1MA1 · R10, R13, R16Ratio, Proportion & Rates of ChangeFoundation & HigherGrades 5–8

Direct & Inverse Proportion

Q: What does the symbol ∝ mean?

"Is proportional to". Replace it with "= k " to form an equation.

If 4 pens cost £3.20, then 10 pens cost £8 — doubling the pens doubles the cost. That is $direct proportion$ . If 4 workers take 12 days to build a wall, then 6 workers take 8 days — more workers, less time. That is $inverse proportion$ . At Higher tier, both extend to squares, cubes and roots, and are written as equations with a constant $k$ .

50 min Ratio & Proportion

The big picture

Direct proportion means one quantity is a constant multiple of the other: $y = k x$ . Inverse proportion means their product is constant: $y = \frac{k}{x}$ . Recognising which applies, then finding $k$ from one pair of values, turns any proportion question into substitution.

Edexcel Higher questions often start " $y$ is inversely proportional to the square of $x$ ", and the marks go to writing the correct equation, finding $k$ , and using it. Graphs also appear: direct proportion is a straight line through the origin; inverse proportion is a reciprocal curve.

What you'll learn

Solve direct and inverse proportion problems using unitary and multiplier methods
Recognise direct and inverse proportion from tables and graphs
Form equations such as $y = k x^{2}$ and $y = \frac{k}{x}$ and use them (Higher)
Describe the effect of changing one variable on another

Direct proportion

Two quantities are in direct proportion if multiplying one by a number multiplies the other by the same number. Their ratio stays constant.

The unitary method finds the value for one unit: if 4 pens cost £3.20, one costs 80p, so 10 cost £8.

As an equation, $y \propto x$ means $y = k x$ . Its graph is a straight line through the origin with gradient $k$ .

y \propto x ⟺ y = k x

Direct proportion.

Your turnSame method, new numbers — try it before moving on.

5 kg of potatoes cost £12.50. How much do 8 kg cost, in pounds?

Your answer

Inverse proportion

Two quantities are inversely proportional if multiplying one by a number divides the other by the same number. Their product stays constant.

If 4 workers take 12 days, the job is $4 \times 12 = 48$ worker-days, so 6 workers take $48 \div 6 = 8$ days.

As an equation, $y \propto \frac{1}{x}$ means $y = \frac{k}{x}$ . Its graph is a reciprocal curve.

y \propto \frac{1}{x} ⟺ y = \frac{k}{x}

Inverse proportion.

Your turnSame method, new numbers — try it before moving on.

6 workers take 10 days to build a wall. How many days would 4 workers take?

Your answer

Why this works

Direct proportion keeps the $ratio$ constant; inverse proportion keeps the $product$ constant. Checking which of $\frac{y}{x}$ or $x y$ stays the same in a table identifies the relationship immediately.

Proportion with powers and roots

" $y$ is directly proportional to the square of $x$ " means $y = k x^{2}$ . " $y$ is inversely proportional to the square root of $x$ " means $y = \frac{k}{x}$ .

The method is always: write the equation with $k$ , substitute the given pair to find $k$ , write the equation with the value of $k$ , then use it.

y = k x^{2}

, so

18 = 9 k

and

k = 2

y = 2 x^{2}

3When

x = 5

y = 2 \times 25

Answer

y = 50

Your turnSame method, new numbers — try it before moving on.

$y$ is directly proportional to $x^{3}$ . When $x = 2$ , $y = 24$ . Find $y$ when $x = 3$ .

Your answer

Tip — Write the equation with $k$ found as a separate line, e.g. " $y = 2 x^{2}$ ". Mark schemes award a mark for it.

The effect of a change

If $y = k x^{2}$ and $x$ is doubled, $y$ is multiplied by $2^{2} = 4$ . If $y = \frac{k}{x ^{2}}$ and $x$ is doubled, $y$ is divided by 4.

For a percentage change, use multipliers: if $x$ increases by 20% in $y = k x^{3}$ , $y$ is multiplied by $1. 2^{3} = 1.728$ , a 72.8% increase.

F = \frac{k}{d ^{2}}

; new distance

1.2 d

2New force

= \frac{k}{1.44 d ^{2}} = 0.694 F

AnswerThe force decreases by about 30.6%.

Your turnSame method, new numbers — try it before moving on.

$y$ is directly proportional to $x^{2}$ . $x$ is increased by 20%. By what percentage does $y$ increase?

Your answer

Think like an examiner

Write the proportionality equation with $k$ before substituting.
Show the value of $k$ and the equation with $k$ replaced.
Use multipliers for percentage changes and explain the conclusion.

Proportion

y = k x

Direct proportion.

y = \frac{k}{x}

Inverse proportion.

y = k x^{n}, y = \frac{k}{x ^{n}}

With powers or roots.

Watch out for these

Writing "

y

inversely proportional to

x^{2}

" as

y = k x^{2}

y = \frac{k}{x ^{2}}

Assuming

y

doubles when

x

doubles in

y = k x^{2}

It quadruples.

Using

\frac{y}{x}

to test inverse proportion.

For inverse proportion,

x y

is constant.

Forgetting to square

x

when finding

k

Substitute into the full equation:

18 = k \times 3^{2}

Stretch yourself

The time $t$ for a pendulum to swing is directly proportional to the square root of its length $l$ . A pendulum of length 64 cm takes 1.6 s. Find the time for a pendulum of length 1 m, and the length needed for a time of 2.4 s.

Hint — Use consistent units. Form $t = k l$ with $l$ in cm.

Questions students ask

How do I tell direct from inverse proportion?

Direct: both increase together and the ratio is constant. Inverse: one increases as the other decreases and the product is constant.

What does the symbol ∝ mean?

"Is proportional to". Replace it with "

= k \times

" to form an equation.

Is every straight-line graph direct proportion?

No — only straight lines through the origin.

Key takeaways

Direct proportion: $y = k x$ , constant ratio, straight line through the origin.
Inverse proportion: $y = \frac{k}{x}$ , constant product, reciprocal curve.
With powers and roots, write the equation, find $k$ , then substitute.
Doubling $x$ in $y = k x^{n}$ multiplies $y$ by $2^{n}$ .
Use multipliers to find the effect of percentage changes.

Exam-style questions

Practise it the way it is examined

3 original questions · 8 marks, written to match the style and mark allocation of the real papers. Work on paper, then open the mark scheme and tick the marks you earned — method marks count even if the final answer slips.

Question 1Grade 53 marks

$d$ is directly proportional to $t^{2}$ . When $t = 2$ , $d = 20$ .

(a)Find a formula for $d$ in terms of $t$ .[2]
(b)Find $d$ when $t = 5$ .[1]

Question 2Grade 72 marks

$P$ is inversely proportional to $V$ . When $V = 40$ , $P = 150$ . Find $V$ when $P = 250$ .[2]

Question 3Grade 83 marks

$y$ is inversely proportional to $x^{2}$ . $x$ is increased by 25%. Find the percentage decrease in $y$ .[3]

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Direct & Inverse Proportion

Direct proportion

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Direct & Inverse Proportion

Direct proportion

Inverse proportion

Proportion with powers and roots

The effect of a change

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Skill drill

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