Learning Objectives
- explain inverse variation;
- recognise inverse-proportion language;
- use the equation y = k/x;
- identify a constant product;
- solve inverse-variation problems.
Introduction
This lesson develops Inverse 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
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...
A fixed journey must be completed.
The time halves. One quantity increases while the other decreases in a predictable way.
What Is Inverse Variation?
Two quantities vary inversely when increasing one causes the other to decrease so that their product remains constant.
| x | 1 | 2 | 4 | 8 |
|---|---|---|---|---|
| y | 24 | 12 | 6 | 3 |
| xy | 24 | 24 | 24 | 24 |
Worked Example 1
y varies inversely as x. When x = 5, y = 12. Find y when x = 8.
Write y = k/x.
12 = k/5, so k = 60.
The equation is y = 60/x.
When x = 8, y = 60/8 = 7.5.
Worked Example 2: Workers and Time
Eight workers complete a fixed task in 15 days. How long would 12 equally efficient workers take?
Workers × days = constant.
k = 8 × 15 = 120.
Days = 120/12 = 10.
The Graph of Inverse Variation
The graph of y = k/x is curved rather than straight. It approaches the axes but does not normally touch them.
🔍 How Do I Recognise Inverse Variation?
Look for:
- “varies inversely as...”
- “is inversely proportional to...”
- a constant product;
- one quantity doubling while the other halves;
- fixed-work, fixed-distance or fixed-volume situations.
✅ Quick Check
🧠 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- y varies inversely as x. If y = 9 when x = 4, find k.
- Given y = 48/x, find:
- y when x = 6;
- x when y = 3.
- Determine whether x: 2, 4, 8 and y: 20, 10, 5 represent inverse variation.
- Six taps fill a tank in 10 hours. Assuming equal flow rates, how long would 15 taps take?
- Time t varies inversely as speed v for a fixed journey. The journey takes 4 hours at 75 km/h. Find the time at 100 km/h.
- y varies inversely as x. When x increases by 3 from 5 to 8, y decreases by 9. Find the relationship between y and x.
✅ Check Your Work
Show practice answers
Foundation
- k = 36.
-
- 8
- 16
- Yes. Every product xy equals 40.
Developing
-
4 hours.
taps × time = constant
6 × 10 = 60
time = 60/15 = 4 -
3 hours.
vt = k
75 × 4 = 300
t = 300/100 = 3
Challenge
-
y = 120/x.
Let y = k/x.
At x = 5, y = k/5.
At x = 8, y = k/8.
k/5 − k/8 = 9
3k/40 = 9
k = 120
Common Mistakes
- Choosing a rule before identifying what the question describes.
- Skipping working or changing notation part-way through a solution.
- Accepting an answer without checking its size, sign or units.
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
- Inverse variation uses y = k/x.
- The product xy is constant.
- When one quantity doubles, the other halves.
- Its graph is curved.