Direct and Inverse Proportion Problems

There are just two possibilities to recognise, and the first job in every question is to decide which one you have.

	Direct proportion	Inverse proportion
In words	As one goes up, the other goes up by the same factor	As one goes up, the other goes down by that factor
Double $x$…	…and $y$ doubles	…and $y$ halves
Equation	$y = kx$	$y = \dfrac{k}{x}$
What stays constant	$\dfrac{y}{x} = k$	$xy = k$
Graph	Straight line through the origin	Curve (hyperbola), never touching the axes
Typical context	Cost of petrol, wages, recipe amounts	Workers and time, speed and journey time

Quick test: ask yourself "if I double the first quantity, does the second double or halve?" More workers → less time = inverse. More petrol → more cost = direct.

Check it really is direct (does doubling one double the other?).
Either find the value for one unit, or find the multiplier between the two situations.
Scale up or down to the amount you need.
Check that the answer is the right sort of size.

Worked Example 1 — Unitary method

$9$ metres of rope costs £$14.85$. Find the cost of $23$ metres.

①One metre $= 14.85 \div 9 = £1.65$

②$23$ metres $= 1.65 \times 23 = £37.95$

Sense check: $23$ m is about $2.5$ times $9$ m, and $£37.95$ is about $2.5$ times $£14.85$ ✓

Worked Example 2 — Multiplier method

$14$ litres of fuel cost £$21.70$. How much fuel can be bought for £$46.50$?

①Multiplier on the money: $46.50 \div 21.70 = 2.1428\ldots$ — messy, so use the unitary method instead.

②Cost per litre $= 21.70 \div 14 = £1.55$

③Litres bought $= 46.50 \div 1.55 = 30$ litres

Answer: $30$ litres

Worked Example 3 — Using the equation

$y$ is directly proportional to $x$. When $x = 12$, $y = 30$. Find $y$ when $x = 20$, and find $x$ when $y = 55$.

①Write $y = kx$.

②Substitute: $30 = k \times 12$, so $k = 2.5$.

③The rule is $y = 2.5x$.

④When $x = 20$: $y = 2.5 \times 20 = 50$.

⑤When $y = 55$: $55 = 2.5x$, so $x = 22$.

The key fact

In inverse proportion the product stays the same: $x_1 y_1 = x_2 y_2 = k$

Check it really is inverse (more of one means less of the other).
Multiply the two known values together to find $k$.
Divide $k$ by the new value to get the answer.
Check: the answer should move in the opposite direction.

Worked Example 4 — Workers and time

It takes $6$ builders $20$ days to build a wall. How long would $8$ builders take, working at the same rate?

①More builders → fewer days, so this is inverse.

②Total work $= 6 \times 20 = 120$ builder-days $= k$.

③Time for $8$ builders $= 120 \div 8 = 15$ days.

Check: $8 > 6$ and $15 < 20$, so the answer moved the right way ✓

Worked Example 5 — Speed and time

A journey takes $2.5$ hours at an average speed of $56$ km/h. How long would the same journey take at $70$ km/h?

①The distance is fixed, so speed and time are inversely proportional.

②$k = 56 \times 2.5 = 140$ (this is the distance in km).

③Time $= 140 \div 70 = 2$ hours.

Worked Example 6 — Using the equation

$y$ is inversely proportional to $x$. When $x = 4$, $y = 15$. Find $y$ when $x = 10$.

①Write $y = \dfrac{k}{x}$.

②$15 = \dfrac{k}{4}$, so $k = 60$.

③Rule: $y = \dfrac{60}{x}$.

④When $x = 10$: $y = \dfrac{60}{10} = 6$.

Not everything that decreases is inverse proportion. "The temperature falls by $2^\circ$ every hour" is a linear decrease, not an inverse one. Inverse proportion requires the product to stay constant.

Situation	Type	Why
Number of tickets and total cost	Direct	Twice as many tickets, twice the cost
Hours worked and pay	Direct	Twice the hours, twice the pay
Number of taps filling a pool and time taken	Inverse	Twice as many taps, half the time
Speed and journey time (fixed distance)	Inverse	Twice the speed, half the time
Number of people sharing a fixed prize	Inverse	Twice as many people, half each
Side length and perimeter of a square	Direct	$P = 4s$
Number of days food lasts and number of animals	Inverse	More animals, fewer days

Worked Example 7 — Testing a table

Decide whether this table shows direct proportion, inverse proportion, or neither.

$x$	$2$	$3$	$12$
$y$	$18$	$12$	$3$

①Test $\dfrac{y}{x}$: $9$, $4$, $0.25$ — not constant, so not direct.

②Test $xy$: $2 \times 18 = 36$, $3 \times 12 = 36$, $12 \times 3 = 36$ — constant!

Inverse proportion, with $y = \dfrac{36}{x}$.

Some questions involve both types at once — typically "workers, days and amount of work".

Worked Example 8 — Two changes at once

$5$ machines can produce $600$ parts in $4$ hours. How many parts can $8$ machines produce in $3$ hours?

①Find the rate for one machine for one hour.

②$5$ machines $\times$ $4$ hours $= 20$ machine-hours produce $600$ parts.

③One machine-hour $= 600 \div 20 = 30$ parts.

④$8$ machines $\times$ $3$ hours $= 24$ machine-hours.

⑤Parts $= 24 \times 30 = 720$.

Answer: $720$ parts

Worked Example 9 — Partial completion

$4$ gardeners can clear a field in $18$ hours. They work for $6$ hours and then $2$ more gardeners join them. How much longer will the job take?

①Total work $= 4 \times 18 = 72$ gardener-hours.

②Work already done $= 4 \times 6 = 24$ gardener-hours.

③Work remaining $= 72 - 24 = 48$ gardener-hours.

④Now there are $6$ gardeners: time $= 48 \div 6 = 8$ hours.

Answer: $8$ more hours (a total of $14$ hours rather than $18$).

Direct

$y = kx$; $\dfrac{y}{x}$ constant; straight line through the origin.

Inverse

$y = \dfrac{k}{x}$; $xy$ constant; hyperbola.

Deciding

Double the first: does the second double (direct) or halve (inverse)?

Unitary method

Find the value for one, then scale.

Inverse shortcut

Multiply the pair you know to get $k$, then divide.

Worker problems

Count total "worker-hours" — that total is the constant.

Testing a table

Try $\dfrac{y}{x}$; if that fails, try $xy$.

Sense check

Inverse answers move the opposite way to the change.

Question 1

$7$ notebooks cost £$10.15$. Find the cost of $12$ notebooks.

▶ Show solution

Direct proportion.

One notebook $= 10.15 \div 7 = £1.45$

Twelve $= 1.45 \times 12 = £17.40$

Question 2

It takes $12$ workers $15$ days to complete a job. How long would $20$ workers take?

▶ Show solution

Inverse proportion.

Total work $= 12 \times 15 = 180$ worker-days.

Time $= 180 \div 20 = 9$ days.

Question 3

$y$ is directly proportional to $x$. When $x = 8$, $y = 22$. (a) Find the formula. (b) Find $y$ when $x = 15$. (c) Find $x$ when $y = 55$.

▶ Show solution

(a) $y = kx$; $22 = 8k$, so $k = 2.75$ and $y = 2.75x$.

(b) $y = 2.75 \times 15 = 41.25$

(c) $55 = 2.75x$, so $x = 20$.

Question 4

$y$ is inversely proportional to $x$. When $x = 6$, $y = 14$. Find $y$ when $x = 21$.

▶ Show solution

$y = \dfrac{k}{x}$; $14 = \dfrac{k}{6}$, so $k = 84$.

$y = \dfrac{84}{21} = 4$

Question 5

Say whether each pair is in direct proportion, inverse proportion, or neither.

(a) The number of pizzas ordered and the total bill.
(b) The number of friends sharing a £$60$ taxi fare and the amount each pays.
(c) A person's age and their height.

▶ Show solution

(a) Direct — twice as many pizzas costs twice as much.

(b) Inverse — twice as many friends means each pays half.

(c) Neither — height increases with age for a while, then stops. There is no constant ratio or product.

Question 6

Decide the type of proportion and find the missing value $p$.

$x$	$4$	$10$	$25$
$y$	$50$	$20$	$p$

▶ Show solution

$\dfrac{y}{x}$: $12.5$ then $2$ — not constant.

$xy$: $4 \times 50 = 200$ and $10 \times 20 = 200$ — constant.

Inverse proportion with $k = 200$, so $p = 200 \div 25 = 8$.

Question 7

A tank is filled by $5$ identical pumps in $36$ minutes. Two of the pumps break down. How long will the remaining pumps take to fill the tank?

▶ Show solution

Inverse proportion.

$k = 5 \times 36 = 180$ pump-minutes.

Now only $3$ pumps work: $180 \div 3 = 60$ minutes.

Question 8

$6$ printers produce $2700$ leaflets in $5$ hours. How many leaflets would $10$ printers produce in $4$ hours?

▶ Show solution

Printer-hours available originally $= 6 \times 5 = 30$.

One printer-hour $= 2700 \div 30 = 90$ leaflets.

New printer-hours $= 10 \times 4 = 40$.

Leaflets $= 40 \times 90 = 3600$.

Question 9

A farmer has enough feed for $45$ sheep for $28$ days. He buys $18$ more sheep. Assuming each sheep eats the same amount, how long will the feed now last?

▶ Show solution

Total feed $= 45 \times 28 = 1260$ sheep-days.

New number of sheep $= 45 + 18 = 63$.

Days $= 1260 \div 63 = 20$ days.

Check: more sheep, fewer days ✓

Question 10

$8$ decorators can paint a block of flats in $21$ days. After $9$ days, $3$ decorators leave.

(a) How much of the work remains? (b) How many more days will the job take? (c) By how many days is the job delayed overall?

▶ Show solution

(a) Total work $= 8 \times 21 = 168$ decorator-days.

Work done $= 8 \times 9 = 72$ decorator-days.

Remaining $= 168 - 72 = 96$ decorator-days.

(b) Decorators left $= 8 - 3 = 5$.

Extra days $= 96 \div 5 = 19.2$ days.

(c) Total time $= 9 + 19.2 = 28.2$ days, compared with the planned $21$ days.

Delay $= 28.2 - 21 = \mathbf{7.2}$ days.