This activity is about finding the missing percentage in a repeated percentage change problem.

You will need a calculator for this activity!

Repeated percentage change

We know that if we invest £3,000 in the bank for 4 years at an interest rate of 3%, to find the total amount after 4 years we would do the calculation:

£3,000 x 1.03⁴

£3,000 being the investment.

1.03 being the multiplier for an increase of 3%

The power 4 being the 4 (periods) years we need to calculate.

We also know that if a car cost £5,000 and it depreciated by 2% every year, to find the value of the car after 5 years we would do the calculation:

£5,000 x 0.98⁵

£5,000 being the initial cost of the car.

0.98 being the multiplier for an decrease of 2% (100 - 2 = 98)

The power 5 being the 5 (periods) years we need to calculate.

In this activity, we will be given all the information, but we will have to find the percentage.

Let's look at a typical question.

Example

Ray invested £2,000 in the bank at an interest rate of X%.

After 3 years his investment was worth £2,315.25.

Calculate the interest rate X.

Answer

We use the same formula as at the top and substitute in what we are given.

2,000 x X³ = 2,315.25

£2,000 the investment.

X being the unknown multiplier for the interest rate.

3 being the number of years the money is invested.

£2,315.25 being the value of Ray's investment after 3 years.. 

We have to solve this equation to find X

2,000 x X³ = 2,315.25

Divide both sides by 2,000

X³ = 2,315.25 ÷ 2,000
X³ = 1.57625

(Write down all the numbers on the calculator display here to cut down on errors)

To find X we need to cube root both sides - the button on the calculator looks like this:

X = 1.05

1.05 is an increase of 5%.

Therefore, the percentage answer is 5%

Let's practise finding these roots on the calculator first, then we can try some of these questions!