Problems Involving Proportion

Deciding whether two quantities are directly proportional, inversely proportional or neither, and solving problems about workers, machines and supplies by changing one quantity at a time.

Before you start

Direct proportion (lesson 6.1): $\frac{y}{x}$ is the same for every pair; doubling $x$ doubles $y$ .
Inverse proportion (lesson 6.4): $xy$ is the same for every pair; doubling $x$ halves $y$ .

Direct, Inverse or Neither?

Deciding

From words: imagine one quantity doubled, with everything else kept the same. If the other is exactly doubled, try direct proportion; if it is exactly halved, try inverse proportion. Then check with numbers.

From a table (lesson 6.5): work out $\frac{y}{x}$ and $xy$ for every column. If $\frac{y}{x}$ is the same throughout, it is direct; if $xy$ is, it is inverse; if neither is, it is neither.

"Both go up" does not prove direct proportion, and "one goes up as the other goes down" does not prove inverse proportion.

Example 1: Deciding from words

Is each pair of quantities directly proportional, inversely proportional, or neither? Give a reason with numbers.
(a) The number of identical notebooks bought, and their total cost.
(b) The number of workers building a wall, all at the same rate, and the number of days they take.
(c) The length of the side of a square, and its area.
(d) The distance of a taxi ride, and the fare ($4.20 at the start and $0.70 a kilometre).
(e) The number of people sharing $180 equally, and the amount each receives.

Solution

(a) Direct. If 2 notebooks cost $3, then 4 cost $6: cost ÷ number is $1.50 for every pair.

(b) Inverse. If 5 workers take 8 days, 1 worker takes 40 days and 10 workers take 4 days: workers × days is 40 every time.

(c) Neither. A side of 2 cm gives 4 cm²; a side of 4 cm gives 16 cm². Doubling the side multiplies the area by 4. (The area is directly proportional to the square of the side, lesson 6.3.)

(d) Neither. 1 km costs $4.90 and 2 km cost $5.60, not $9.80: fare ÷ distance is 4.9, then 2.8.

(e) Inverse. 4 people receive $45 each and 6 people $30 each: number × share is 180 every time.

(a) direct (b) inverse (c) neither (d) neither (e) inverse

Workers and Time for One Job

For one job, with everyone working at the same rate, the time taken is inversely proportional to the number of workers. Work out first how long one worker would take.

Example 2: Workers who join partway

8 workers can finish a job in 15 days. After they have worked for 3 days, 4 more workers join them. How many more days does the job take?

Solution

1 worker would take $8 \times 15 = 120$ days for the whole job.

In the first 3 days, 8 workers do as much as 1 worker in $8 \times 3 = 24$ days.

What is left would take 1 worker $120 - 24 = 96$ days.

Now there are $8 + 4 = 12$ workers: $96 \div 12 = 8$ days.

The job takes 8 more days (11 days in all).

Combined Problems: One Quantity at a Time

Some problems change three quantities at once: the number of workers, the amount of work, and the time. The time is inversely proportional to the number of workers for a fixed amount of work, and directly proportional to the amount of work for a fixed number of workers. So change one quantity at a time, keeping the others fixed.

One quantity at a time

1. Write what you know as a sentence: "5 men paint 2 houses in 3 days".

2. Change one quantity, keeping the others fixed. Before you work it out, ask: should the answer be larger or smaller? More workers take less time (divide); more work takes more time (multiply).

3. Repeat until every quantity has its new value.

Example 3: Men, houses and days

5 men can paint 2 identical houses in 3 days. Assuming that all the men work at the same rate, how long will 6 men take to paint 7 such houses?

Solution

5 men paint 2 houses in 3 days.

1 man paints 2 houses in $3 \times 5 = 15$ days. (Fewer men, more days.)

1 man paints 1 house in $15 \div 2 = 7.5$ days. (Less work, fewer days.)

1 man paints 7 houses in $7.5 \times 7 = 52.5$ days. (More work, more days.)

6 men paint 7 houses in $52.5 \div 6 = 8.75$ days. (More men, fewer days.)

6 men take $8\frac{3}{4}$ days.

Example 4: How many machines?

4 identical machines make 600 items in 5 hours. How many such machines are needed to make 900 items in 3 hours?

Solution

4 machines make 600 items in 5 hours.

1 machine makes 600 items in $5 \times 4 = 20$ hours.

1 machine makes 900 items in $20 \times \frac{900}{600} = 30$ hours. (More items, more hours.)

To make 900 items in 3 hours: $30 \div 3 = 10$ machines. (Less time, more machines.)

10 machines

Example 5: Food for animals

A lorry-load of feed can feed 1000 chickens on a farm for 20 days. Assuming that every chicken eats the same amount each day, how many lorry-loads are needed to feed 550 chickens for 400 days?

Solution

1 lorry-load feeds 1000 chickens for 20 days.

For 550 chickens for 20 days: fewer chickens need less feed, so $\frac{550}{1000} = 0.55$ of a lorry-load.

For 550 chickens for 400 days: 400 days is $\frac{400}{20} = 20$ times as long, so $0.55 \times 20 = 11$ lorry-loads.

11 lorry-loads

Example 6: Machines and time both change

3 identical machines make 600 items in 4 hours. How many items do 5 such machines make in 6 hours?

Solution

1 machine makes $600 \div 3 = 200$ items in 4 hours. (Fewer machines, fewer items.)

1 machine makes $200 \div 4 = 50$ items in 1 hour.

5 machines make $50 \times 5 = 250$ items in 1 hour, and $250 \times 6 = 1500$ items in 6 hours.

1500 items

Common Mistakes

Multiplying at an inverse step. If 5 men take 3 days, then 6 men take $3 \times \frac{5}{6} = 2.5$ days, not $3 \times \frac{6}{5} = 3.6$ days: more men cannot take longer.
Changing two quantities in one step. Change one, keep the others fixed, and write the step as a sentence.
Calling quantities proportional without saying what is fixed: the days are inversely proportional to the men only for a fixed number of houses.
Deciding from "both go up" or "one goes up, the other goes down". Check with numbers.
Learning "chickens × days" as a rule to apply. It works only because each step is a proportion you can explain.

Practice Questions

For each step, decide first whether the answer should be larger or smaller. The questions get harder as you go.

Question 1

Is each pair directly proportional, inversely proportional, or neither? Give a reason.
(a) The mass of rice bought at a fixed price per kilogram, and its cost.
(b) The steady speed of a car on a 60 km journey, and the time taken.
(c) A temperature in degrees Celsius, $C$ , and the same temperature in degrees Fahrenheit, $F = 1.8C + 32$ .
(d) The number of identical pipes filling a tank, and the time taken.

ShowHide solution

(a) Direct: cost ÷ mass is the price per kilogram, the same every time.

(b) Inverse: speed × time is 60 km every time (30 km/h for 2 h, 60 km/h for 1 h).

(d) Inverse: twice as many pipes fill the tank in half the time.

(a) direct (b) inverse (c) neither (d) inverse

Question 2

15 workers can harvest a field in 8 days, all working at the same rate. (a) How long would 12 workers take? (b) How many workers are needed to finish in 5 days?

ShowHide solution

1 worker would take $15 \times 8 = 120$ days.

(a) $120 \div 12 = 10$ days. (b) $120 \div 5 = 24$ workers.

(a) 10 days (b) 24 workers

Question 3

In 6 hours, 4 men can dig 3 identical trenches. Assuming that all the men work at the same rate, how long will 8 men take to dig 10 such trenches?

ShowHide solution

1 man digs 3 trenches in $6 \times 4 = 24$ hours.

1 man digs 1 trench in $24 \div 3 = 8$ hours.

1 man digs 10 trenches in $8 \times 10 = 80$ hours.

8 men dig 10 trenches in $80 \div 8 = 10$ hours.

10 hours

Question 4

12 workers can put together 12 identical chairs in 9 minutes. Assuming that they all work at the same rate, how long will 8 workers take to put together 32 chairs?

ShowHide solution

1 worker puts together 12 chairs in $9 \times 12 = 108$ minutes.

1 worker puts together 1 chair in $108 \div 12 = 9$ minutes.

1 worker puts together 32 chairs in $9 \times 32 = 288$ minutes.

8 workers put together 32 chairs in $288 \div 8 = 36$ minutes.

36 minutes

Question 5

6 identical printers print 3000 leaflets in 5 hours. How many such printers are needed to print 4800 leaflets in 4 hours?

ShowHide solution

1 printer prints 3000 leaflets in $5 \times 6 = 30$ hours.

1 printer prints 4800 leaflets in $30 \times \frac{4800}{3000} = 48$ hours.

To do it in 4 hours: $48 \div 4 = 12$ printers.

12 printers

Question 6

4 identical machines make 900 bottles in 3 hours. How many bottles do 6 such machines make in 5 hours?

ShowHide solution

1 machine makes $900 \div 4 = 225$ bottles in 3 hours. (Fewer machines, fewer bottles.)

1 machine makes $225 \div 3 = 75$ bottles in 1 hour.

6 machines make $75 \times 6 = 450$ bottles in 1 hour, and $450 \times 5 = 2250$ bottles in 5 hours.

2250 bottles

Question 7

One bag of fish food feeds 60 fish in a pond for 15 days. Assuming that every fish eats the same amount each day, how many bags are needed to feed 80 fish for 90 days?

ShowHide solution

1 bag feeds 60 fish for 15 days.

For 80 fish for 15 days: more fish need more food, so $\frac{80}{60} = \frac{4}{3}$ bags.

For 80 fish for 90 days: 90 days is $\frac{90}{15} = 6$ times as long, so $\frac{4}{3} \times 6 = 8$ bags.

8 bags

Question 8

A ship carries enough food for 40 sailors for 30 days. After 10 days, 10 more sailors come on board. How many more days will the food last, if everyone eats the same amount each day?

ShowHide solution

After 10 days, the food left would last the 40 sailors another $30 - 10 = 20$ days.

That food would last 1 sailor $40 \times 20 = 800$ days.

For $40 + 10 = 50$ sailors: $800 \div 50 = 16$ days.

16 more days

Question 9

12 workers can finish a job in 20 days. After 5 days, 3 of the workers leave. How many more days does the job take?

ShowHide solution

1 worker would take $12 \times 20 = 240$ days for the whole job.

In 5 days, 12 workers do as much as 1 worker in $12 \times 5 = 60$ days; $240 - 60 = 180$ days of work are left.

$12 - 3 = 9$ workers take $180 \div 9 = 20$ more days.

20 more days (25 days in all)

Question 10

4 identical taps fill 2 identical tanks in 3 hours. How long do 6 such taps take to fill 9 such tanks?

ShowHide solution

1 tap fills 2 tanks in $3 \times 4 = 12$ hours.

1 tap fills 1 tank in $12 \div 2 = 6$ hours.

1 tap fills 9 tanks in $6 \times 9 = 54$ hours.

6 taps fill 9 tanks in $54 \div 6 = 9$ hours.

9 hours

Question 11

Kumar is told that 8 men build 3 sheds in 6 days. He writes: "12 men build 3 sheds in $6 \times 12 \div 8 = 9$ days." Find his mistake and the correct answer.

ShowHide solution

More men should take fewer days, but his answer is more. He multiplied by $\frac{12}{8}$ at an inverse step.

1 man builds 3 sheds in $6 \times 8 = 48$ days, so 12 men take $48 \div 12 = 4$ days.

4 days

Question 12

Li Ting's height and mass both increase as she grows. At 4 years old she was 100 cm tall with a mass of 16 kg; at 13 she is 155 cm tall with a mass of 45 kg. Are her height and mass in direct proportion? Give a reason in numbers.

ShowHide solution

Mass ÷ height at 4: $\frac{16}{100} = 0.16$ kg per cm. At 13: $\frac{45}{155} \approx 0.29$ kg per cm.

These are not the same, so the mass is not directly proportional to the height, although both increase.

No: mass ÷ height is 0.16 kg per cm at 4 and about 0.29 kg per cm at 13.

Key Points

To decide from words, double one quantity with everything else fixed: exactly doubled suggests direct, exactly halved suggests inverse; then check every pair with numbers.
To decide from a table, work out $\frac{y}{x}$ and $xy$ for every column; if neither is the same throughout, it is neither.
For one job, 1 worker takes (workers × time); divide that by the new number of workers, or by the new time.
When workers join or leave partway, find the work done, then share out the work that is left.
For combined problems, change one quantity at a time and ask before each step whether the answer should be larger or smaller.
Say what is fixed: time is inversely proportional to the number of workers only for a fixed amount of work.