Ratio and Proportions

Lessons

Senior 1 · Mathematics

Ratio and Proportions

5 lessons

Lesson 4 of 5

Inverse proportion

22 min

For umuganda, a village plans to dig a drainage ditch. The leader works out that 6 people would need 8 days. With 12 people the job is done in 4 days, and with 24 people in only 2. This time more people means less time: doubling one quantity halves the other. The total amount of work, 48 person-days, stays the same.

Workers $n$	2	3	4	6	8	12
Days $t$	24	16	12	8	6	4
$n \times t$	48	48	48	48	48	48

Definition

Inverse proportion: two quantities $x$ and $y$ are inversely proportional if their product is always the same number $k$ : $xy = k \quad\Longleftrightarrow\quad y = \frac{k}{x}.$ We write $y \propto \frac{1}{x}$ . $k$ is the constant of proportionality.

Example: workers $n$ and days $t$ : $n t = 48$ , so $t = \frac{48}{n}$ .

The rule, and why it holds

If $x y = k$ and $x$ is multiplied by some number, then $y$ must be divided by the same number to keep the product equal to $k$ . So doubling $x$ halves $y$ , and tripling $x$ divides $y$ by 3. In the ditch example the product is the total work, which does not depend on how many people share it.

	Direct proportion	Inverse proportion
Formula	$y = k x$	$y = \dfrac{k}{x}$
What stays constant	the ratio $\dfrac{y}{x}$	the product $x y$
When $x$ doubles	$y$ doubles	$y$ halves
Graph	straight line through the origin	a curve that falls, never touching either axis

The graph of inverse proportion

Plot the table and join the points with a smooth curve, not with straight segments. The curve falls steeply at first and then levels out. It never reaches either axis: $t = 0$ would need infinitely many workers, and $n = 0$ workers never finish. Each shaded rectangle under the curve has the same area, $n \times t = 48$ .

Worked example

A borehole pump fills a village water tank in 6 hours when it pumps 400 litres per hour. How long would it take at 300 litres per hour?

The tank holds the same amount either way, so rate $\times$ time is constant: $400 \times 6 = 2{,}400 \text{ litres} = 300 \times t \;\Rightarrow\; t = \frac{2{,}400}{300} = 8 \text{ hours}.$ A slower rate gives a longer time, which is what inverse proportion predicts.

Worked example

A school's supply of maize flour would feed 240 boarders for 15 days. For how many days would it feed 300 boarders?

Is it direct or inverse? More students eat the food faster, so the days go down: inverse.
Constant product: $240 \times 15 = 3{,}600$ student-days of food.
For 300 students: $t = \dfrac{3{,}600}{300} = 12$ days.

Worked example

$y$ is inversely proportional to $x$ , and $y = 9$ when $x = 4$ . Find $y$ when $x = 6$ , and $x$ when $y = 12$

$k = xy = 4 \times 9 = 36$ , so $y = \dfrac{36}{x}$ .

When $x = 6$ : $y = \dfrac{36}{6} = 6$ . When $y = 12$ : $x = \dfrac{36}{12} = 3$ .

Worked example

Decide whether each is direct proportion, inverse proportion or neither

Situation	Test	Answer
Mass of rice and its cost at a fixed price	cost $\div$ mass constant	direct
Speed of a bus and the time for a 150 km trip (roughly the road from Kigali to Rubavu)	speed $\times$ time $= 150$ km	inverse
Number of people sharing a 12,000 Frw bill equally, and each share	people $\times$ share $= 12,000$	inverse
A person's age and height	neither ratio nor product is constant	neither

Important

Before you calculate, ask: "If one quantity doubles, what happens to the other?" If it doubles too, the proportion is direct: keep the ratio constant. If it halves, the proportion is inverse: keep the product constant. If neither happens, it is not a proportion at all.

Warning

Don't use the direct method on an inverse problem. "6 workers take 8 days, so 12 workers take 16 days" doubles both quantities, as if the job got bigger. Always check the answer makes sense: more workers must finish sooner.

Warning

"One goes down as the other goes up" is not enough for inverse proportion. The product must stay the same. A phone's battery level falls as the hours of use go up, but hours $\times$ battery level is not constant, so it is not inverse proportion.

Tip

Give the constant a meaning, such as person-days of work, litres in the tank or student-days of food. Once you know what it is, the calculation writes itself: divide the constant by the new value.

Corbettmaths: "Inverse Proportion". Watch how the constant is found first from the known pair, then used to find the missing value.

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Practice

Say first whether the proportion is direct or inverse.

$y$ is inversely proportional to $x$ , and $y = 10$ when $x = 3$ . Find $y$ when $x = 5$ .
4 painters paint a school block in 9 days. How long would 6 painters take, working at the same rate?
A car at 60 km per hour takes 3 hours for a journey. How long does the same journey take at 90 km per hour?
5 notebooks cost 2,500 Frw. How much do 8 notebooks cost?
Complete the table for inverse proportion: $x = 2, 4, 5, \square, 20$ and $y = 10, \square, 4, 2, \square$ , where each box is a missing value.
A bag of feed lasts 30 hens for 12 days. The farmer buys 6 more hens. How long will the bag last now?
Does $x = 1, 2, 3, 4$ with $y = 12, 6, 4, 2$ show inverse proportion? Explain.

Show answersHide answers

$k = 30$ , so $y = \frac{30}{5} = 6$ .
Inverse: $4 \times 9 = 36$ painter-days, so $\frac{36}{6} = 6$ days.
Inverse: distance $= 180$ km, so $\frac{180}{90} = 2$ hours.
Direct: 1 notebook costs 500 Frw, so 8 cost 4,000 Frw.
$k = 20$ : $y = 5$ when $x = 4$ ; $x = 10$ when $y = 2$ ; $y = 1$ when $x = 20$ .
$30 \times 12 = 360$ hen-days; $\frac{360}{36} = 10$ days.
No: the products are 12, 12, 12 and 8. The last pair breaks the pattern (it would need $y = 3$ ).

Sources and credits

8.9: Use Direct and Inverse Variation, from Elementary Algebra by Lynn Marecek, MaryAnne Anthony-Smith and Andrea Honeycutt Mathis, OpenStax, provided by LibreTexts, licensed under CC BY 4.0. Adapted by Ganzaa.
Video: Inverse Proportion, corbettmaths, embedded from YouTube.

The umuganda, water-tank, boarding-school and feed examples, the graph, the practice items and the quiz questions were written by Ganzaa. The figures are models for practice.

1The rule, and why it holds

2The graph of inverse proportion

The rule, and why it holds

The graph of inverse proportion