Solved Exercises on Real-Life Proportional Problems in Grade 7

Master real-life proportional problems: unit rates, scaling, conversion factors, and practical applications through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Unit Rate Shopping Problem
Exercise 1
A store sells 5 pounds of apples for $12.50. At this rate, how much would 8 pounds of apples cost? What is the unit rate?
Definition:

Unit Rate: A rate expressed as a quantity of 1 (e.g., cost per pound, miles per hour). Proportional Relationship: When quantities change at a constant rate.

Method for solving unit rate problems:
  1. Find the unit rate (cost per 1 unit)
  2. Set up a proportion or use the unit rate to find the unknown
  3. Verify the answer makes sense in context
Unit Rate
$12.50 ÷ 5 = $2.50/pound
8 Pounds Cost
8 × $2.50 = $20.00
Step 1: Find the unit rate

Unit rate = Total cost ÷ Total quantity

Unit rate = $12.50 ÷ 5 pounds = $2.50 per pound

Step 2: Set up the proportional relationship

Cost = Unit rate × Quantity

Cost = $2.50 × 8 pounds = $20.00

Step 3: Verify the answer

Check: 5 pounds cost $12.50 → $12.50 ÷ 5 = $2.50/pound

8 pounds cost $20.00 → $20.00 ÷ 8 = $2.50/pound ✓

Unit rate: $2.50/pound | 8 pounds cost: $20.00
Final answer:

The unit rate is $2.50 per pound. Eight pounds of apples would cost $20.00.

Applied rules:

• Unit Rate Formula: Unit rate = Total amount ÷ Number of units

• Proportional Multiplication: Total cost = Unit rate × Quantity

• Constant Rate: The unit rate remains the same regardless of quantity

Key Concept:

Unit rates allow us to compare prices and make decisions across different quantities. The unit rate stays constant in proportional relationships.

2 Speed and Distance Problem
Exercise 2
A car travels 180 miles in 3 hours at a constant speed. How far would it travel in 5 hours? What is the speed of the car?
Definition:

Speed: Distance traveled per unit of time (distance ÷ time). Constant Speed: Maintains the same rate of travel, creating a proportional relationship between distance and time.

Speed
180 miles ÷ 3 hours = 60 mph
Distance in 5 hours
60 mph × 5 hours = 300 miles
Step 1: Find the speed (unit rate)

Speed = Distance ÷ Time

Speed = 180 miles ÷ 3 hours = 60 miles per hour

Step 2: Apply the speed to the new time

Distance = Speed × Time

Distance = 60 mph × 5 hours = 300 miles

Step 3: Verify using proportions

180 miles / 3 hours = ? miles / 5 hours

? = (180 × 5) ÷ 3 = 900 ÷ 3 = 300 miles ✓

Speed: 60 mph | Distance in 5 hours: 300 miles
Final answer:

The car's speed is 60 mph. It would travel 300 miles in 5 hours.

Applied rules:

• Distance-Speed-Time Formula: Distance = Speed × Time

• Constant Rate: At constant speed, distance and time are proportional

• Unit Rate: Speed is the unit rate of distance over time

🚗
Travel Planning
Calculate arrival times and fuel needs
✈️
Flight Calculations
Estimate flight times and distances
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Exercise Tracking
Monitor pace and performance
3 Recipe Scaling Problem
Exercise 3
A recipe calls for 2 cups of flour to serve 6 people. How many cups of flour are needed to serve 15 people? What is the rate of flour per person?
Definition:

Recipe Scaling: Adjusting ingredient amounts proportionally to serve different numbers of people. Rate per Person: Amount of ingredient needed per serving.

Flour per person
2 cups ÷ 6 people = 1/3 cup/person
Flour for 15 people
1/3 × 15 = 5 cups
Step 1: Find the rate of flour per person

Flour per person = Total flour ÷ Number of people

Flour per person = 2 cups ÷ 6 people = 1/3 cup per person

Step 2: Calculate flour needed for 15 people

Total flour = Flour per person × Number of people

Total flour = 1/3 cup × 15 people = 5 cups

Step 3: Verify using proportions

2 cups / 6 people = ? cups / 15 people

? = (2 × 15) ÷ 6 = 30 ÷ 6 = 5 cups ✓

Rate: 1/3 cup per person | 15 people: 5 cups
Final answer:

The rate is 1/3 cup of flour per person. To serve 15 people, 5 cups of flour are needed.

Applied rules:

• Unit Rate Calculation: Ingredient per person = Total ingredient ÷ Total people

• Proportional Scaling: New amount = Unit rate × New quantity

• Constant Ratio: The ratio of ingredient to people remains constant

Key Concept:

Recipe scaling is a practical application of proportional relationships. Each ingredient maintains the same ratio to the number of servings.

Real-Life Proportional Problems: Rules and Methods
\(\text{Unit Rate} = \frac{\text{Total Amount}}{\text{Number of Units}}\)
Unit Rate Formula
Unit Rate
\(r = \frac{a}{b}\)
Amount per unit
Proportional Relationship
\(y = kx\)
Direct variation
Conversion Factor
\(1 = \frac{a}{b} \text{ or } \frac{b}{a}\)
Equivalent ratio
Key definitions:

Unit Rate: A rate expressed as a quantity of 1 (e.g., miles per hour, cost per item)

Proportional Relationship: A relationship where two quantities change at a constant rate

Conversion Factor: A ratio used to convert from one unit to another while maintaining proportionality

Problem-solving methodology:
  1. Identify Quantities: Determine what quantities are proportional
  2. Find Unit Rate: Calculate the rate per unit
  3. Set Up Proportion: Create a proportional relationship
  4. Solve: Use multiplication or cross-products to find the unknown
  5. Verify: Check that the answer makes sense in context
Tip 1: Always identify the unit rate first - it's the foundation of proportional problems.
Tip 2: In word problems, look for phrases like "per," "each," or "for every" to identify unit rates.
Tip 3: Set up proportions with consistent units on top and bottom.
Tip 4: Check your answer by verifying the unit rate remains constant.
Real-Life Applications: Shopping, cooking, travel, work, measurements, scaling.
Common Pitfalls: Mixing units, forgetting to keep rates consistent, incorrect setup of proportions.
Proportional Problem Rules:

• Constant Rate: The unit rate remains unchanged

• Direct Variation: As one quantity increases, the other increases proportionally

• Unit Consistency: Units must match in proportions

• Cross Products: In a/b = c/d, ad = bc

Solution: Exercises 4 to 5
4 Work Rate Problem
Exercise 4
A printer prints 240 pages in 8 minutes. At this rate, how many pages can it print in 15 minutes? How long would it take to print 450 pages?
Definition:

Work Rate: The amount of work completed per unit of time. When work rate is constant, work done is proportional to time spent.

Pages per minute
240 ÷ 8 = 30 pages/min
Pages in 15 min
30 × 15 = 450 pages
Time for 450 pages
450 ÷ 30 = 15 min
Step 1: Find the printing rate

Rate = Pages ÷ Time

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

Step 2: Calculate pages in 15 minutes

Pages = Rate × Time

Pages = 30 pages/min × 15 min = 450 pages

Step 3: Calculate time for 450 pages

Time = Pages ÷ Rate

Time = 450 pages ÷ 30 pages/min = 15 minutes

15 min: 450 pages | 450 pages: 15 min
Final answer:

The printer can print 450 pages in 15 minutes. It would take 15 minutes to print 450 pages.

Applied rules:

• Work Rate Formula: Rate = Work ÷ Time

• Proportional Relationship: Work = Rate × Time

• Constant Rate: Rate remains the same under normal conditions

5 Currency Conversion Problem
Exercise 5
If $1 USD equals 0.85 EUR, how many euros can you get for $120 USD? How many USD do you need to get 85 EUR?
Definition:

Currency Conversion: Converting between different currencies using exchange rates. Exchange rates represent proportional relationships between currencies.

Exchange Rate
1 USD = 0.85 EUR
120 USD to EUR
120 × 0.85 = 102 EUR
85 EUR to USD
85 ÷ 0.85 = 100 USD
Step 1: Identify the exchange rate

Exchange rate: 1 USD = 0.85 EUR

This means for every 1 USD, you get 0.85 EUR

Step 2: Convert $120 USD to EUR

Euros = USD × Exchange rate

Euros = $120 × 0.85 = 102 EUR

Step 3: Convert 85 EUR to USD

USD = Euros ÷ Exchange rate

USD = 85 EUR ÷ 0.85 = 100 USD

$120 USD = 102 EUR | 85 EUR = $100 USD
Final answer:

You can get 102 EUR for $120 USD. You need $100 USD to get 85 EUR.

Applied rules:

• Exchange Rate: Fixed proportional relationship between currencies

• Conversion Formula: New currency = Original × Exchange rate

• Reverse Conversion: Original = New currency ÷ Exchange rate

Comprehensive Summary: Real-Life Proportional Problems
\(\text{Unit Rate} = \frac{\text{Total Amount}}{\text{Number of Units}}, \quad y = kx\)
Core Proportional Formulas
Core Definitions:

Unit Rate: A rate expressed per single unit of measure (e.g., $3.50 per pound, 60 miles per hour, 2 cups per batch).

Proportional Relationship: A relationship where two quantities change at a constant rate (y = kx).

Conversion Factor: A ratio that expresses equivalent measurements in different units (1 foot = 12 inches).

Problem-Solving Steps:
  1. Read Carefully: Identify the quantities that are proportional
  2. Find Unit Rate: Calculate the rate per single unit
  3. Set Up Relationship: Create the proportional equation or ratio
  4. Solve: Use multiplication, division, or cross-products to find the unknown
  5. Check: Verify the answer is reasonable and units match
Quick Tip: Look for keywords like "per," "each," "for every," or "rate" to identify unit rates.
Memory Aid: In proportional problems, "what you do to one side, you do to the other."
Verification: Always check that your unit rate remains consistent throughout the problem.
Unit Awareness: Keep track of units - they should cancel appropriately in calculations.
Real-Life Applications: Shopping (unit pricing), cooking (recipe scaling), travel (speed/distance), work (production rates), conversions (currency, measurements).
Common Scenarios: Gas mileage, hourly wages, grocery shopping, construction projects, manufacturing production.
Key Rules and Properties:

• Constant Rate: In proportional relationships, the rate of change remains constant

• Direct Variation: As one quantity increases, the other increases by the same factor

• Unit Consistency: Units must match when setting up proportions

• Cross Products Property: In a/b = c/d, the cross products are equal (ad = bc)

• Reciprocal Relationships: If a/b = k, then b/a = 1/k

🛒
Shopping
Compare unit prices to find best deals
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Cooking
Scale recipes for different serving sizes
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Travel
Calculate distance, time, and fuel needs
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Finance
Currency conversion, tax calculations
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Business
Production rates, pricing models
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Construction
Material calculations, scale drawings

Questions & Answers

Question: How do I know when to multiply and when to divide in proportional problems? I keep getting confused.

Answer: Great question! Here's a simple way to remember:

  • When finding TOTAL: Multiply unit rate × number of units
  • When finding UNIT RATE: Divide total ÷ number of units
  • When finding NUMBER OF UNITS: Divide total ÷ unit rate

Example: If apples cost $2 per pound:

  • To find cost of 5 pounds: $2 × 5 = $10 (multiply)
  • To find unit rate from $10 for 5 pounds: $10 ÷ 5 = $2 (divide)
  • To find pounds for $10: $10 ÷ $2 = 5 pounds (divide)

Think about what makes sense: if you want more of something, you usually multiply. If you're breaking down to find a rate, you divide.

Question: What's the difference between a unit rate and a regular rate? When should I use each?

Answer: Here's the distinction:

  • Regular Rate: A comparison of two quantities (e.g., 120 miles in 2 hours)
  • Unit Rate: A rate where the second quantity is 1 (e.g., 60 miles per 1 hour)

When to use each:

  • Unit Rate: Use when comparing different options or when you need to scale up/down (better gas mileage, better price per unit)
  • Regular Rate: Use when describing a situation as it happened (trip took 3 hours to cover 180 miles)

Example: If a car goes 240 miles in 4 hours:

  • Regular rate: 240 miles in 4 hours
  • Unit rate: 60 miles per hour

The unit rate is more useful for planning future trips!

Question: I sometimes get mixed up with which quantities to put in the numerator and denominator. How do I decide?

Answer: The key is to follow the phrase "something per something else." The first part goes in the numerator, the second in the denominator.

  • Miles per hour: Miles ÷ Hours (miles in numerator)
  • Cost per item: Cost ÷ Items (cost in numerator)
  • Students per teacher: Students ÷ Teachers (students in numerator)

Memory trick: The word "per" means "divided by." So whatever comes after "per" goes in the denominator.

Context matters: Think about what makes sense. If you're calculating how much of one thing corresponds to one unit of another, that's your unit rate.

For example, if you want to know how much something costs per pound, you divide the total cost by the total pounds.