(Higher Tier)

You might be asked to write an equation for two quantities that are in proportion to one another.

So, if a and b are in proportion to one another you write:

a ? b

? is the symbol for proportionality.

For example, say 7 apples cost 49p and you wanted to find out what 12 would cost.

a = the cost

b = the number of apples

k = the cost of one apple

a = k x b

49 = k x 7

k = 49 7

k = 7

So, a = 7b

Now you need to substitute 12 with b:

a = 7 x 12

a = 84p

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Indirect proportion

If two amounts are indirectly proportional then they move in the opposite directly to each other. If one goes up, the other goes down. The equation for this is:

a ? ¹/_b

For example, the time it takes to distribute a number of leaflets is indirectly proportional to the number of people delivering them.

If it takes 5 people 7 hours to deliver the leaflets, how long would it take 7 people?

t = the time taken to deliver the leaflets

d = the number of people delivering

t and d are indirectly proportional:

t ? ¹/_d

t = k x ¹/_d

5 = k x ¹/7

So, the equation you can use is:

t = ³⁵/_d

This means the answer is:

t = ³⁵/₇

t = 5

So it would take 5 hours for 7 people to deliver the leaflets.

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Repeated proportional change

If you multiply a number continuously by the same number then the proportion is constant (1 is the exception).

Compound interest is a good example of this.

For instance, say you borrowed £800 from a bank for 3 years with an interest of 5% compound interest. How much interest would you have paid by the end?

Every year 5% is added. This means that each year you multiply by 1.05 (100% + 5%).

So, £300 is borrowed for three years. This works out at:

800 x (1.05 x 1.05 x 1.05)

= 800 x 1.05³ = 926.10

To find the interest take £800 away from your answer: 926.10 – 800 = £126.10

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