Repeated percentage change

Repeated percentage change occurs when a value is increased or decreased by a percentage multiple times, often used for compound growth or decay. Each time, the change is applied to the new value.

Multipliers

A percentage increase or decrease can be shown as a decimal, which is called a multiplier. To find a multiplier, start with 100%, then add/subtract the percentage you are changing.

Add 40%

$$100\mathrm{\%}+40\mathrm{\%}=140\mathrm{\%}$$

Next, divide it by 100 (to convert it to a decimal).

$$140\mathrm{\%}÷100=1.4$$

1.4 is now our multiplier. If you multiply anything by 1.4, it will increase it by 40%.

Multiple multipliers

If we want to increase 60 by 40% then 40% then 40% we will need to multiply it by 1.4 three times.

$$60\times 1.4\times 1.4\times 1.4$$

We can use indices for this.

$$60\times {1.4}^3=164.64$$

Compound interest

If a bank pays interest on your savings each year, we can calculate the amount of money in the account after a number of years using compound interest.

If we pay in £300 to a bank account that earns 5% per annum (per year) for 3 years, we can find the amount we will end up with.

Find the multiplier.

$$100\mathrm{\%}+5\mathrm{\%}=105\mathrm{\%}$$

$$105\mathrm{\%}÷100=1.05$$

Increase by this amount three times. Remember we can use indices.

$$300\times {1.05}^3=\pounds 347.29$$

Depreciation

If the amounts are going down, we call it depreciation.

If the value of a £200 TV decreases by 10% each year for 4 years, we can find its value after the 4 years.

Find the multiplier.

$$100\mathrm{\%}-10\mathrm{\%}=90\mathrm{\%}$$

$$90\mathrm{\%}÷100=0.9$$

Decrease by this amount three times.

$$200\times {0.9}^4=\pounds 131.22$$

Always check that the answer makes sense. The TV is now worth less, so this makes sense!