Direct and Inverse Proportion

Two amounts are in direct proportion when they change in the same way: if one doubles, the other doubles too. If 1 kg of rice costs £2, then 2 kg costs £4 and 5 kg costs £10.

Two amounts are in inverse proportion when one goes up as the other goes down: if one doubles, the other halves. If 2 people take 6 hours to paint a room, then 4 people take 3 hours.

In the exam you are usually asked to work out the cost of a different number of items, or how long a job takes with more or fewer workers or machines, for 2 or 3 marks. Shorter questions ask you to choose the graph that shows direct or inverse proportion, for 1 mark. You may also be given a formula and asked to use it.

The surest method is to find the value of one item first, then multiply by the number you want. It works whatever the numbers are.

\text{value of one} = \text{total} \div \text{number of items}

Then multiply the value of one by the number of items you want.

For example, if 3 identical mugs cost £12, one mug costs \pounds 12\div3=\pounds 4, so 7 mugs cost \pounds 4\times7=\pounds 28

It also works backwards. To find how many items a total buys, divide the total by the value of one. With mugs at £4 each, £36 buys 36\div4=9 mugs.

In a table, divide each amount by the number it goes with. If you get the same answer every time, the two amounts are in direct proportion. That answer is the value of one.

On a graph, direct proportion is a straight line through the origin, the point (0,0). It must start at (0,0), because 0 items cost £0.

Line A shows direct proportion: 2 hours cost £4 and 4 hours cost £8, twice as much. Line B does not go through (0,0). It has a fixed charge of £4, so 2 hours cost £8 but 4 hours cost £12, which is not twice as much.

When people or machines share a job, more of them finish it faster. If each one works at the same rate, doubling the number of workers halves the time, and three times as many workers take a third of the time.

The total amount of work stays the same. If 3 people take 8 hours to clean a hall, that is 3\times8=24 hours of work. One person alone would take 24 hours, and 6 people would take 24\div6=4 hours.

\text{number of workers} \times \text{time} = \text{total work}

Find the total work first. Then divide it by the new number of workers to find the time, or by the new time to find the number of workers.

Supplies that are shared work in the same way. Food that lasts 5 campers for 6 days is 5\times6=30 days of food for one camper, so it lasts 3 campers for 30\div3=10 days.

Before you work anything out, ask: if one amount doubles, what happens to the other? If it doubles too, it is direct proportion. If it halves, it is inverse proportion.

• The number of stamps bought and their total cost: direct.
• The number of litres of fuel in a car and how far it can go: direct.
• The number of painters and the time to paint a house: inverse.
• The speed of a train and the time it takes for the same journey: inverse.

Some amounts are in neither kind of proportion. A child's height does not double when their age doubles.

The graph of inverse proportion is a curve. It goes down steeply at first, then less steeply, and it never touches either axis.

Each point multiplies to the same total, the 24 litres in the tank: 1\times24, 2\times12, 3\times8 and 4\times6 all make 24. Doubling the flow rate from 2 to 4 litres per minute halves the time from 12 minutes to 6 minutes.

You may be given a formula to use. A formula such as C=3n, where the cost C pounds is 3 times the number of items n, shows direct proportion. A formula such as t=\frac{24}{n}, where the time t is 24 divided by the number of people n, shows inverse proportion.

The question may say "t is inversely proportional to n" and then give you the formula. You don't have to make the formula yourself: put the number you are given into the formula and work it out.

• Treating inverse proportion as direct proportion. More workers means less time, so multiply to find the total work, then divide.
• Multiplying the total by the new number. If 4 tickets cost £30, then 6 tickets do not cost \pounds 30\times6; find the cost of 1 ticket first.
• Thinking any straight line shows direct proportion. The line must also go through (0,0).
• Using numbers the formula doesn't need. When you are given a formula, put the right value into it; not every number in the question is used.
• Rounding the number of workers down. If 13.3 workers are needed, 13 will not finish in time, so the answer is 14
• Writing money with one decimal place. Write £42.50, not £42.5.