**(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* = 7*b*

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 1.05^{3} = 926.10

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

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