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Sec 2 Direct and Inverse Proportion | Constant Ratio, Constant Product and Graphs

Secondary 2 Direct and Inverse Proportion is the canonical eduKateSingapore guide to proportional relationships: how two quantities change together, how to recognise whether one is directly or inversely proportional to the other, how to find the constant of proportionality, and how to move between tables, equations, graphs and real-world rate problems. For students searching direct and inverse proportion, Sec 2 proportion, proportionality, constant of proportionality, unit rate or proportional graphs, the central idea is not a formula list. It is a relationship question: when one quantity changes, what must happen to the other?

Direct proportion means the ratio between two quantities stays constant. If one quantity doubles, the other doubles. Inverse proportion means the product stays constant. If one quantity doubles, the other halves. These are different mathematical structures, and confusing them creates errors in speed, work-rate, density, scale, cost and many later algebraic topics.

This page belongs to the Secondary Mathematics Topic Library. It grows from Sec 1 Ratio, Rate and Speed, links to the broader world-level What is a Rate? guide, and prepares students for graphs, equations, variation and upper-secondary applications.

The fastest distinction: ratio constant or product constant?

  • Direct proportion: y/x is constant, so y = kx.
  • Inverse proportion: xy is constant, so y = k/x.
  • Neither: no single constant preserves the required relationship.

A student should be able to say which quantity is constant before calculating. That verbal step prevents method selection from becoming guesswork.

Direct proportion: same multiplicative scale

Suppose y is directly proportional to x. If x=3 gives y=12, then y/x = 12/3 = 4. The constant of proportionality is k=4, so y=4x.

Now any value can be found. If x=7, y=28. If y=60, x=15.

The relationship is multiplicative: every x-value is scaled by the same factor 4.

Worked example: cost

Five identical notebooks cost $17.50. At the same unit price, what do 12 notebooks cost?

Cost is directly proportional to number of notebooks. Unit cost = 17.50 ÷ 5 = $3.50. Cost of 12 = 12 × 3.50 = $42.

Algebraically, C=3.5n.

Direct proportion as a graph

If y is directly proportional to x, the graph of y against x is a straight line through the origin. The gradient is the constant of proportionality k.

This connects directly to Sec 2 Linear Graphs. A straight line that does not pass through the origin is linear, but not directly proportional.

Direct proportion diagnostic

  • Does (0,0) fit the relationship?
  • Is y/x the same for every pair?
  • Does doubling x double y?
  • Does the graph pass through the origin?

Inverse proportion: same product

Suppose y is inversely proportional to x. If x=4 gives y=15, then xy = 60. Therefore y=60/x.

If x=10, y=6. If x doubles from 4 to 8, y halves from 15 to 7.5.

The product stays fixed even though the ratio does not.

Worked example: equal work

A task takes 6 workers 10 hours, assuming all workers work at the same constant rate and the total work is fixed. If 12 workers do the same work, the time is 5 hours.

Workers × time = constant work-units: 6 × 10 = 60. Then 12 × t = 60, so t=5.

The mathematical model assumes equal productivity and no coordination losses; real workplaces may not behave perfectly this way.

Inverse proportion as a graph

The graph y=k/x is curved rather than a straight line. As x grows, y approaches zero but does not become zero merely because x is large. The first quadrant branch is common in positive real-world examples such as fixed-work rate problems.

At this level, the important point is recognising that a constant product creates a different shape from a constant ratio.

A table test

When the wording is unclear, a table can expose the relationship.

  • Calculate y/x for several rows. If constant, test direct proportion.
  • Calculate xy for several rows. If constant, test inverse proportion.
  • If neither is constant, the relationship is neither of these simple forms.

Example table

x: 2, 4, 8. y: 24, 12, 6.

Ratios y/x: 12, 3, 0.75 — not constant.

Products xy: 48, 48, 48 — constant.

Therefore y is inversely proportional to x.

The constant of proportionality

The constant k stores the specific scale of the relationship.

  • Direct: k = y/x.
  • Inverse: k = xy.

Two different real-world systems can both be directly proportional but have different constants. A taxi charging $0.80 per kilometre and a printing service charging $0.12 per page share the same structure, not the same k.

From words to equation

y is directly proportional to x

Model: y = kx

C is directly proportional to mass m

Model: C = km

t is inversely proportional to n

Model: t = k/n

pressure P is inversely proportional to area A

Model: P = k/A

Worked example: find k, then solve

y is directly proportional to x. When x=8, y=30. Find y when x=14.

30=8k, so k=3.75.

When x=14, y=3.75×14=52.5.

Worked inverse example: find k, then solve

y is inversely proportional to x. When x=5, y=18.

k=xy=90, so y=90/x.

When x=12, y=7.5.

A common trap: additive change is not proportion

Suppose y=3x+5. Increasing x by 1 increases y by 3, but y is not directly proportional to x because the graph does not pass through the origin and y/x is not constant.

Linear relationship and direct proportion are related but not identical ideas.

A second trap: ‘more people, less time’ is not enough

The fact that one quantity rises while another falls does not automatically prove inverse proportion. Inverse proportion requires a constant product under the model.

Check the mathematics rather than relying on verbal direction alone.

Unit rate and direct proportion

Unit-rate methods are direct proportion in practical clothing. If 7 kg costs $31.50, then 1 kg costs $4.50 and cost C=4.5m.

This is the same structure students met in Ratio, Rate and Speed, now written more algebraically.

Scale and maps

Scale drawings use direct proportionality between drawing length and actual length. A fixed scale 1:200 means every drawing length is multiplied by 200 to obtain actual length in the same unit.

Area and volume do not scale by the same factor: if lengths scale by k, areas scale by k² and volumes by k³. That distinction belongs to similarity and mensuration rather than simple direct proportion.

Proportion and percentage

Percentage questions are often proportional, but not every percentage-change problem is a direct-proportion model across stages. A 20% increase changes the multiplier from one state to another. Use Sec 1 Percentage when the base itself changes.

Graph interpretation worked example

A straight line through (0,0), (3,12), and (5,20) represents y against x.

Gradient = 4, so y=4x.

Because the line passes through the origin and y/x=4, the relationship is directly proportional.

Inverse table worked example

A fixed journey distance of 120 km is considered under several constant speeds.

At 40 km/h, time=3 h. At 60 km/h, time=2 h. At 80 km/h, time=1.5 h.

Speed × time = 120 in every case. For this fixed-distance model, time is inversely proportional to speed.

The proportion error taxonomy

  • Structure error: chooses direct when product is constant, or inverse when ratio is constant.
  • Constant error: calculates k incorrectly.
  • Unit error: mixes incompatible units.
  • Graph error: calls any straight line direct proportion.
  • Language error: assumes opposite movement means inverse proportion.
  • Scale error: uses a length scale factor directly for area or volume.
  • Context error: applies an ideal inverse model where assumptions are not stated or reasonable.

A six-step exam routine

  1. Name the two quantities.
  2. Check units.
  3. Test whether ratio or product is constant.
  4. Write y=kx or y=k/x.
  5. Use a known pair to find k.
  6. Substitute the required value and interpret the answer.

Transfer drill: same surface words, different structure

Scenario A

The cost of identical tickets rises with number bought at a fixed ticket price. Direct proportion.

Scenario B

The time taken for a fixed amount of work falls as the number of equally productive workers increases. Inverse proportion under the stated model.

Scenario C

A delivery service charges $5 plus $2 per kilometre. Linear, not directly proportional, because of the fixed $5 charge.

Scenario D

A rectangle has fixed width 5 cm while length varies. Area is directly proportional to length. Perimeter is not directly proportional to length because P=2l+10.

Practice set

Practice 1

Question: y ∝ x, y=18 when x=6. Find y when x=15.

Answer: k=3; y=45.

Practice 2

Question: y ∝ 1/x, y=20 when x=3. Find y when x=12.

Answer: k=60; y=5.

Practice 3

Question: Table x:2,4,6; y:7,14,21.

Answer: Direct proportion, k=3.5.

Practice 4

Question: Table x:2,4,8; y:16,8,4.

Answer: Inverse proportion, k=32.

Practice 5

Question: y=5x+2.

Answer: Linear but not directly proportional.

Practice 6

Question: Fixed 240 km journey: speed 80 km/h.

Answer: time=3 h.

How parents and tutors should diagnose proportion errors

Do not record only “wrong answer”. Ask which of four layers failed:

  • Did the student identify direct versus inverse correctly?
  • Did the student derive k correctly?
  • Were the units aligned?
  • Was the final substitution and interpretation correct?

A student who repeatedly chooses the wrong structure needs comparison practice, not more arithmetic drills.

Frequently asked questions

What is direct proportion?

A relationship where y/x is constant, usually written y=kx.

What is inverse proportion?

A relationship where xy is constant, usually written y=k/x.

Is every straight-line graph a direct proportion graph?

No. Direct proportion requires the line to pass through the origin.

How do I decide between direct and inverse?

Test whether ratio y/x or product xy remains constant.

Why is inverse proportion curved on a graph?

Because equal changes in x do not produce equal changes in y; y changes according to k/x.

Where does this sit in Atlas?

This is the canonical Sec 2 Direct and Inverse Proportion owner under Secondary Mathematics Topic Library.

The final Proportion rule

Ask what remains invariant. Direct proportion preserves a ratio. Inverse proportion preserves a product. Once the invariant is visible, equations, tables, graphs and real-world rate questions become different representations of the same relationship.

Explore the connected learning guides

Choose the question that brought you here. Open one useful guide, try a small task, and stop when you have what you need.

Take one question further

The same learning habit can travel across subjects, while each subject keeps its own methods. These routes help you notice a difficulty, understand one part of it, and return to something you can do.

A word is familiar, but using it is difficult.

Move from recognising a word to retrieving it in a new context. Understand vocabulary plateaus.

Try it without the guide: Choose one word you already know. Close the guide and use it in a new sentence. Explain why it fits; try another context tomorrow.

A piece of writing has ideas, but the reader loses the thread.

Make the order of events and the links between sentences clear. Explore composition writing.

Try it without the guide: Choose one short paragraph. Read the relevant explanation, close it, and revise the paragraph. Ask someone to tell you what happened and why.

The Mathematics seems familiar, but marks still disappear.

Find the first point where the working stops being reliable. Find Secondary 4 A-Math mark leakage.

Try it without the guide: For a Secondary 4 A-Math question you have attempted, locate the first uncertain line. Repair that step, then try a comparable question without the worked answer.

A Science fact is remembered, but the explanation is incomplete.

Connect the evidence to a scientific idea and the resulting change. Follow the Primary Science learning route.

Try it without the guide: Choose a familiar Primary Science example. Explain the evidence, the idea and the result without notes. Then change one condition and explain your prediction.

Two accounts of the world seem to disagree.

Check the question, source, date and evidence before combining claims. Explore the World Knowledge research library.

Try it without the guide: Take one claim. Find the source best placed to support it, note its date, and state what remains uncertain. Return to your original question.

There is plenty of help, but independence is hard to see.

Check what the learner can understand and do after support is removed. Understand how education works.

Try it without the guide: Choose one small task the child has practised. Agree on a calm, brief attempt without prompts. Use what happens to choose one next step, then stop.

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