Solved Exercises on Understanding Proportional Relationships in Grade 7

Master proportional relationships: ratios, rates, constant of proportionality, and real-world applications through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Constant of Proportionality
Exercise 1
If 5 apples cost $3.75, find the constant of proportionality and write an equation relating the number of apples to their cost.
Definition:

Constant of proportionality: The ratio between two proportional quantities, represented by k in y = kx

Finding the constant:
  1. Identify the relationship between quantities
  2. Set up the ratio (y/x)
  3. Calculate the constant k
  4. Write the equation y = kx
Ratio
$3.75/5
Constant
k = $0.75
Equation
C = 0.75n
Step 1: Identify the relationship

Cost (C) is proportional to number of apples (n)

Step 2: Set up the ratio

k = Cost ÷ Number of apples = $3.75 ÷ 5 = $0.75 per apple

Step 3: Write the equation

C = k × n, so C = 0.75n

k = $0.75 per apple, Equation: C = 0.75n
Final answer:

The constant of proportionality is $0.75 per apple, and the equation is C = 0.75n

Applied rules:

• Direct proportion: y = kx where k is constant

• Unit rate: Divide dependent variable by independent variable

• Rate of change: The constant k represents the rate of change

2 Unit Rate Comparison
Exercise 2
Store A sells 8 oranges for $4.80. Store B sells 12 oranges for $7.20. Which store offers the better deal per orange?
Definition:

Unit rate: The rate expressed as a quantity of 1, such as cost per item or distance per unit time

Store A
$0.60/orange
Store B
$0.60/orange
Comparison
Equal
Step 1: Calculate Store A's unit rate

Cost per orange = $4.80 ÷ 8 = $0.60 per orange

Step 2: Calculate Store B's unit rate

Cost per orange = $7.20 ÷ 12 = $0.60 per orange

Step 3: Compare the unit rates

Both stores charge $0.60 per orange

Both stores offer the same deal: $0.60 per orange
Final answer:

Both stores offer the same unit rate of $0.60 per orange

Applied rules:

• Unit conversion: Divide total cost by total quantity

• Comparison: Compare unit rates to determine better value

• Equivalent ratios: Equal unit rates mean equivalent proportions

3 Proportional Table Completion
Exercise 3
Complete the table showing a proportional relationship:
Hours Worked: 2, 4, ?, 8
Earnings: $15, ?, $45, ?
Definition:

Proportional table: A table where the ratio between corresponding values remains constant

Find k
k = $7.50/hour
Complete
4→$30, 6→$45, 8→$60
Step 1: Find the constant of proportionality

k = Earnings ÷ Hours = $15 ÷ 2 hours = $7.50 per hour

Step 2: Complete missing values

For 4 hours: Earnings = $7.50 × 4 = $30

For ? hours with $45: Hours = $45 ÷ $7.50 = 6 hours

For 8 hours: Earnings = $7.50 × 8 = $60

Step 3: Verify all ratios are equal

$15/2 = $30/4 = $45/6 = $60/8 = $7.50 per hour ✓

Completed table: Hours: 2, 4, 6, 8; Earnings: $15, $30, $45, $60
Final answer:

Hours: 2, 4, 6, 8; Earnings: $15, $30, $45, $60

Applied rules:

• Constant ratio: In a proportional relationship, y/x = k for all pairs

• Missing values: Use the constant to find unknown values

• Verification: Check that all ratios equal the constant k

Proportional Relationships Guide
y = kx
Direct Proportion
Constant
k = y/x
Constant of proportionality
Unit Rate
Rate = Total/Quantity
Per-unit comparison
Cross Product
a/b = c/d → ad = bc
Proportion equality
Key definitions:

Proportional relationship: Two quantities where the ratio of corresponding values is constant

Constant of proportionality: The constant value k in the equation y = kx

Direct variation: When one variable increases, the other increases at a constant rate

Proportionality identification:
  1. Check ratios: Calculate y/x for each pair of values
  2. Verify consistency: All ratios should be equal
  3. Find k: The common ratio is the constant of proportionality
  4. Write equation: Express as y = kx
Tip 1: Plot the points on a graph - they should form a straight line through the origin.
Tip 2: When comparing deals, always calculate the unit rate to make fair comparisons.
Tip 3: In proportional relationships, when x = 0, y must also equal 0.
Characteristics: Linear relationship passing through origin, constant rate of change, equal ratios.
Applications: Unit pricing, scaling recipes, map scales, speed-distance-time, earning calculations.
Solution: Exercises 4 to 5
4 Speed and Distance
Exercise 4
A car travels at a constant speed of 65 miles per hour. Write an equation relating distance (d) to time (t). How far will the car travel in 3.5 hours?
Definition:

Constant speed: Distance is directly proportional to time when speed is constant

Equation
d = 65t
After 3.5h
d = 227.5 mi
Step 1: Identify the relationship

Distance (d) is proportional to time (t) when speed is constant

Step 2: Identify the constant of proportionality

Speed = 65 miles per hour, so k = 65

Step 3: Write the equation

d = k × t, so d = 65t

Step 4: Calculate distance for 3.5 hours

d = 65 × 3.5 = 227.5 miles

Equation: d = 65t; Distance in 3.5 hours: 227.5 miles
Final answer:

The equation is d = 65t, and the car will travel 227.5 miles in 3.5 hours

Applied rules:

• Distance-speed-time: d = rt where r is the rate/speed

• Constant rate: Creates a proportional relationship

• Unit consistency: Ensure units match (miles, hours, etc.)

5 Recipe Scaling
Exercise 5
A recipe for 4 people requires 1.5 cups of flour. How much flour is needed for 10 people? Write the equation showing the relationship.
Definition:

Recipe scaling: Ingredients scale proportionally with the number of servings

Find k
k = 0.375 cups/person
For 10 people
3.75 cups
Equation
F = 0.375n
Step 1: Find the constant of proportionality

k = Flour ÷ People = 1.5 cups ÷ 4 people = 0.375 cups per person

Step 2: Write the equation

F = k × n, so F = 0.375n, where F = cups of flour, n = number of people

Step 3: Calculate for 10 people

F = 0.375 × 10 = 3.75 cups

Step 4: Verify the solution

Original: 1.5 cups for 4 people → 1.5/4 = 0.375 cups per person

New: 3.75 cups for 10 people → 3.75/10 = 0.375 cups per person ✓

Flour needed for 10 people: 3.75 cups; Equation: F = 0.375n
Final answer:

3.75 cups of flour are needed for 10 people, with the equation F = 0.375n

Applied rules:

• Scaling: Ingredients scale directly with the number of servings

• Unit rate: Find the amount needed per serving

• Multiplication: Multiply unit rate by desired number of servings

Proportional Relationships Summary
y = kx, where k = y/x
Proportional Relationship
Key definitions:

Proportional relationship: A relationship between two variables where their ratio is constant

Constant of proportionality: The constant value k in the equation y = kx

Direct variation: When one variable increases, the other increases by the same factor

Identification methods:
  1. Table method: Check if y/x is the same for all pairs
  2. Graph method: Look for a straight line passing through (0,0)
  3. Equation method: Check if it can be written as y = kx
  4. Real-world context: Look for "per", "for every", or "at a constant rate"
Tip 1: Always verify by checking multiple pairs of values in tables.
Tip 2: The graph of a proportional relationship always passes through the origin (0,0).
Tip 3: Unit rates are the same as the constant of proportionality.
Tip 4: Cross multiply to solve proportion problems: a/b = c/d → ad = bc.
Characteristics: Linear graph through origin, constant rate of change, y = kx form.
Real-world examples: Speed/time, cost/quantity, ingredient/servings, distance/scale.
Essential formulas to remember:

• Proportional equation: y = kx

• Constant of proportionality: k = y/x

• Proportion: a/b = c/d → ad = bc

• Unit rate: Rate per 1 unit = Total ÷ Quantity

• Scale factor: New value = Original value × Scale factor

Proportional Relationships Visualization
Exercise 6: Multiple Proportional Scenarios
Compare different proportional relationships:
Cost vs Quantity: $0.75 per item
Distance vs Time: 65 mph
Ingredients vs Servings: 0.375 cups per person

Analysis: The chart shows how different proportional relationships have different constants of proportionality.

  • Cost vs Quantity: y = 0.75x (steeper slope indicates higher rate)
  • Distance vs Time: y = 65x (much steeper due to larger constant)
  • Ingredients vs Servings: y = 0.375x (gentler slope)

Questions & Answers

Question: In Exercise 1, how do I know that $3.75 ÷ 5 = $0.75? Could I have made the equation the other way around (n = 0.75C)?

Answer: Great question about the setup! Let's break this down:

Division calculation: $3.75 ÷ 5 = $0.75 because we're finding the cost per apple. We divide the total cost by the number of apples.

Variable relationship: In this context, we want to find the cost (dependent variable) based on the number of apples (independent variable). So we use C = kn where k = $0.75 per apple.

Alternative equation: You could write n = (1/k)C, which would be n = (1/0.75)C = (4/3)C ≈ 1.33C. This tells you how many apples you can buy for a given cost, which is a valid but different relationship.

The choice depends on what you're trying to predict. Usually, we express the dependent variable (what we want to find) in terms of the independent variable (what we control or know).

Question: In Exercise 2, both stores had the same unit rate. Does this ever happen in real life, and what should I do if it does?

Answer: Yes, equal unit rates do occur in real life! Here's what to consider:

  • Price matching: Competitors often match each other's prices
  • Standard pricing: Some items have industry-standard prices
  • Promotional periods: Sales may temporarily equalize prices

When unit rates are equal:

  • Consider other factors like quality, location, or service
  • Look for bulk discounts or special offers
  • Check if there are minimum purchase requirements

In math problems, equal unit rates demonstrate that the proportional relationships are identical, meaning the same rate applies in both situations.

Question: How can I quickly tell if a table shows a proportional relationship without calculating all the ratios?

Answer: Here are some quick checks for proportional relationships:

  1. Origin check: Does (0,0) appear in the table? In proportional relationships, when x = 0, y must also equal 0
  2. Pattern check: If x doubles, does y also double? If x triples, does y triple?
  3. Unit rate consistency: Calculate y/x for the first two pairs - if they're equal, calculate one more to confirm

Example: If you have (2, 5) and (4, 10), notice that when x doubled from 2 to 4, y also doubled from 5 to 10. This suggests a proportional relationship with k = 2.5.

However, always verify with at least one additional pair to be certain, as coincidental patterns can occur.

Question: Why does the graph of a proportional relationship always pass through the origin (0,0)?

Answer: This happens because of the mathematical definition of proportional relationships:

Equation perspective: In y = kx, when x = 0, then y = k(0) = 0. So the point (0,0) is always on the graph.

Real-world perspective: Think about our examples:

  • If you buy 0 apples, you pay $0
  • If you work 0 hours, you earn $0
  • If you travel for 0 hours, you cover 0 distance

Physical meaning: In any proportional situation, having zero of the input variable results in zero of the output variable. This fundamental relationship is why the line always passes through the origin.

This is a key characteristic that distinguishes proportional relationships from other linear relationships (which may have a y-intercept other than 0).

Question: What's the difference between rate and unit rate? Are they the same as the constant of proportionality?

Answer: These terms are related but have distinct meanings:

  • Rate: A ratio comparing two different kinds of quantities (e.g., 65 miles per hour, $3 per 2 apples)
  • Unit rate: A rate where the second quantity is 1 unit (e.g., 65 miles per 1 hour, $1.50 per 1 apple)
  • Constant of proportionality: The constant k in y = kx, which is numerically equal to the unit rate in proportional relationships

Connection: In proportional relationships, the constant of proportionality k is equal to the unit rate. For example, if y = 0.75x, the constant k = 0.75, which means 0.75 dollars per 1 unit, or $0.75 per unit.

Example: If 6 apples cost $3.60, the rate is "$3.60 per 6 apples", the unit rate is "$0.60 per apple", and the constant of proportionality is k = 0.60 in the equation C = 0.60n.