Direct and inverse proportion
🎯 In this topic you will
- Understand the relationship between two quantities when they are in direct or inverse proportion
🧠 Key Words
- inverse proportion
- proportion
Show Definitions
- inverse proportion: A relationship where one value increases as another decreases so their product stays constant.
- proportion: A statement that two ratios are equal, or a part compared to the whole.
You already know that two quantities are in direct proportion when their ratios stay the same as they increase or decrease. For example, when you buy bottles of milk, the more bottles you buy, the more it will cost you. The two quantities, number of bottles and total cost, are in direct proportion.
When two quantities are in inverse proportion, as one quantity increases the other quantity decreases in the same ratio. For example, suppose it takes two people 20 minutes to wash a car. If you increase the number of people, the time taken to wash the car will decrease.
🧠 Think like a Mathematician
Task: Decide if each situation describes direct proportion, inverse proportion, or neither. Justify your reasoning.
Questions:
👀 show answer
- a) Direct proportion – more packets means higher cost at the same price per packet.
- b) Neither – value depends on many factors, not just age (location, condition, market demand).
- c) Inverse proportion – higher speed means less time for the same distance ($\text{time} = \dfrac{\text{distance}}{\text{speed}}$).
- d) Direct proportion – more packets means greater total mass if each packet has the same weight.
- e) Neither – more players does not guarantee more runs; performance varies.
- f) Inverse proportion – more painters means less time needed, assuming equal work rate.
- g) Neither (though often negative correlation) – cars usually lose value with age, but not in a simple proportional way.
❓ EXERCISES
2. Two litres of fruit juice cost $\$3.50$. Work out the cost of
a. $4$ litres b. $10$ litres
c. $1$ litre d. $5$ litres of fruit juice.
👀 Show answer
Unit price: $\$3.50 \div 2=\$1.75$ per litre.
2a. $4\times\$1.75=\$7.00$
2b. $10\times\$1.75=\$17.50$
2c. $1\times\$1.75=\$1.75$
2d. $5\times\$1.75=\$8.75$
3. Here is a recipe for rice pudding.
a. How much sugar is needed for $8$ people?
b. How much rice is needed for $6$ people?
c. How much milk is needed for $10$ people? Give your answer in litres.
👀 Show answer
The specific base quantities aren’t visible in the provided excerpt. Use direct proportion from the recipe that serves $P$ people:
3a. If the recipe uses $S$ grams of sugar for $P$ people, then for $8$ people: $\displaystyle S\times \frac{8}{P}\,$ grams.
3b. If the recipe uses $R$ grams of rice for $P$ people, then for $6$ people: $\displaystyle R\times \frac{6}{P}\,$ grams.
3c. If the recipe uses $M$ millilitres of milk for $P$ people, then for $10$ people: $\displaystyle M\times \frac{10}{P}\,$ mL $=\left(M\times \frac{10}{P}\right)\!\div 1000$ litres.
4. Four horses can eat a bale of hay in two days. Copy and complete the working for these questions.
a. How long does it take one horse to eat a bale of hay?
b. How long does it take eight horses to eat a bale of hay?
👀 Show answer
Eating time is inversely proportional to the number of horses.
4a. From $4$ horses $\to 2$ days, for $1$ horse time $=2\times 4=8$ days.
4b. From $4$ horses $\to 2$ days, doubling to $8$ horses halves the time: $2\div 2=1$ day.
5. It takes Dieter $36$ seconds to run a certain distance. Copy and complete the working for these questions.
a. Dieter halves his speed. How long will it take him to run the same distance?
b. Dieter runs three times as fast. How long will it take him to run the same distance?
👀 Show answer
Time $\propto \dfrac{1}{\text{speed}}$ for a fixed distance.
5a. Half speed $\Rightarrow$ time doubles: $36\times 2=72$ seconds $\;(=\,\mathbf{72\ s})$.
5b. Three times speed $\Rightarrow$ time thirds: $36\div 3=12$ seconds $\;(=\,\mathbf{12\ s})$.
6. It takes Julia $40$ minutes to drive to work at an average speed of $60\ \text{km/h}$.
a. Julia drives at an average speed of $120\ \text{km/h}$. How long will it take Julia to drive to work?
b. It takes Julia $80$ minutes to drive to work. What is Julia’s average speed?
👀 Show answer
Distance to work: $60\times \dfrac{40}{60}=40$ km.
6a. Time $=\dfrac{40}{120}$ h $=\dfrac{1}{3}$ h $=20$ minutes.
6b. Time $=80$ min $=\dfrac{4}{3}$ h. Speed $=\dfrac{40}{\tfrac{4}{3}}=30\ \text{km/h}$.
7. It costs a fixed amount to hire a villa in Spain. Up to $12$ people can stay in the villa. Antonio hires the villa for his family of four people. The cost per person is €$300$. Copy and complete the table.
| $Number\ of\ people$ | $4$ | $12$ | $2$ | $1$ | $6$ | $10$ | $5$ |
|---|---|---|---|---|---|---|---|
| $Cost\ per\ person\ (€)$ | $300$ |
👀 Show answer
| $Number\ of\ people$ | $4$ | $12$ | $2$ | $1$ | $6$ | $10$ | $5$ |
|---|---|---|---|---|---|---|---|
| $Cost\ per\ person\ (€)$ | $300$ | $100$ | $600$ | $1200$ | $200$ | $120$ | $240$ |
Total hire cost is fixed. From $4$ people at €$300$ each: total €$1200$.
Cost per person $=\dfrac{1200}{n}$ for $n$ people.
Thus: $n=12\Rightarrow 100$; $n=2\Rightarrow 600$; $n=1\Rightarrow 1200$; $n=6\Rightarrow 200$; $n=10\Rightarrow 120$; $n=5\Rightarrow 240$ (all in euros).
🧠 Think like a Mathematician
Task: Explore different ways to solve an inverse proportion problem and decide which method you prefer.
Scenario: 9 men can build a house in 28 days. How long will it take 12 men to build the house?
Methods:
- 9 men → 28 days
- 1 man → 28 × 9 = 252 days
- 12 men → 252 ÷ 12 = 21 days
- $\dfrac{12}{9} = \dfrac{4}{3}$ (12 men is 4/3 of 9 men)
- Time is inversely proportional, so divide: $28 \div \tfrac{4}{3} = 28 \times \tfrac{3}{4} = 21$ days
Questions:
👀 show answer
- a) Both methods are correct. - Bengt’s is a step-by-step unitary method, clear but slightly longer. - Susu’s uses the idea of inverse proportion more directly, shorter but requires confidence with fractions.
- b) Another method: use the formula for inverse proportion: $\text{men} \times \text{days} = \text{constant}$. So $9 \times 28 = 12 \times x$. $x = \dfrac{9 \times 28}{12} = 21$ days.
- c) Preference may vary: - Bengt’s is easier for learners who like step-by-step logic. - Susu’s or the formula method is faster if you are confident with inverse proportion.
❓ EXERCISES
9. It takes $6$ people $4$ hours to sort and pack a load of eggs. How long will it take $10$ people to sort and pack the same number of eggs?
Give your answer in hours and minutes.
👀 Show answer
Total work $=6 \times 4=24$ person-hours.
For $10$ people: time $=\dfrac{24}{10}=2.4$ hours $=2$ hours $24$ minutes.
10. Arun and Marcus are looking at this question:
Question:
At a theme park, there are $36$ people on a roller coaster.
The ride takes $4$ minutes.
How long does the ride take when there are $18$ people on the roller coaster?
Arun says:
“I think the ride will take $2$ minutes as the number of people has halved. $36 \div 2=18$, so $4 \div 2=2$ minutes.”
Marcus says:
“I think the ride will still take $4$ minutes because that is the length of the ride and it doesn’t matter how many people are on the roller coaster.”
a. What do you think? Justify your answer.
b. Discuss your answer to part a with a partner.
👀 Show answer
The ride length is fixed at $4$ minutes regardless of the number of riders. The number of people does not affect the ride duration.
Correct answer: Marcus is right. The ride still takes $4$ minutes.
11. In a science experiment, Camila measures how far a ball bounces when she drops it from different heights. The table shows her results.
| $Height\ when\ dropped\ (cm)$ | $50$ | $100$ | $150$ | $200$ | $250$ |
|---|---|---|---|---|---|
| $Height\ of\ bounce\ (cm)$ | $40$ | $80$ | $120$ | $160$ | $200$ |
a. Do Camila’s results show that the height of the drop and the height of the bounce are in direct proportion? Explain your answer.
b. How high does the ball bounce when it is dropped from a height of $120\ \text{cm}$?
c.
i. Make a copy of this coordinate grid and plot the points in the table on the grid.
ii. What do you notice about the points?
iii. Is it possible to draw a straight line through all the points?
iv. Camila drops the ball and it bounces back up $180\ \text{cm}$. Use your graph to work out the height from which she dropped the ball.
👀 Show answer
11a. Yes. The ratio $\dfrac{bounce}{drop}$ is constant: $\dfrac{40}{50}=\dfrac{80}{100}=\dfrac{120}{150}=\dfrac{160}{200}=\dfrac{200}{250}=0.8$. A constant ratio means the quantities are in direct proportion with constant of proportionality $k=0.8$.
11b. If $bounce = 0.8\times drop$, then for $drop=120$: $bounce=0.8\times 120=96\ \text{cm}$.
11c(i). Plot the points $(50,40)$, $(100,80)$, $(150,120)$, $(200,160)$, $(250,200)$ on the grid.
11c(ii). The points lie on a straight line through the origin.
11c(iii). Yes—since the relation is linear and passes through the origin, one straight line fits all points.
11c(iv). From $bounce=0.8\times drop$ and $bounce=180$: $drop=\dfrac{180}{0.8}=225\ \text{cm}$.
12. In a science experiment, Abnar measures the increase in the length of a string when it has different masses attached. The table shows his results.
| Mass (g) | 20 | 30 | 40 | 50 |
|---|---|---|---|---|
| Increase in length (mm) | 12 | 18 | 24 | 30 |
a. What type of proportion do Abnar’s results show? Explain your answer.
b. Draw a graph to show Abnar’s results. Draw a straight line through all the points.
c. Use your graph to work out
i. the increase in length when a mass of 45 g is attached
ii. the mass attached when the increase in length is 20 mm.
d. Is the following statement true or false? Explain your answer.
‘When two quantities are in direct proportion, you can draw a straight-line graph to show the relationship between the two quantities.’
👀 Show answer
12a. The results show direct proportion. The ratio (increase ÷ mass) is constant: 12/20 = 18/30 = 24/40 = 30/50 = 0.6. So increase = 0.6 × mass.
12b. Plot the points (20, 12), (30, 18), (40, 24), (50, 30). They lie on a straight line through the origin.
12c(i). For 45 g: increase = 0.6 × 45 = 27 mm.
12c(ii). If increase = 20 mm: mass = 20 ÷ 0.6 = 33⅓ g (≈ 33.3 g).
12d. True. Direct proportion gives a straight-line graph through the origin because one quantity is a constant multiple of the other.
⚠️ Be careful!
In standard form, the coefficient must satisfy $1 \le |a| < 10$. Don’t write $32 \times 10^{5}$; rewrite it as $3.2 \times 10^{6}$