Direct & Indirect Proportion

Telling direct from indirect (inverse) proportion, setting up each correctly, and solving the classic exam problems — unit costs, workers and time, speed and journey time.

Two quantities are in proportion when changing one changes the other in a fixed way. Grade 9 asks you to spot which way — get that wrong and the whole answer is wrong, so decide before you calculate.

1The two kinds

Direct proportion gives a straight line through the origin; indirect proportion gives a curve that falls away.

Direct: more → more · yx = k so y = kx
Indirect (inverse): more → less · x × y = k

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Ask yourself in words: "if I double the first quantity, does the second double or halve?" Double → direct. Halve → indirect.

2Direct proportion

Worked ExampleIf 6 pens cost R30, what do 10 pens cost?

1
More pens cost more money, so this is DIRECT.
Decide the type first.
2
Find one unit: 30 ÷ 6 = R5 per pen.
The unitary method is the safest route.
3
10 pens = 10 × 5 = R50.
Multiply by the new amount.

3Indirect (inverse) proportion

Worked Example4 workers build a wall in 12 days. How long would 6 workers take?

1
More workers means FEWER days, so this is INDIRECT.
The classic inverse situation.
2
Find the constant: 4 × 12 = 48 worker-days.
For inverse proportion multiply, don't divide.
3
6 workers: 48 ÷ 6 = 8 days.
Divide the constant by the new number.

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Treating a workers-and-time problem as direct gives 6 × 3 = 18 days — more workers taking longer, which is nonsense. Always sanity-check the direction of your answer.

Worked ExampleA car takes 3 hours at 80 km/h. How long at 120 km/h?

1
Faster speed means LESS time → indirect.
2
Constant: 80 × 3 = 240 km.
That constant is simply the distance.
3
240 ÷ 120 = 2 hours.

Practice exercises

Work each one out, then click to reveal the answer.

📄 Worksheet ✅ Memo 📝 Exam paper 🗝️ Marking guide

1
If 5 books cost R125, what do 8 books cost?
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One book R25 → R200 (direct)
2
If 3 kg of rice costs R45, what does 7 kg cost?
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R15/kg → R105 (direct)
3
8 workers finish a job in 6 days. How long would 4 workers take?
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48 ÷ 4 = 12 days (indirect)
4
5 taps fill a tank in 12 minutes. How long would 10 taps take?
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60 ÷ 10 = 6 minutes (indirect)
5
A journey takes 4 hours at 90 km/h. How long at 120 km/h?
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360 ÷ 120 = 3 hours (indirect)
6
Is 'number of workers vs time taken' direct or indirect?
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Indirect — more workers, less time.
7
Is 'litres of petrol vs cost' direct or indirect?
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Direct — more litres, more cost.
8
y is directly proportional to x, and y = 20 when x = 4. Find k.
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k = 20 ÷ 4 = 5, so y = 5x