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VARIATION · LESSON 03

The Constant of Variation

Understand the fixed value that connects directly proportional quantities.

⏱️ 18–22 min 📘 Developing ✏️ Worked examples included

🍫 Start With Chocolate Costs

2 bars cost $6. How much will 5 bars cost? Move the slider to visualize the question before using algebra.

k = 6 ÷ 2 = 3, so C = 3(5) = $15
Bars5 bars
Total cost$15
Bars, b12345
Cost, C ($)3691215

Learning Objectives

You should be able to:

Introduction

This lesson develops The Constant of Variation from meaning and patterns before moving to calculations. Look for what the quantities, shapes or symbols represent, then connect that idea to the method.

Key Notes

Read the definitions and key facts below carefully. Each rule is most useful when you can explain why it fits the situation, not simply recall it.

Visual Exploration

Notice→Connect→Explain

Use the diagrams, tables or examples in this lesson to identify what changes, what stays fixed and which relationship connects the values.

🤔 Think About This...

Two shops sell identical exercise books.

Shop A

5 books cost US$10.

Shop B

5 books cost US$15.

Both costs vary directly with the number of books. What makes the two relationships different?

The price per book is different. That fixed value is the constant of variation.

What Is the Constant of Variation?

In a direct relationship y = kx, the number k tells us how much y is produced by one unit of x.

k = y ÷ x
Key idea: k is the fixed ratio connecting the two variables.
Relationship Meaning of k
Cost = k × number of books Price per book
Distance = k × time Constant speed
Pay = k × hours worked Hourly rate

Worked Example 1: Finding k

y varies directly as x. When x = 9, y = 54. Find k and write the equation.

1

Use k = y ÷ x.

2

k = 54 ÷ 9 = 6.

3

The equation is y = 6x.

Worked Example 2: Interpreting k

Earnings E vary directly as hours h. A worker earns US$42 for 6 hours.

1

E = kh.

2

42 = 6k, so k = 7.

3

E = 7h.

4

k = 7 means the worker earns US$7 per hour.

Finding k from a Table

x 3 5 8
y 12 20 32
y ÷ x 4 4 4

Since every ratio equals 4, the constant is k = 4 and the relationship is y = 4x.

Finding k from a Graph

For a direct-variation graph y = kx, k is also the gradient.

k = gradient = change in y ÷ change in x

A straight line through the origin also passes through (4, 10). Find k.

1

k = y ÷ x.

2

k = 10 ÷ 4 = 2.5.

3

The equation is y = 2.5x.

Comparing Constants

Consider y = 2x and y = 5x.

A larger positive value of k gives a steeper direct-variation graph.

🔍 How Do I Recognise a Question About k?

The question may ask you to:

  • find the constant of proportionality;
  • find the constant of variation;
  • write the formula connecting two variables;
  • find a unit price, hourly rate or constant speed;
  • use one pair of values to predict another.

⚠️ Common Mistakes

Mistake 1: Dividing x by y instead of y by x. For y = kx, use k = y ÷ x.

Mistake 2: Giving k without its meaning in a word problem.

Mistake 3: Using only one table ratio without checking the others when testing proportionality.

✅ Quick Check

If y = 28 when x = 7, find k.
What does k mean in Cost = k × mass?
Which line is steeper: y = 3x or y = 8x?

🧠 Let’s Think Together

Before calculating, identify the information given, the result required and the mathematical relationship that connects them. Predict whether the answer should be larger, smaller or unchanged, then use that prediction to check the result.

Practice Questions

Foundation
  1. Find k and write the equation:
    1. y = 35 when x = 5;
    2. y = 18 when x = 12.
  2. In the equation C = 2.4m, where C is cost in dollars and m is mass in kilograms, explain the meaning of 2.4.
  3. Determine whether the table represents direct variation:
    x 2 4 7
    y 9 18 31.5
Developing
  1. Distance d varies directly as time t. A cyclist travels 48 km in 3 hours.
    1. Find k.
    2. Explain the meaning of k.
    3. Find the distance in 5.5 hours.
  2. A direct-variation graph passes through the origin and the point (6, 15). Find its equation.
Challenge
  1. Two quantities satisfy y = kx. When x increases by 8, y increases by 30. Find k.

✅ Check Your Work

Show practice answers

Foundation

    1. k = 7, so y = 7x.
    2. k = 1.5, so y = 1.5x.
  2. US$2.40 per kilogram.
  3. Yes. Every value of y ÷ x equals 4.5, so y = 4.5x.

Developing

    1. k = 16.
    2. The cyclist’s constant speed is 16 km/h.
    3. 88 km.
    d = kt
    48 = 3k, so k = 16.
    d = 16(5.5) = 88.
  2. y = 2.5x.
    k = 15 ÷ 6 = 2.5.

Challenge

  1. k = 3.75.
    Since y = kx, a change of 8 in x causes a change of 8k in y.
    8k = 30
    k = 30 ÷ 8 = 3.75

Exam Focus

Show the mathematical relationship you are using, keep notation consistent and give a clear final answer. Use a quick estimate, diagram or inverse operation to check that the result is reasonable.

Summary