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Proportionality

Proportionality

Direct and inverse proportion

Simple Extended 8 min read

On this page
  1. In this note
  2. Formula bank
  3. Direct and inverse proportion
  4. The three-step method
  5. Spotting proportion in a table
  6. Word problems
  7. Ways they test it
  8. Mistakes to avoid
  9. Summary
  10. Practice
  11. Can I do this?
  12. Quiz

In this note

Syllabus 2.8: direct and inverse proportion Extended
  • 2.8Understand direct proportion \(y\propto x\) and inverse proportion \(y\propto\dfrac1x\), and use the symbol \(\propto\).
  • 2.8Write the relationship as an equation with a constant of proportionality \(k\), find \(k\), and use the equation.
  • 1.11Solve word problems by proportional reasoning (more workers → fewer days).

Before you start: ratio and proportion. Next: proportion with powers and roots.

Formula bank

Direct proportionLearn it\(y\propto x\ \Rightarrow\ y=kx\)\(\dfrac yx=k\) is constant; graph a straight line through (0, 0).Use it when "directly proportional", more → more.
Inverse proportionLearn it\(y\propto\dfrac1x\ \Rightarrow\ y=\dfrac kx\)\(xy=k\) is constant; graph a curve that never meets the axes.Use it when "inversely proportional", more → less.
Workers and daysLearn itworkers × days = total work (constant)Use it when same rate, different team size.

Direct and inverse proportion

Direct: double x → double y, so the ratio \(\tfrac yx\) is always the same number \(k\): \(y=kx\). Inverse: double x → HALVE y, so the product \(xy\) is always \(k\): \(y=\tfrac kx\). The symbol \(\propto\) means "is proportional to" - replace it by \(=k\times\) and you have an equation.

Graphsy = kxy = k/x
Direct: a straight line through (0, 0). Inverse: a curve that gets close to both axes.

The three-step method

General methodHow to answer any proportion question

  1. Write the equation: \(y=kx\) (direct) or \(y=\dfrac kx\) (inverse).
  2. Find \(k\): substitute the pair of values you are given.
  3. Write the equation again with the number for \(k\) (this is "y in terms of x").
  4. Substitute the new value; solve if the unknown is x.

Example \(y\propto x\). \(y=12\) when \(x=4\). Find \(y\) when \(x=10\).

\(y=kx\); find k.

  1. \(12=4k\Rightarrow k=3\)
  2. \(y=3x\)
  3. \(y=3\times10\)

Answer 30

Example \(y\propto\dfrac1x\). \(y=6\) when \(x=5\). Find \(x\) when \(y=0.5\).

\(y=\tfrac kx\); \(k=xy\).

  1. \(k=30\), \(y=\tfrac{30}x\)
  2. \(0.5=\tfrac{30}x\Rightarrow x=60\)

Answer 60

Try it yourself

1
\(y\propto x\), y = 20 when x = 4. y when x = 7?
Show answer

Answer 35

2
\(y\propto\tfrac1x\), y = 8 when x = 3. y when x = 6?
Show answer

Answer 4

3
\(y\propto\tfrac1x\), y = 6 when x = 5. Write y in terms of x.
Show answer

Answer \(y=\tfrac{30}x\)

Spotting proportion in a table

General methodHow to tell direct from inverse in a table

  1. Work out y ÷ x for each pair: all the same → direct.
  2. Work out x × y for each pair: all the same → inverse.

Example Which table shows inverse proportion? (i) x: 2, 4, 6; y: 5, 10, 15. (ii) x: 2, 4, 8; y: 12, 6, 3.

Test the ratio and the product.

  1. (i) \(y\div x=2.5\) every time: direct
  2. (ii) \(xy=24\) every time: inverse

Answer (ii)

Word problems

General methodHow to solve a proportion word problem

  1. Decide: more → more (direct) or more → less (inverse)?
  2. Direct: find the value of ONE (unitary method), then multiply.
  3. Inverse: find the total (workers × days), then divide.
  4. Sense check: should the answer be bigger or smaller?

Example 6 workers build a wall in 10 days. How long would 4 workers take?

Fewer workers → more days: inverse.

  1. 6 × 10 = 60 worker-days
  2. 60 ÷ 4

Answer 15 days

Example 5 kg of rice costs $8.50. Find the cost of 8 kg.

More rice → more cost: direct.

  1. $8.50 ÷ 5 = $1.70 per kg
  2. × 8

Answer $13.60

Try it yourself

1
5 people paint a hall in 12 hours. How long for 4 people?
Show answer

Answer 15 hours

Ways they test it

Type 1 Find k and use it

"\(F\) is inversely proportional to \(d\). \(F=20\) when \(d=3\). Find \(F\) when \(d=8\)." (3 marks)

Spot it "proportional" with one pair of values.

  1. Equation with k (M1).
  2. k (M1).
  3. Answer (A1).

For example \(F=\tfrac kd\)

  1. \(k=60\)
  2. \(60\div8\)

Answer 7.5

Type 2 y in terms of x

"Find a formula for y in terms of x."

Spot it "in terms of".

  1. Give the equation with the number for k.

Type 3 Word problem

"8 machines take 15 hours. How long would 12 machines take?" (2 marks)

Spot it a rate in context.

  1. Direct or inverse?
  2. Unitary or total.

For example 8 machines, 15 h

  1. 120 machine-hours
  2. \(120\div12\)

Answer 10 hours

Type 4 From a change

"When x increases by 5, y increases by 15. Find y when x = 12."

Spot it "increases by".

  1. A change of 5 in x is \(5k\) in y.

Mistakes to avoid

  • ✗ Inverse treated as direct✓ \(y=\tfrac kx\)Read the word 'inversely'.
  • ✗ Answering before finding k✓ find k firstThe method marks are for k.
  • ✗ 'y in terms of x' answered with a number✓ \(y=\tfrac{30}x\)They want the formula.

Summary

  • ∝ becomes \(=k\times\).
  • Equation → find k → use it.
  • Direct: ratio fixed, line through the origin. Inverse: product fixed, curve.
  • Words: more → more is direct; more → less is inverse.

If you see …

  • "inversely"\(y=\tfrac kx\).
  • workers / machines and timemultiply for the total, then divide.

Practice

Practice: easy, medium, hard

Easy

1
\(y\propto x\), y = 12 when x = 3. Find k.
Show answer

Answer 4

2
\(y\propto x\), y = 12 when x = 3. Find y when x = 9.
Show working
  1. \(y=4x\)

Answer 36

3
\(y\propto\tfrac1x\), y = 9 when x = 4. Find y when x = 3.
Show working
  1. \(k=36\)

Answer 12

Medium

1
\(P\) is directly proportional to \(d\). \(P=126\) when \(d=18\). Find \(P\) when \(d=25\).
Guide me

What is it asking? P

What do we know? \(P=kd\)

Plan k = 126 ÷ 18

Show working
  1. \(k=7\)
  2. \(7\times25\)

Answer 175

2
8 machines take 15 hours to do a job. How long would 12 machines take?
Guide me

What is it asking? hours

What do we know? more machines → less time

Plan machine-hours = 120

Show working
  1. 120 ÷ 12

Answer 10 hours

3
\(F\) is inversely proportional to \(d\). \(F=20\) when \(d=3\). Find \(d\) when \(F=4\).
Guide me

What is it asking? d

What do we know? \(F=\tfrac kd\)

Plan k = 60

Show working
  1. \(4=\tfrac{60}d\)

Answer 15

Hard

1
\(y\) is directly proportional to \(x\). When \(x\) increases by 5, \(y\) increases by 15. Find \(y\) when \(x=12\).
Guide me

What is it asking? y

What do we know? a change, not a pair

Plan \(5k=15\)

Show working
  1. \(k=3\)
  2. \(3\times12\)

Answer 36

PartAnswer / workingMarksGuidance
k = 3M1
36A1
Total: 2 marks
2
\(y\) is inversely proportional to \(x\). When \(x=4\), \(y=t\). Find \(y\) in terms of \(t\) when \(x=6\).
Guide me

What is it asking? y in terms of t

What do we know? \(k=xy=4t\)

Plan \(y=\tfrac{4t}6\)

Show working
  1. \(k=4t\)
  2. \(y=\tfrac{4t}{6}\)

Answer \(\tfrac{2t}3\)

PartAnswer / workingMarksGuidance
\(k=4t\)M1
\(\tfrac{4t}6\)M1
\(\tfrac{2t}3\)A1
Total: 3 marks
3
\(y\) is inversely proportional to \(x\). \(y=5\) when \(x=4\). Find \(x\) when \(y=2\).
Guide me

What is it asking? x

What do we know? \(k=20\)

Plan \(2=\tfrac{20}x\)

Show working
  1. \(x=20\div2\)

Answer 10

PartAnswer / workingMarksGuidance
\(y=\tfrac kx\)M1
k = 20M1
10A1
Total: 3 marks

Can I do this?

Anything unticked: revise that section first. All ticked: take the quiz.

Quiz

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