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VARIATION · LESSON 02

Direct Variation

Learn how quantities change together in the same ratio.

⏱️ 18–22 min 📘 Foundation ✏️ Worked examples included

💼 Start With Wages

A worker earns $3 per hour. Move the slider and watch the hours, wages, equation, table, coordinates and graph update together.

y = 3x → y = 3(5) = 15
Hours5
Wages$15
x (hours)12345
y (wages $)3691215

The highlighted coordinate is (5, 15). Every plotted point follows the same rule.

Learning Objectives

You should be able to:

Introduction

This lesson develops Direct 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...

One kilogram of tomatoes costs US$2.

1 kgUS$2
2 kgUS$4
3 kgUS$6
5 kgUS$10
What happens to the cost when the mass is multiplied by 2, 3 or 5?

The cost is multiplied by exactly the same factor. This is direct variation.

What Is Direct Variation?

Two quantities vary directly when they increase or decrease in the same ratio.

Key idea: If x is multiplied by a number, y is multiplied by the same number.
y ∝ x

The symbol ∝ means “is proportional to”. To calculate with the relationship, we replace it with an equation:

y = kx

The fixed number k is called the constant of variation or constant of proportionality.

How Can We Test Direct Variation?

For direct variation, the ratio y ÷ x remains constant.

x 2 4 6 10
y 6 12 18 30
y ÷ x 3 3 3 3
The constant ratio shows that y = 3x.

Worked Example 1: Finding a Missing Value

y varies directly as x. When x = 4, y = 20. Find y when x = 7.

1

Write y = kx.

2

Substitute x = 4 and y = 20: 20 = 4k.

3

k = 5.

4

The relationship is y = 5x.

5

When x = 7, y = 5(7) = 35.

Worked Example 2: A Real-Life Problem

The cost C dollars of printing flyers varies directly as the number n printed. Printing 80 flyers costs US$12. Find the cost of 150 flyers.

1

C = kn.

2

12 = 80k, so k = 12 ÷ 80 = 0.15.

3

C = 0.15n.

4

For 150 flyers, C = 0.15(150) = US$22.50.

Interpret k in context. Here k = 0.15 means each flyer costs US$0.15.

Direct Variation Graph

The graph of y = kx is a straight line passing through the origin.

x = 0 ⟹ y = k(0) = 0

This is why the origin is an essential feature of a direct-proportion graph.

A straight-line graph that does not pass through the origin is linear, but it is not direct variation.

🔍 How Do I Recognise Direct Variation in an Exam?

Look for phrases such as:

  • “varies directly as...”
  • “is directly proportional to...”
  • “increases in the same ratio...”
  • “when one doubles, the other doubles...”

Also check for a constant ratio and a graph through the origin.

⚠️ Common Mistakes

Mistake 1: Writing y = x + k.
Direct variation uses multiplication: y = kx.

Mistake 2: Forgetting to find k before calculating a new value.

Mistake 3: Assuming any straight-line relationship is direct variation.

✅ Quick Check

If y ∝ x and y = 18 when x = 6, find k.
If y = 4x, what happens to y when x is tripled?
Does y = 3x + 2 represent direct variation? Explain.

🧠 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. State whether each table represents direct variation:
    1. x: 1, 2, 3; y: 4, 8, 12
    2. x: 1, 2, 3; y: 5, 9, 13
  2. y varies directly as x. If y = 24 when x = 8, find k.
  3. Given y = 7x, find:
    1. y when x = 5;
    2. x when y = 56.
Developing
  1. The cost C of mangoes varies directly as their mass m. If 3 kg costs US$4.50, find:
    1. the equation connecting C and m;
    2. the cost of 8 kg.
  2. The distance d travelled at constant speed varies directly as time t. A vehicle covers 135 km in 1.5 hours. Find the distance covered in 4 hours.
Challenge
  1. y varies directly as x. When x increases from 6 to 15, y increases by 27. Find the equation connecting y and x.

✅ Check Your Work

Attempt every question before opening the answers.

Show practice answers

Foundation

    1. Yes. y ÷ x = 4 for every pair.
    2. No. The ratios are 5, 4.5 and 13/3, so they are not constant.
  2. k = 3.
    y = kx
    24 = 8k
    k = 3
    1. y = 35.
    2. x = 8.

Developing

    1. C = 1.5m.
    2. US$12.
    4.50 = 3k, so k = 1.50.
    For 8 kg: C = 1.50(8) = 12.
  2. 360 km.
    d = kt
    135 = 1.5k, so k = 90 km/h.
    d = 90(4) = 360 km.

Challenge

  1. y = 3x.
    Let y = kx.
    The change in y is k(15) − k(6) = 9k.
    9k = 27, so k = 3.

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