Solved Exercises on Constant of Proportionality for Grade 7

Master constant of proportionality: direct proportion, unit rates, and proportional relationships through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Direct Proportion with Distance
Exercise 1
A car travels 120 miles in 3 hours at a constant speed. Write an equation relating distance (d) to time (t), and find the constant of proportionality.
Definition:

Constant of Proportionality: The constant value (k) in the equation y = kx, where y is directly proportional to x. It represents the rate of change.

Finding the constant of proportionality method:
  1. Identify the proportional relationship
  2. Write the general form y = kx
  3. Substitute known values for x and y
  4. Solve for k
  5. Write the specific equation
Given Data
d = 120 miles, t = 3 hours
Proportional Form
d = kt
Substitute Values
120 = k(3)
Solve for k
k = 40
Step 1: Identify the relationship

Distance is directly proportional to time when speed is constant

Step 2: Write the general equation

d = kt (where k is the constant of proportionality)

Step 3: Substitute the known values

120 = k × 3

Step 4: Solve for k

k = 120 ÷ 3 = 40

Step 5: Write the specific equation

d = 40t

Constant of proportionality: 40 miles per hour
Final answer:

The equation is d = 40t, and the constant of proportionality is 40 miles per hour.

Applied rules:

• Direct proportion: y = kx where k is constant

• Unit rate: k represents the rate of change

• Consistency: Units must be consistent (miles/hour)

2 Unit Rate Problem
Exercise 2
If 5 apples cost $7.50, find the constant of proportionality and write an equation for the cost (c) in terms of the number of apples (n).
Definition:

Unit Rate: A rate with a denominator of 1, representing the amount per single unit.

Given Values
5 apples = $7.50
Unit Rate
$7.50 ÷ 5 = $1.50
Constant k
$1.50 per apple
Equation
c = 1.50n
Step 1: Identify the proportional relationship

Cost is directly proportional to the number of apples

Step 2: Find the unit rate (constant of proportionality)

k = Total cost ÷ Number of apples = $7.50 ÷ 5 = $1.50 per apple

Step 3: Write the equation

c = kn where c is cost and n is number of apples

c = 1.50n

Step 4: Verify the equation

For n = 5: c = 1.50(5) = $7.50 ✓

Constant: $1.50 per apple, Equation: c = 1.50n
Final answer:

The constant of proportionality is $1.50 per apple, and the equation is c = 1.50n.

Applied rules:

• Unit rate: k = y/x for proportional relationship y = kx

• Cost calculation: Total cost = Unit price × Quantity

• Verification: Check equation with original data

3 Proportional Tables
Exercise 3
Complete the table and find the constant of proportionality if y is directly proportional to x.
x y
2 10
5 25
7 ?
? 40
Definition:

Proportional Table: A table where the ratio of corresponding values is constant, indicating a direct proportional relationship.

First Ratio
10/2 = 5
Second Ratio
25/5 = 5
Constant k
5
Equation
y = 5x
Step 1: Find the constant of proportionality

k = y/x for any pair: k = 10/2 = 5

Verify: k = 25/5 = 5 ✓

Step 2: Complete the table using y = 5x

When x = 7: y = 5(7) = 35

When y = 40: 40 = 5x, so x = 8

Step 3: Verify all ratios equal k

10/2 = 5, 25/5 = 5, 35/7 = 5, 40/8 = 5 ✓

Constant: 5, Completed table: (7,35) and (8,40)
Final answer:

The constant of proportionality is 5. The completed table has y = 35 when x = 7, and x = 8 when y = 40.

Applied rules:

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

• Direct substitution: Use y = kx to find missing values

• Verification: All ratios must equal the same constant

Rules and methods, laws,...
y = kx \text{ where } k = \frac{y}{x}
Proportional Relationship Formula
Direct Proportion
y = kx
Constant Rate
Constant k
k = y/x
Unit Rate
Inverse Proportion
y = k/x
Not Direct
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 at a constant rate

Complete methodology:
  1. Identify the relationship: Determine if y is directly proportional to x
  2. Find k: Calculate k = y/x using known values
  3. Verify: Check that k is the same for all data points
  4. Apply: Use y = kx to find missing values
  5. Interpret: Understand the meaning of k in context
Tip 1: The constant of proportionality is always the unit rate (per 1).
Tip 2: In a graph, proportional relationships pass through the origin (0,0).
Tip 3: All points on a proportional graph lie on the same straight line.
Tip 4: If ratios y/x are not equal, the relationship is not proportional.
Properties of Proportional Relationships: Pass through origin, constant ratio, straight line graph, real-world applications.
Common Applications: Unit conversions, scaling, recipes, speed-distance-time, pricing.
Formulas to know by heart:

• Proportional equation: y = kx

• Constant: k = y/x

• Verification: All ratios y/x must equal k

Solution: Exercises 4 to 5
4 Rate and Time Problem
Exercise 4
A printer prints 240 pages in 4 minutes. How many pages will it print in 7 minutes? What is the constant of proportionality?
Definition:

Rate Problem: A problem involving a constant rate of change between two quantities.

Given Data
240 pages in 4 minutes
Constant k
240 ÷ 4 = 60
Equation
p = 60t
For 7 minutes
p = 60 × 7 = 420
Step 1: Identify the relationship

Pages printed is directly proportional to time

Step 2: Find the constant of proportionality

k = Pages ÷ Time = 240 ÷ 4 = 60 pages per minute

Step 3: Write the equation

p = 60t where p is pages and t is time in minutes

Step 4: Calculate pages for 7 minutes

p = 60 × 7 = 420 pages

Constant: 60 pages per minute, Answer: 420 pages
Final answer:

The constant of proportionality is 60 pages per minute. The printer will print 420 pages in 7 minutes.

Applied rules:

• Rate calculation: Divide total by time to find unit rate

• Direct proportion: y = kx with constant k

• Unit consistency: Ensure units match throughout

5 Weight and Cost Problem
Exercise 5
The cost of apples is directly proportional to their weight. If 2.5 kg of apples cost $7.50, find the constant of proportionality and calculate the cost of 4.2 kg of apples.
Definition:

Direct Proportionality: When two quantities increase or decrease at the same rate.

Given Values
Cost = $7.50, Weight = 2.5 kg
Constant k
$7.50 ÷ 2.5 = $3.00
Equation
c = 3.00w
For 4.2 kg
c = 3.00 × 4.2 = $12.60
Step 1: Identify the proportional relationship

Cost (c) is directly proportional to weight (w)

Step 2: Calculate the constant of proportionality

k = Cost ÷ Weight = $7.50 ÷ 2.5 kg = $3.00 per kg

Step 3: Write the equation

c = 3.00w where c is cost in dollars and w is weight in kg

Step 4: Calculate cost for 4.2 kg

c = 3.00 × 4.2 = $12.60

Constant: $3.00 per kg, Cost: $12.60
Final answer:

The constant of proportionality is $3.00 per kg. The cost of 4.2 kg of apples is $12.60.

Applied rules:

• Proportional equation: y = kx where k is the unit rate

• Unit rate: Cost per unit weight

• Substitution: Replace variable with known value

Key Statistical Concepts and Properties
k = \(\frac{y}{x}\) \text{ where } y = kx
Constant of Proportionality Formula
Key definitions:

Constant of Proportionality: The constant value that relates two proportional quantities

Direct Proportion: When two quantities increase or decrease together at the same rate

Unit Rate: A rate with a denominator of 1, representing the amount per single unit

Complete methodology:
  1. Identify proportionality: Determine if y is proportional to x
  2. Calculate k: Use k = y/x with known values
  3. Verify consistency: Check that k is the same for all data points
  4. Apply the model: Use y = kx for predictions
  5. Interpret meaning: Understand what k represents in context
Tip 1: The constant of proportionality is always the rate per unit.
Tip 2: Proportional relationships form straight lines through the origin.
Tip 3: If the ratio y/x changes, the relationship is not proportional.
Tip 4: Always include units when stating the constant of proportionality.
Properties of Proportional Relationships: Pass through origin, constant rate, linear graph, predictable behavior.
Applications: Used in scaling, unit conversions, recipes, physics, and economics.
Important Statistical Rules:

• y = kx for direct proportionality

• k = y/x for any point (x,y) on the line

• All points satisfy the same constant ratio

• Graph passes through (0,0) and is a straight line

Statistical Visualization: Proportionality Comparison
Exercise 6: Data Visualization
Compare proportional relationships:
Car: Distance = 60 × Time (k = 60 mph)
Printer: Pages = 30 × Minutes (k = 30 ppm)
Apples: Cost = 2.50 × Weight (k = $2.50/kg)

Analysis: The chart shows different proportional relationships with varying constants.

  • Car has the highest rate of change (60)
  • Printer has moderate rate (30)
  • Apples have the lowest rate (2.50)

Questions & Answers

Question: How do I know if a relationship is proportional from a table of values?

Answer: To determine if a relationship is proportional from a table, check these conditions:

Conditions for proportionality:

  • Constant ratio: Calculate y/x for each pair and verify they're all equal
  • Passes through origin: When x = 0, y must equal 0
  • Linear relationship: The graph would form a straight line

Example of proportional table:

x y y/x
1 5 5
2 10 5
3 15 5

Since all ratios equal 5, this is proportional with k = 5.

Question: What's the difference between the constant of proportionality and slope?

Answer: In a proportional relationship, the constant of proportionality and slope are the same value!

Relationship between k and slope:

  • Proportional case: y = kx, slope = k, passes through origin
  • General linear: y = mx + b, slope = m, y-intercept = b
  • Special case: When b = 0, m = k (proportional relationship)

For example:

  • y = 4x: k = 4 (constant of prop.), slope = 4
  • y = 4x + 2: slope = 4, but not proportional (b ≠ 0)

The constant of proportionality is a special case of slope for relationships that pass through the origin!

Question: Can the constant of proportionality be a fraction or decimal?

Answer: Yes, absolutely! The constant of proportionality can be any real number, including fractions and decimals.

Examples of fractional constants:

  • Decimal: y = 0.5x (k = 0.5)
  • Fraction: y = (3/4)x (k = 3/4)
  • Improper fraction: y = (5/2)x (k = 2.5)

Real-world examples:

  • Price per pound: $2.75 per lb (k = 2.75)
  • Speed: 0.6 miles per minute (k = 0.6)
  • Recipe: 3/4 cup sugar per batch (k = 3/4)

The constant of proportionality represents a rate, so it often involves decimals or fractions in real-world applications!