Percentage change - WJECFurther examples for Higher tier

Example 1

If you increase an amount by 10%, then decrease the new amount by 10%, what is the percentage change?

Solution

Let’s try an amount first, such as £100.

Method 1: The Multiplier Method

If we are increasing by 10% the multiplier would be × 1.10

We are then decreasing this by 10% so multiplier here is × 0.90

Combining them 1.10 × 0.90 = 0.99

This means there would be a 1% decrease, or it would be 99% of the original amount.

Method 2

10% of £100 = £10

£100 + £10 = £110

Now decrease this by 10%.

10% of £110 = £11

£110 − £11 = £99

Overall change:

£100 − £99 = £1 and this is 1% of £100.

Method 3

Now decrease this by 10%.

Overall change:

Example 2

What happens if you increase an amount by 100%, then decrease the result by 100%?

Solution

Method 1

For a 100% increase the multiplier would be × 2.00

For a 100% decrease the multiplier would be × 0.00

Combining them 2 × 0 = 0

Method 2

100% of £100 = £100

£100 + £100 = £200

Now decrease this by 100%.

100% of £200 = £200

£200 − £200 = £0

Overall change:

£100 − £0 = £100 and this is 100% of £100

So the outcome doesn’t work in quite the same way for 100%.

Example 3

The value of an investment fell by 11% for two years in a row then went up by 11% for two years in a row.

1. If it was originally worth £4,800, how much was it worth after the four years?
2. What is the amount lost or gained?
3. What would happen if the decreases and increases were reversed?

Solution 1

At the start, it was worth £4,800

The value fell by 11% for two years in a row, so it was then worth:

£4800 × 0.89² = £3,802.08

It then went up by 11% for two years in a row, so it was then worth:

£3802.08 × 1.11² = £4684.542768 = £4,684.54 (to the nearest penny).

The investment worth £4,800 originally is worth £4,684.54 after four years.

Solution 2

Actual change:

£4,800 − £4,684.54 = −£115.46

This is a loss of £115.46

Solution 3

If the losses and gains are reversed:

At the start it was worth £4,800

The value rose by 11% for two years in a row, so it was then worth:

£4,800 × 1.11² = £5,914.08

It then went down by 11% for two years in a row, so it was then worth:

£5,914.08 × 0.89² = £4,684.542768 = £4,684.54 (to the nearest penny).

This is exactly the same as the first case so the investment worth £4,800 originally is worth £4,684.54 after four years.

Method

You can demonstrate the fact there is no change by using the multipliers in one step.

Down 11% for two years: multiplier = × 0.89²

Up 11% for two years: multiplier = × 1.11²

If the value goes down first, then up: 4800 × 0.89² × 1.11²

If the value goes up first, then down: 4800 × 1.11² × 0.89²

Therefore:

× 0.89² × 1.11² = × 1.11² × 0.89²