure final answer has correct units
Tip 1: Write out all units during calculations to track cancellations.
Tip 2: Use conversion factors that equal 1 (like 60 min/1 hr).
Tip 3: Break multi-step problems into smaller, manageable parts.
Tip 4: Check that your final units match what's requested.
Real-Life Applications: Unit conversions (mph to m/s), compound rates (cost per person-hour), multi-step calculations (production planning).
Common Pitfalls: Incorrect unit arrangements, forgotten conversions, mismatched units in calculations.
Complex Rate Rules:

• Unit Preservation: Conversion factors must equal 1 to preserve the rate value

• Dimensional Analysis: Units must cancel properly to reach the desired unit

• Sequential Operations: Perform rate operations in logical sequence

• Verification: Check that the magnitude of your answer makes sense

Solution: Exercises 4 to 5
4 Flow Rate Conversion
Exercise 4
Water flows through a pipe at a rate of 500 gallons per minute. Convert this rate to cubic feet per hour. (1 cubic foot = 7.48 gallons)
Definition:

Flow Rate Conversion: Converting fluid flow rates between different volume and time units.

Starting Rate
500 gal/min
Conversion Factors
1 ft³/7.48 gal, 60 min/1 hr
Final Rate
4,011.23 ft³/hr
Step 1: Write the original rate

500 gallons per minute = 500 gal/min

Step 2: Identify conversion factors

Convert gallons to cubic feet: 1 ft³/7.48 gal

Convert minutes to hours: 60 min/1 hr

Step 3: Set up the calculation

(500 gal/min) × (1 ft³/7.48 gal) × (60 min/1 hr)

Step 4: Perform the calculation

(500 × 1 × 60)/(1 × 7.48 × 1) ft³/hr

= 30,000/7.48 ft³/hr ≈ 4,011.23 ft³/hr

500 gal/min ≈ 4,011.23 ft³/hr
Final answer:

The flow rate is approximately 4,011.23 cubic feet per hour.

Applied rules:

• Dimensional Analysis: Arrange factors to cancel gallons and minutes

• Unit Matching: Ensure all time units are consistent

• Precision: Round appropriately based on context

💧
Volume
gal ↔ ft³
⏱️
Time
min ↔ hr
🌊
Flow Rate
gal/min ↔ ft³/hr
5 Combined Labor Rate
Exercise 5
Three workers can complete a job in 8 hours. Worker A earns $15/hour, Worker B earns $18/hour, and Worker C earns $20/hour. What is the total labor cost for completing the job?
Definition:

Combined Labor Rate: Calculating total costs when multiple entities work together at different individual rates.

Individual Rates
A:$15, B:$18, C:$20/hour
Time Worked
8 hours each
Total Cost
$424
Step 1: Identify individual hourly rates

Worker A: $15 per hour

Worker B: $18 per hour

Worker C: $20 per hour

Step 2: Calculate individual costs

Worker A cost: $15 × 8 = $120

Worker B cost: $18 × 8 = $144

Worker C cost: $20 × 8 = $160

Step 3: Calculate total cost

Total cost = $120 + $144 + $160 = $424

Total labor cost: $424
Final answer:

The total labor cost for completing the job is $424.

Applied rules:

• Individual Calculation: Calculate each worker's cost separately

• Summation: Add individual costs to get total cost

• Time Consistency: All workers work the same duration

Comprehensive Summary: Complex Rates
\(\text{New Rate} = \text{Original Rate} \times \prod_{i=1}^{n} \frac{u_i}{v_i}, \quad \text{where} \frac{u_i}{v_i} = 1\)
Complex Rate Conversion Formula
Core Definitions:

Complex Rate: A rate involving multiple units or requiring conversion between different unit systems.

Rate Conversion: The process of changing a rate from one unit system to another while preserving the underlying relationship.

Compound Rate: A rate formed by combining multiple related rates through multiplication.

Complex Rate Problem-Solving Steps:
  1. Identify All Units: List every unit present in the problem
  2. Plan Conversion Path: Determine which units need to be converted and in what order
  3. Set Up Conversion Factors: Create fractions equal to 1 that will cancel unwanted units
  4. Arrange Factors: Position conversion factors so units cancel appropriately
  5. Calculate Sequentially: Perform calculations step-by-step to avoid errors
Quick Tip: Write units next to every number during calculations to track cancellations.
Memory Aid: Conversion factors must equal 1 (like 60 min/1 hr or 1 km/1000 m).
Strategy: For complex problems, break them into smaller unit conversion steps.
Verification: Check that your final units match what's requested and the magnitude makes sense.
Real-Life Applications: Engineering (flow rates), science (unit conversions), business (compound costs), physics (velocity conversions).
Common Scenarios: Travel (mph to m/s), cooking (cups to liters), finance (compound interest rates), manufacturing (production rates).
Key Rules and Properties:

• Unit Equality: Conversion factors must equal 1 (numerator = denominator in different units)

• Dimensional Analysis: Units must cancel properly to yield desired result

• Sequential Operations: Perform conversions in logical order

• Rate Preservation: The actual rate remains constant, only units change

• Verification Principle: Final units must match those required by the problem

🚗
Speed
mph ↔ m/s
💧
Flow
gal/min ↔ L/s
💰
Cost
$/hr ↔ $/day
📏
Distance
mi ↔ km

Questions & Answers

Question: I get confused about which conversion factor to use. How do I know whether to use 5280 ft/1 mi or 1 mi/5280 ft?

Answer: Great question! The key is to arrange the conversion factor so that the units you want to cancel are in both the numerator and denominator.

Rule: Put the unit you want to eliminate in the opposite position from where it appears in the original measurement.

Example: Converting 60 miles to feet:

  • Start with: 60 miles
  • Want to eliminate "miles"
  • "Miles" is currently in the numerator
  • So put "miles" in the denominator of the conversion factor: 5280 ft/1 mi
  • 60 miles × 5280 ft/1 mi = 316,800 ft

The "miles" cancel out, leaving you with feet!

Always ask: "What units do I want to cancel?" Then arrange the conversion factor accordingly.

Question: What's the difference between a compound rate and a regular rate? Can you give more examples?

Answer: Here's the distinction:

  • Regular Rate: A simple comparison of two different units (e.g., 60 miles per hour)
  • Compound Rate: A rate that combines multiple related rates (e.g., cost per person-hour)

More examples of compound rates:

  • Person-hours: If 5 workers work 8 hours each, that's 40 person-hours
  • Cost per person-mile: Airfare divided by passengers and distance
  • Calories per serving per hour: Energy expenditure during exercise
  • Dollars per square foot per year: Rent rates for commercial spaces

Compound rates often combine rates that have meaningful relationships in real-world contexts.

They're useful for making complex comparisons and calculations more manageable.

Question: When solving multi-step rate problems, I sometimes lose track of what I'm calculating. How can I stay organized?

Answer: Organization is crucial for complex rate problems! Here are some strategies:

1. Write down all given information:

  • List all rates, quantities, and units
  • Circle what you're asked to find

2. Plan your approach:

  • Identify intermediate steps needed
  • Determine what units need to match
  • Sketch a path from given to unknown

3. Show your work clearly:

  • Label each step with what you're calculating
  • Write units with every number
  • Box your final answer with correct units

4. Check your work:

  • Verify units cancel correctly
  • Ensure the answer makes sense in context
  • Check that final units match the question

Taking time to organize prevents costly mistakes in complex problems!