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Topic

Proportion and inverse proportion

Proportion links two quantities. In direct proportion, more of one quantity means more of the other. In inverse proportion, more of one quantity means less of the other.

Topic explanation

Direct proportion is used when both quantities increase or decrease together at the same rate. If 4 pencils cost £2.40, one pencil costs £0.60, so 10 pencils cost £6.

Inverse proportion is used when one quantity increases while the other decreases because the total stays the same. If 4 workers take 6 hours, the job is 24 worker-hours.

Before calculating, decide whether the situation is direct or inverse. Shopping, recipes and same-speed travel are usually direct; workers, machines, taps and sharing a fixed bill are usually inverse.

Quick methods

Find one, then scale
For direct proportion, find the value for one unit, then multiply.
Same rate
Costs, distance, recipes and fuel often scale directly when the rate stays the same.
Same total job
For inverse proportion, keep the total work the same, then divide by the new amount.
Check the direction
More items means more cost, but more workers usually means less time.

Worked examples

Example 1

4 pencils cost £2.40. How much do 10 pencils cost?

  1. This is direct proportion because more pencils cost more money at the same rate.
  2. Find one pencil: £2.40 divided by 4 = £0.60.
  3. Find 10 pencils: 10 × £0.60 = £6.00.

So: 10 pencils cost £6.00.

Example 2

4 workers take 6 hours to complete a job. How long would 8 workers take?

  1. This is inverse proportion because more workers take less time for the same job.
  2. Find the total work: 4 workers × 6 hours = 24 worker-hours.
  3. Divide by the new number of workers: 24 divided by 8 = 3 hours

So: 8 workers would take 3 hours.

Example 3

More people share the same bill, so each person pays less. Is this direct or inverse proportion?

  1. The total bill stays the same.
  2. As the number of people increases, each share decreases.
  3. One quantity goes up while the other goes down.

So: This is inverse proportion.

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