Overview
What you’ll learn
- Recognise direct proportion and write it as y ∝ x, y = kx.
- Find the constant of proportionality from one pair of values and use it to predict others.
- Recognise inverse proportion and write it as y ∝ 1/x, y = k/x.
- Identify a proportional relationship from a table by testing ratios or products.
- Describe the graphs of direct and inverse proportion.
- Solve real-world problems involving direct, inverse and squared proportion.
In 30 seconds
- 1Direct proportion: both quantities rise and fall in the same ratio — written y ∝ x, with equation y = kx.
- 2Inverse proportion: one rises as the other falls so that the product stays constant — y ∝ 1/x, with equation y = k/x.
- 3Find k from one known pair of values, then reuse the equation for every other question.
- 4A direct-proportion graph is a straight line through the origin; an inverse-proportion graph is a curve approaching both axes.
- 5Table test: if the ratios y/x match, it is direct; if the products xy match, it is inverse.
- 6Proportional forms can wear other shapes too, such as y ∝ x² — spot the pattern, find k, substitute.
Notes · 1 of 6
Direct proportion
Double one, double the other: the ratio never moves.
Two quantities are in direct proportion when one is always a fixed multiple of the other. Six pens cost three times what two pens cost, because each pen has the same price. In symbols, y ∝ x, and the ∝ becomes an equation with a constant: y = kx.
| Pens (x) | 2 | 4 | 6 | 10 |
|---|---|---|---|---|
| Cost (y) | $3 | $6 | $9 | $15 |
Every y/x here equals 1.5 — the price per pen, which is exactly the constant k. The unitary method (find one, then scale) and the k-method are the same idea wearing different clothes: k = y ÷ x.
Notes · 2 of 6
The constant k does the predicting
One honest pair of values fixes the whole relationship.
- 11. Write the proportion form
y ∝ x becomes y = kx.
- 22. Substitute the known pair
Put in one full (x, y) pair from the question.
- 33. Solve for k
Divide y by x.
- 44. Predict
Use y = kx with the new value — and state the finished equation.
| Step | Working |
|---|---|
| Substitute | 15 = k × 5 |
| Solve | k = 3 |
| Equation | y = 3x |
| Predict at x = 12 | y = 3 × 12 = 36 |
Notes · 3 of 6
Inverse proportion
More hands, less time: the product is the thing that never changes.
Two quantities are in inverse proportion when one increases by the same factor that the other decreases — doubling the workers halves the time. The fixed quantity is the product: y ∝ 1/x, written as the equation y = k/x, or equivalently xy = k.
| Workers (x) | 2 | 4 | 6 | 8 |
|---|---|---|---|---|
| Days (y) | 24 | 12 | 8 | 6 |
Every product xy equals 48 — the total work in worker-days. That product is the k of the inverse relationship. To predict, compute k from one pair, then either divide (y = k/x) or multiply (x = k/y).
Notes · 4 of 6
Proportional graphs
Direct is a straight line through zero; inverse is a curve dodging both axes.
y = kx is the equation of a straight line through the origin with gradient k: zero x gives zero y, and the line climbs steadily. y = k/x is a very different animal: it sweeps down a curve that approaches both axes without touching them, because dividing by a huge x sends y near 0, and x = 0 itself is forbidden.
| Relationship | Equation | Graph | Passes through origin? |
|---|---|---|---|
| Direct | y = kx | straight line, gradient k | yes |
| Inverse | y = k/x | curve approaching both axes | no (x = 0 impossible) |
| Squared form | y = kx² | upward curve starting at origin | yes |
Notes · 5 of 6
Direct or inverse? Test the table
Matching ratios point to direct; matching products point to inverse.
- 11. Divide across
Compute y ÷ x for every row: all equal → direct proportion.
- 22. Multiply across
Compute x × y for every row: all equal → inverse proportion.
- 33. Beware neither
If neither test passes, the data is not proportional at all.
- 44. Write the equation
State y = kx or y = k/x with your k before answering further parts.
| Row | y ÷ x | x × y |
|---|---|---|
| (1, 24) | 24 | 24 |
| (2, 12) | 6 | 24 |
| (3, 8) | 2.67… | 24 |
Notes · 6 of 6
Proportion in the real world
Recipes, fuel, timetables and squared surprises — the k-routine handles them all.
| Context | Relationship | Why |
|---|---|---|
| Cost of rice by the kilogram | direct | unit price is fixed |
| Workers and days for one job | inverse | total worker-days is fixed |
| Speed and time over one journey | inverse | distance is fixed |
| Distance fallen and time | y ∝ x² | physics doubling rule — treat as a squared form |
The universal routine survives every context: name the variables, choose the form, find k from one pair, answer with the equation. Only the final step — rejecting impossible values like negative times — needs human judgement.
Revise
Key terms
The words examiners expect you to use precisely.
- Direct proportion
- A relationship where two quantities increase or decrease in the same ratio: y = kx.
- Inverse proportion
- A relationship where one quantity rises as the other falls with a constant product: y = k/x.
- Constant of proportionality
- The fixed number k linking the variables; y ÷ x for direct, x × y for inverse.
- Unitary method
- Solving by finding the value for one unit first, then scaling.
- Proportion symbol (∝)
- The “is proportional to” sign; it becomes an equals sign once k is found.
- Ratio
- A comparison of two quantities by division; constant ratio signals direct proportion.
Show 6 more key termsShow fewer key terms
- Constant product
- The fixed value of x × y in inverse proportion, such as worker-days.
- Origin
- The point (0, 0); a direct-proportion graph always passes through it.
- Squared proportion
- A form such as y ∝ x², where one quantity varies with the square of the other.
- Worker-days
- The product of workers and days — the fixed total work in an inverse-proportion job.
- Scale factor
- The factor by which one quantity is multiplied, forcing the other to follow its rule.
- Sanity check
- A quick direction test: in direct proportion both quantities climb together; in inverse they trade off.
Revise
Common mistakes
Where students most often lose marks in this chapter — and what to do instead.
Mistake: Treating “more workers, fewer days” as direct proportion.
The quantities move in opposite directions, which is the signature of inverse proportion.
Instead: Multiply workers by days to get the constant k, then divide for the new case.
Mistake: Adding the same amount instead of scaling: “x grows by 2, so y grows by 2”.
Proportion is multiplicative — ratios, not differences, stay constant.
Instead: Compute k from a pair, then multiply; check with a second pair before trusting the rule.
Show 8 more mistakesShow fewer mistakes
Mistake: Forgetting to find k and guessing the answer by cross-multiplying blindly.
Working without the equation hides mistakes and earns no method marks.
Instead: Write y = kx (or y = k/x), substitute the known pair, find k, then answer.
Mistake: Claiming the graph of y = k/x touches the x-axis.
Dividing by a huge x makes y tiny but never zero; x = 0 is impossible.
Instead: The inverse curve approaches both axes without touching them.
Mistake: Using one worker-day value after the team size changes, e.g. keeping 6 days for 12 workers.
Changing one quantity forces the other to adjust so the product stays fixed.
Instead: Recompute: k = 6 × 8 = 48, so 12 workers need 48 ÷ 12 = 4 days.
Mistake: Reporting a negative or zero answer for time, workers or cost.
Those quantities have meaningful ranges that the algebra ignores.
Instead: Keep the root that fits the story and reject the rest with a reason.
Mistake: Halving k when the question doubles x in a y ∝ x² problem.
Squaring doubles the sensitivity: x × 2 multiplies y by 4.
Instead: Substitute into y = kx² with the new x; never eyeball squared forms.
Mistake: Thinking “12 workers instead of 8, so the job takes 4 fewer days”.
Inverse proportion trades by division, not subtraction — the fixed product decides the new time.
Instead: k = 8 × 6 = 48, so 12 workers need 48 ÷ 12 = 4 days; the drop is 2 days, not 4.
Mistake: Confirming inverse proportion just because y decreases as x increases.
Plenty of falling tables are not proportional at all: (1, 12), (2, 11), (3, 10) falls but the products 12, 22, 30 are not constant.
Instead: Inverse requires a constant product x × y — test the products, not the trend.
Mistake: Scaling down when the story scales up: “2 pens cost $3, so 6 pens cost $1”.
Six pens are three times as many, so the cost must triple, not shrink.
Instead: Multiply by the same scale factor: 6 = 2 × 3, so the cost is 3 × $3 = $9.
Check yourself
Quick quiz
18 questions with instant explanations. About 5 minutes.
Question 1 of 3
- Question 1: current question
- Question 2: not answered
- Question 3: not answered
Find the equation for direct proportion
y is proportional to x, and y = 12 when x = 3. What is the equation linking y and x?
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Practise
Topical questions
Exam-style practice in three tiers, from recall to data questions.
- 5Foundation
- 6Standard
- 4Challenging
15 questions worth 59 marks, each with a model answer, mark scheme and examiner tips.
Practise topical questionsExam practice
Practice paper
A 30-mark paper set out like a school paper, with a separate mark scheme. Give yourself 40 min.
Question paper
Chapter 5 Practice Paper
30 marks · 40 min · 5 pages
10 multiple-choice questions and structured questions with answer lines. Chapter 5.
Mark scheme
PROChapter 5 Practice Paper — mark scheme
Answers · Marking points · 3 pages
Model answers with a mark for each creditworthy point — for marking at home or in class.
Where this chapter is tested
Each school sets its own scope — check your assessment outline.
- Term 3Coming soonWeighted Assessment 3 (WA3)
- Term 4Coming soonEnd-of-Year Examination (EOY)
Questions
Frequently asked questions
What is Sec 2 Math Chapter 5 about?
Chapter 5 covers direct and inverse proportion: writing y ∝ x as y = kx and y ∝ 1/x as y = k/x, finding the constant k from one pair of values, testing tables, comparing the two graph shapes, and solving real-world problems including squared forms.
How do I tell direct from inverse proportion in a table?
Run two tests. If every y ÷ x gives the same number, the table is direct proportion. If every x × y gives the same number, it is inverse. If neither holds, the data is not proportional.
What does the graph of inverse proportion look like?
A smooth curve that approaches both axes without ever touching them: large x sends y close to zero, and x = 0 itself is impossible because you cannot divide by it. Direct proportion, in contrast, is a straight line through the origin.
How do worker-day problems work?
The total work is fixed: workers × days = k. Find k from the given case, then divide k by the new number of workers to get the new time. Twelve workers with k = 48 finish in 48 ÷ 12 = 4 days.
Up next
Chapter 6: Congruence and Similarity