Inverse Proportion

What it means for one quantity to be inversely proportional to another, how to test a table by working out $xy$ for every pair, and how to solve problems about taps, workers, speeds and shares.

Before you start

Lesson 6.1: $y$ is directly proportional to $x$ when $\frac{y}{x}$ is the same for every pair. "Both go up together" is not enough.
Speed, time and distance: distance = speed × time, so a journey of 120 km at 40 km/h takes 120 ÷ 40 = 3 hours.

What Inverse Proportion Means

A car makes a journey of 120 km. The table shows the time, $y$ hours, that the journey takes at a steady speed of $x$ km/h. The last row multiplies each speed by its time.

$x$ (km/h)	$10$	$20$	$30$	$40$	$60$	$120$
$y$ (h)	$12$	$6$	$4$	$3$	$2$	$1$
$xy$	$120$	$120$	$120$	$120$	$120$	$120$

For every pair, $xy = 120$ : $10 \times 12 = 120$ , $20 \times 6 = 120$ , and so on to $120 \times 1 = 120$ . (Speed × time is the distance, which is 120 km every time.) This is what makes the time inversely proportional to the speed.

Inverse proportion

$y$ is inversely proportional to $x$ if $xy$ is the same for every pair of values of $x$ and $y$ . We write $y \propto \frac{1}{x}$ (" $y$ varies inversely as $x$ ").

The value of $xy$ is the constant of proportionality, $k$ . For the journey, $k = 120$ .

Because $xy$ stays the same, multiplying $x$ by a number divides $y$ by the same number:

• when the speed is doubled, from 10 to 20 km/h, the time is halved, from 12 to 6 hours;

• when the speed is multiplied by 3, from 20 to 60 km/h, the time is divided by 3, from 6 to 2 hours;

• when the speed is halved, from 120 to 60 km/h, the time is doubled, from 1 to 2 hours.

So for positive quantities, as one increases the other decreases. But that alone is not the definition: in Example 1(b) one quantity goes down as the other goes up, and they are not inversely proportional.

Example 1: Deciding from a table

Is $y$ inversely proportional to $x$ in each table?
(a) Identical pumps emptying a pond: $x$ pumps take $y$ hours.
(b) A 10-litre tank of water: after $x$ litres have been used, $y$ litres are left.

Table (a)

$x$ pumps	$2$	$3$	$4$	$6$
$y$ hours	$18$	$12$	$9$	$6$

Table (b)

$x$ litres used	$1$	$2$	$3$	$4$	$5$
$y$ litres left	$9$	$8$	$7$	$6$	$5$

Solution

(a) Work out $xy$ for every pair:

$x$ pumps	$2$	$3$	$4$	$6$
$y$ hours	$18$	$12$	$9$	$6$
$xy$	$36$	$36$	$36$	$36$

$xy = 36$ for every pair, so the time is inversely proportional to the number of pumps, with $k = 36$ : one pump alone would take 36 hours.

(b) Work out $xy$ for every pair:

$x$ litres used	$1$	$2$	$3$	$4$	$5$
$y$ litres left	$9$	$8$	$7$	$6$	$5$
$xy$	$9$	$16$	$21$	$24$	$25$

$xy$ is not the same, so $y$ is not inversely proportional to $x$ , even though $y$ goes down as $x$ goes up. What stays the same here is the sum: $x + y = 10$ . Doubling the litres used from 2 to 4 takes the litres left from 8 to 6, not to 4.

(a) Yes: $xy = 36$ for every pair. (b) No: $xy$ is not the same for every pair.

The Points on a Graph

Plot the journey times with the speed across and the time up. Speed can take any value, not only the ones in the table, so the points are joined by a smooth curve.

The points do not lie on a straight line: they lie on a curve that falls steeply at first and then more slowly. The curve does not pass through the origin; a speed of 0 would never finish the journey.

A falling curve does not by itself show inverse proportion: the test is always $xy$ . Lesson 6.5 writes this curve as an equation, $y = \frac{120}{x}$ .

Solving Problems

Two methods

The product method. Since $xy$ is the same for every pair, $x_1y_1 = x_2y_2$ . Put in the three values you know and solve for the fourth.

The unitary method. Work out what one unit would take (more time, so multiply), then what the new number takes (divide).

Both need an assumption, which a question usually states: all the taps (workers, machines) work at the same rate.

Example 2: Taps filling a pool

10 identical taps can fill a pool in 4 hours. How long will 8 such taps take to fill it?

Solution

Product method. The number of taps and the time are inversely proportional, so $x_1y_1 = x_2y_2$ :

$10 \times 4 = 8 \times y$

$40 = 8y$ , so $y = 5$ .

Unitary method. 1 tap would take 10 times as long: $4 \times 10 = 40$ hours. 8 taps take $\frac{1}{8}$ of that: $40 \div 8 = 5$ hours.

Check the direction: fewer taps (8 instead of 10) should take longer, and 5 hours is longer than 4.

8 taps take 5 hours.

Example 3: Workers and days

6 workers can build a wall in 10 days. Assuming that they all work at the same rate, how long would 4 workers take to build the same wall?

Solution

1 worker would take $6 \times 10 = 60$ days.

4 workers take $60 \div 4 = 15$ days.

Product method: $6 \times 10 = 4 \times y$ , so $y = \frac{60}{4} = 15$ .

4 workers take 15 days.

Example 4: How many workers are needed?

The wall of Example 3 must be built in 4 days. How many workers are needed?

Solution

1 worker would take 60 days. To finish in 4 days, the number of workers is $60 \div 4 = 15$ .

Product method: $6 \times 10 = n \times 4$ , so $n = \frac{60}{4} = 15$ .

15 workers are needed, which is 9 more than the 6 of Example 3.

Example 5: Food supplies and equal shares

(a) A camp has enough food for 30 people for 12 days. How long would the same food last 40 people, eating at the same rate?
(b) $120 is shared equally among $x$ people, and each receives $ $y$ . Complete the table.

People, $x$	$4$	$5$	$6$	$8$
Each, $ $y$	?	?	?	?

Solution

(a) The food would last 1 person $30 \times 12 = 360$ days. For 40 people: $360 \div 40 = 9$ days.

More people, so fewer days: the number of days is inversely proportional to the number of people.

(b) Each share is $\frac{120}{x}$ : $\frac{120}{4} = 30$ , and so on. The product $xy$ is the sum shared, 120, every time.

People, $x$	$4$	$5$	$6$	$8$
Each, $ $y$	$30$	$24$	$20$	$15$
$xy$	$120$	$120$	$120$	$120$

(a) 9 days. (b) $30, $24, $20, $15.

Common Mistakes

Treating an inverse proportion as direct. For 8 taps in Example 2, " $4 \times \frac{8}{10} = 3.2$ hours" is wrong: fewer taps cannot be quicker. Ask first: should the answer be larger or smaller?
"One goes up and the other goes down, so it is inverse proportion." The litres used and left in a tank do that, but $xy$ is not the same. Work out $xy$ .
Checking $xy$ for two pairs only. Every pair must give the same product.
Mixing up the unitary steps. One tap takes longer than ten (multiply); eight taps take less time than one (divide).
Forgetting the assumption. The methods work only if every tap, worker or machine works at the same rate.

Practice Questions

Decide first whether the answer should be larger or smaller, then use the product or the unitary method. The questions get harder as you go.

Question 1

Is $y$ inversely proportional to $x$ ? If it is, give the constant of proportionality.

$x$	$2$	$4$	$5$	$10$
$y$	$25$	$12.5$	$10$	$5$

ShowHide solution

$x$	$2$	$4$	$5$	$10$
$y$	$25$	$12.5$	$10$	$5$
$xy$	$50$	$50$	$50$	$50$

$xy = 50$ for every pair.

Yes, $y \propto \frac{1}{x}$ , with $k = 50$ .

Question 2

Is $y$ inversely proportional to $x$ ? Give a reason.

$x$	$2$	$3$	$4$	$6$
$y$	$9$	$6$	$4.5$	$2$

ShowHide solution

$x$	$2$	$3$	$4$	$6$
$y$	$9$	$6$	$4.5$	$2$
$xy$	$18$	$18$	$18$	$12$

The first three pairs give 18, but $6 \times 2 = 12$ .

No: $xy$ is not the same for every pair (the last pair gives 12).

Question 3

$y$ is inversely proportional to $x$ , and $y = 12$ when $x = 5$ . Without working out $k$ , find $y$ when (a) $x = 10$ , (b) $x = 15$ , (c) $x = 2.5$ .

ShowHide solution

(a) $x$ is doubled, so $y$ is halved: $y = 6$ .

(b) $x$ is multiplied by 3, so $y$ is divided by 3: $y = 4$ .

Check: $10 \times 6$ , $15 \times 4$ and $2.5 \times 24$ are all $60 = 5 \times 12$ .

(a) $y = 6$ (b) $y = 4$ (c) $y = 24$

Question 4

A tank can be filled by 4 identical taps in 70 minutes. How long do 7 such taps take to fill it?

ShowHide solution

$4 \times 70 = 7 \times y$ , so $280 = 7y$ and $y = 40$ .

Or: 1 tap takes $4 \times 70 = 280$ minutes, so 7 taps take $280 \div 7 = 40$ minutes.

40 minutes

Question 5

At a steady 80 km/h, a journey takes 3 hours. (a) How long does it take at 60 km/h? (b) What steady speed would make it take 2.5 hours?

ShowHide solution

Speed × time is the distance: $80 \times 3 = 240$ km, the same for every speed.

(a) $60 \times y = 240$ , so $y = 4$ hours.

(b) $x \times 2.5 = 240$ , so $x = 96$ km/h.

(a) 4 hours (b) 96 km/h

Question 6

8 painters take 9 days to paint a block of flats. Assuming that they all work at the same rate, how long would 12 painters take?

ShowHide solution

1 painter would take $8 \times 9 = 72$ days. 12 painters take $72 \div 12 = 6$ days.

6 days

Question 7

A farmer has enough feed for 45 cows for 16 days. He sells some of the cows, and the feed now lasts 24 days. How many cows did he sell?

ShowHide solution

The feed would last 1 cow $45 \times 16 = 720$ days.

For it to last 24 days, the number of cows is $720 \div 24 = 30$ .

He sold $45 - 30 = 15$ cows.

15 cows

Question 8

A prize of $360 is shared equally among $x$ winners, and each receives $ $y$ . Copy and complete the table.

$x$	$2$	$3$	?	$9$
$y$ ($)	?	?	$45$	?

ShowHide solution

$xy = 360$ every time: $y = \frac{360}{2} = 180$ , $y = \frac{360}{3} = 120$ , $x = \frac{360}{45} = 8$ , $y = \frac{360}{9} = 40$ .

$x$	$2$	$3$	$8$	$9$
$y$ ($)	$180$	$120$	$45$	$40$
$xy$	$360$	$360$	$360$	$360$

$180, $120, 8 winners, $40

Question 9

Rectangles are drawn, each with an area of 24 cm². A rectangle has sides of $x$ cm and $y$ cm. Complete a table of $y$ for $x = 2, 3, 4, 6, 8, 12$ , and plot the points. Do they lie on a straight line?

ShowHide solution

The area is $xy = 24$ , so $y = \frac{24}{x}$ .

$x$ (cm)	$2$	$3$	$4$	$6$	$8$	$12$
$y$ (cm)	$12$	$8$	$6$	$4$	$3$	$2$
$xy$	$24$	$24$	$24$	$24$	$24$	$24$

The other sides are 12, 8, 6, 4, 3 and 2 cm. The points lie on a curve, not a straight line; $xy = 24$ for every pair, so $y$ is inversely proportional to $x$ .

Question 10

Wei Ming is told that 8 identical machines print a school's examination papers in 9 hours. He works out the time for 12 machines as $9 \times \frac{12}{8} = 13.5$ hours. Explain his mistake and find the correct time.

ShowHide solution

He treated the time as directly proportional to the number of machines. More machines should take less time, not more.

1 machine would take $8 \times 9 = 72$ hours, so 12 machines take $72 \div 12 = 6$ hours.

The time is inversely proportional to the number of machines; 12 machines take 6 hours.

Question 11

Nadia is reading a 200-page book. She says: "As the number of pages I have read goes up, the number of pages left goes down, so they are inversely proportional." Test her claim with 50, 100 and 150 pages read.

ShowHide solution

Pages left: 150, 100, 50.

$xy$ : $50 \times 150 = 7500$ , $100 \times 100 = 10\,000\text{,}$ $150 \times 50 = 7500$ . These are not the same.

What stays the same is the sum: pages read + pages left = 200.

She is wrong: $xy$ is not the same for every pair, so the two are not inversely proportional.

Question 12

A ferry crossing at a steady 24 km/h takes 45 minutes. The company wants the crossing to take 36 minutes. At what steady speed must the ferry travel?

ShowHide solution

For the same crossing, speed × time is the same: $24 \times 45 = v \times 36$ .

$1080 = 36v$ , so $v = 30$ .

Check the direction: less time needs a higher speed, and 30 km/h is higher than 24 km/h.

30 km/h

Key Points

$y$ is inversely proportional to $x$ ( $y \propto \frac{1}{x}$ ) if $xy$ is the same for every pair of values; that value is $k$ .
To test a table, work out $xy$ for every pair.
When $y \propto \frac{1}{x}$ , multiplying $x$ by a number divides $y$ by the same number.
"One goes up as the other goes down" is not enough: the litres used and left in a tank do that and are not inversely proportional.
Solve problems with $x_1y_1 = x_2y_2$ , or by the unitary method (one unit first: multiply, then divide).
Check the direction of every answer, and state the assumption that everything works at the same rate.