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Ratio, proportion and percentages

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# 2.2 Direct proportion

In a recipe the quantity of each ingredient needed depends upon the number of portions. As the number of portions increases, the quantity required increases. The quantity per portion is the same. This is called direct proportion. The quantity is said to be directly proportional to the number of portions. If 2 potatoes are required for one portion, 4 will be required for two portions etc. A useful method for direct proportion problems is to find the quantity for one and multiply by the number you want.

## Example 7

John lives with three cats. His daughter asks him to look after her cat for a week while she goes away. John normally buys two tins of cat food a day for the three cats. How many tins should he buy for the four cats for a week?

There are several ways to do this. Here is one.

3 cats eat 2 tins a day

1 cat eats tin a day

4 cats eat tins a day

4 cats eat tins a week

So John should buy 19 tins for the week.

Note it helped to simplify the problems by considering 1 cat, rather than going straight to 4 cats.

## Example 8

Debbie is checking her phone bill. Her mobile phone calls have all been charged at the same rate, 30 pence per minute. (Call charges are rounded to the nearest penny and charged to the nearest second.)

• (a) She wants to check the cost of a call to her friend. The call lasted 7 minutes and 34 seconds. How much should she have been charged?

• (b) How long, at this rate, can she speak to her friend if the call charge is to cost no more than £2.50?

(a) 1 minute cost 30 p. (i.e. 60 seconds for 30 p).

• 1 second for p = 0.5 p.

• 7 minutes 34 seconds is 454 seconds, costing 454 × 0.5 p = 227 p.

• The call should have been charged at £2.27.

(b) 0.5 p will allow her to talk for 1 second.

• 1 p will allow her to talk for 2 seconds.

• £2.50 (250 p) will allow her to talk for 250 × 2 seconds, = 500 seconds, i.e. 8 minutes 20 seconds.