# Calculate Compound Interest

In this worksheet, students will calculate compound interest, which assumes that the percentage increase or decrease is accumulating over time, including problems involving interest and depreciation.

Key stage:  KS 4

GCSE Subjects:   Maths

GCSE Boards:   AQA, Eduqas, Pearson Edexcel, OCR

Curriculum topic:   Ratio, Proportion and Rates of Change

Curriculum subtopic:   Ratio, Proportion and Rates of Change, Discrete Growth and Decay

Difficulty level:

### QUESTION 1 of 10

Compound interest means a percentage of money that is added to or taken away from an initial figure over a period of time.

Sometimes you love it BUT sometimes you hate it!

For compound interest calculations, the interest earned in each period needs to be added to the starting amount.

So compound interest for any year is interest paid on the total sum of money invested PLUS interest earned from the previous year.

Let's look at this in action now in an example.

e.g. Find the compound interest when £400 is invested for 3 years at an interest rate of 8%.

Interest for year 1: £400 × 0.08 = £32

Total for year 1: £400 + £32 = £432

Interest for year 2: £432 × 0.08 = £34.56

Total for year 2: £432 + £34.56 = £466.36

Interest for year 3: £466.36 × 0.08 = £37.31

Total for year 3: £466.36 + £37.31 = £503.67

Not a bad profit there - what would you spend it on?

Compound interest calculations are also used to find the depreciation of something.

Depreciation occurs when items lose some of their value after being bought.

If you bought a car two years ago which you were now trying to sell, you would expect to receive less than you bought it for, right?

Let's review an example focused on depreciation now.

e.g. Jim bought a speed boat that cost £18,000. Each year the price decreased in value by 5% at the beginning of each year. Calculate the value of the speed boat in 3 years time.

Year 1: £18,000 × 0.05 = 900    -->    £18,000 - 900 = £17,100

Year 2: £17,100 × 0.05 = £855    -->    £17,100 - £855 = £16,254

Year 3: £16,254 × 0.05 = £812.25    -->    £16,254 - £812.25 = £15,441.75

It was not Jim's wisest decision to buy that boat, was it?

Keep your eyes peeled for a 'Eureka moment!' in this activity.

You will need to have a calculator handy at this point to learn about an alternative way of calculating compound interest.

In this activity, we will calculate compound interest using the method above, which assumes that the percentage increase or decrease is accumulating over time. We will focus on problems involving interest and depreciation.

Sam has invested some money so that he can buy a new laptop.

He invested £500 in his bank, which offered a compound interest rate of 3%.

How much money did he have at the end of each time period below?

A student borrows £400 so that he can buy a bike.

Interest is added to the loan at 5% per annum (per year).

How much does the student owe after 4 years, if no repayments are made?

£473.50

£490.21

£486.92

£486.20

Between 2012 and 2014, a bank paid a compound interest rate of 2% on savings.

Between 2014 and 2015, the bank paid a compound interest rate of 3% on savings.

If we invested £800 in 2012, how much money do we have in 2015?

£473.50

£490.21

£486.92

£486.20

Linda invests £575 for a period of 3 years with a compound interest rate of 5%.

How much does Linda have at the end of 3 years?

£656.45

£665.64

£667.28

£655.55

Colin rents a house for £620 per month.

In the agreement, it states that the rent will automatically increase by 4% each year.

What will Colin's rent be at the end of his 7th year of renting the house?

Note, all answer options have been rounded to 2 decimal places

£816.28

£815.88

£815.96

£816.25

Match each investment on the left to the correct end amount on the right.

## Column B

Investment of £40 over 4 years -interest rate 3%
£314.93
Investment of £250 over 3 years - interest rate ...
£68.34
Investment of £87 over 9 years - interest rate 6%
£762.49
Investment of £642 over 5 years -interest rate 3...
£1307.10
Investment of £66 over 7 years - interest rate 0....
£45.02
Investment of £1250 over 3 years - interest rate ...
£146.98

Eloise has owned her horse for 3 years but she has now grown too big to ride it and needs a new one.

She paid £7,000 for it 3 years ago.

The value of the horse has decreased by 3% each year.

What is the value of the horse now?

## Column B

Investment of £40 over 4 years -interest rate 3%
£314.93
Investment of £250 over 3 years - interest rate ...
£68.34
Investment of £87 over 9 years - interest rate 6%
£762.49
Investment of £642 over 5 years -interest rate 3...
£1307.10
Investment of £66 over 7 years - interest rate 0....
£45.02
Investment of £1250 over 3 years - interest rate ...
£146.98

Match each scenario on the left to the correct current value on the right.

Note, all answer options have been rounded to 2 decimal places

## Column B

A car was bought for £5000, it depreciated 3% eac...
£2292.46
A bicycle was bought for £250, it depreciated 5% ...
£398.13
An iPhone was bought for £450, it depreciated 4% ...
£269.13
A computer was bought for £1200, it depreciated 1...
£845.96
A van was bought for £3200, it depreciated 8% eac...
£4293.67
An iPad was bought for £325, it depreciated 9% ea...
£225.63

How long would it take for an invested sum of £4500 to be worth at least £5000 with a compound interest rate of 4% per year?

2 years

4 years

3 years

5 years

James has invested his money unwisely.

Each year the value of his assets depreciates by 10%.

If James started with £5000, how many years will it take until his assets are valued at less than half their original worth?

2 years

4 years

3 years

5 years

• Question 1

Sam has invested some money so that he can buy a new laptop.

He invested £500 in his bank, which offered a compound interest rate of 3%.

How much money did he have at the end of each time period below?

EDDIE SAYS
Let's start by applying the process we learnt in the Introduction. Be sure to use the amount calculated for the previous year to add the next year's interest to. Year 1: £500 × 0.03 = £15 Total: £500 + £15 = £515 Year 2: £515 × 0.03 =£15.45 Total: £515 + £15.45 = £530.45 Year 3: £530.45 × 0.03 = £15.9135 Total £530.45 + £15.9135 = £546.36 (to 2 d.p) No wonder Sam is smiling, it looks like he's made a good investment!
• Question 2

A student borrows £400 so that he can buy a bike.

Interest is added to the loan at 5% per annum (per year).

How much does the student owe after 4 years, if no repayments are made?

£486.20
EDDIE SAYS
Here we go again! £400 × 0.05 = £20 Add this onto the original amount = £420 £420 × 0.05 = £21 Add this onto the amount from the previous year = £441 £441 × 0.05 = £22.05 Add this onto the amount from the previous year = £463.05 £463.05 × 0.05 = £23.15 Add this onto the amount from the previous year = £486.20
• Question 3

Between 2012 and 2014, a bank paid a compound interest rate of 2% on savings.

Between 2014 and 2015, the bank paid a compound interest rate of 3% on savings.

If we invested £800 in 2012, how much money do we have in 2015?

EDDIE SAYS
We need to be careful when calculating our return - we don't want to miss out, do we? 2012 Interest: 800 × 0.02 = £16 New total: £800 + £16 = £816 2013 Interest: 816 × 0.02 = £16.32 New total: £816 + £16.32 = £832.32 2014 Interest: £832.32 × 0.03 = £24.97 (Note, the interest rate went up at the beginning of this year, so we need to change the amount we are multiplying by.) New total: £832.32 + £24.97 = £857.29 2015 Interest: £857.29 × 0.03 = £25.72 New total: £857.29 + £25.72 = £883.01
• Question 4

Linda invests £575 for a period of 3 years with a compound interest rate of 5%.

How much does Linda have at the end of 3 years?

£665.64
EDDIE SAYS
So, here is the eagle-eye moment mentioned in the Introduction! You have to know how to calculate compound interest in the way you have been, which is important as you may be asked to do this without a calculator. However, there is a quicker way to do this with a calculator. We know that Linda wants to invest the money at 5% over 3 years. So, in our calculator, we can type: 575 × 1.053 ('power of button', '3') = £665.64 The 'power of' button looks different on different calculators, so make sure you use your own calculator all the time as you will become familiar with the buttons. Now we are going to practise using this shortcut in the rest of this activity.
• Question 5

Colin rents a house for £620 per month.

In the agreement, it states that the rent will automatically increase by 4% each year.

What will Colin's rent be at the end of his 7th year of renting the house?

Note, all answer options have been rounded to 2 decimal places

£815.88
EDDIE SAYS
Let's use our calculator to simplify this process. £620 × 1.057 = £815.88 (to 2 d.p.) It's easy when you know how, isn't it?!
• Question 6

Match each investment on the left to the correct end amount on the right.

## Column B

Investment of £40 over 4 years ...
£45.02
Investment of £250 over 3 years ...
£314.93
Investment of £87 over 9 years -...
£146.98
Investment of £642 over 5 years ...
£762.49
Investment of £66 over 7 years -...
£68.34
Investment of £1250 over 3 years...
£1307.10
EDDIE SAYS
Let's work through these, one at a time, using our calculator. £40 × 1.034 = £45.02 £250 × 1.083 = £314.93 £87 × 1.069 = £146.98 £642 × 1.0355 = £762.49 £66 × 1.0057 = £68.34 £1250 × 1.0153 = £1307.10
• Question 7

Eloise has owned her horse for 3 years but she has now grown too big to ride it and needs a new one.

She paid £7,000 for it 3 years ago.

The value of the horse has decreased by 3% each year.

What is the value of the horse now?

EDDIE SAYS
There are two ways of calculating this. The longer way Depreciation in year 1: £7000 × 0.03 = £210 New value after year 1: £7000 - 210 = £6790 D in Y2: £6790 × 0.03 = £203.70 NV after Y2: £6790 - £203.70 = £6586.30 D in Y3: £6586.30 × 0.03 = £197.59 NV after Y3: £6586.30 - £197.59 = £6388.71 The quicker way As the percentage is decreasing, we take 3% away from 100% which give us 97%. £7000 × 0.973 = £6388.71 (to 2 d.p.) Which option did you use?
• Question 8

Match each scenario on the left to the correct current value on the right.

Note, all answer options have been rounded to 2 decimal places

## Column B

A car was bought for £5000, it d...
£4293.67
A bicycle was bought for £250, i...
£225.63
An iPhone was bought for £450, i...
£398.13
A computer was bought for £1200,...
£845.96
A van was bought for £3200, it d...
£2292.46
An iPad was bought for £325, it ...
£269.13
EDDIE SAYS
It's much easier when you know how the calculator can work for you, but it's important to know the long way round to help understand the process and so you can provide any explanations you may have to give. Let's work these out, one by one, using our calculator again. As these are all depreciation sums, we need to subtract our annual rate from 1 to find our multiplier. 5000 × 0.975 = £4293.67 250 × 0.952 = £225.63 450 × 0.963 = £398.13 1200 × 0.893 = £845.96 3200 × 0.924 = £2292.46 325 × 0.912 = £269.13
• Question 9

How long would it take for an invested sum of £4500 to be worth at least £5000 with a compound interest rate of 4% per year?

3 years
EDDIE SAYS
Where to start here? Let's find the compound interest for year 1: 4500 × 0.04 = £180 We know that the sum will earn £180 in year 1, if we add this back on to our starting amount, we reach £4680. Y2 interest: £4680 × 0.04 = £187.20 Y2 total: £4680 + £187.20 = £4867.20 Y3 interest: £4867.20 × 0.04 = £194.69 (to 2 d.p.) Y3 total: £4867.20 + £194.69 = £5061.89 Or, we could just use our calculator to find: 4500 × 1.04? By trying a few different powers, and working out if we need to increase or decrease from where we are at. Either way, we find that after 3 years, the £4500 investment would be worth at least £5000...it will be worth £5061.89 to be exact!
• Question 10

James has invested his money unwisely.

Each year the value of his assets depreciates by 10%.

If James started with £5000, how many years will it take until his assets are valued at less than half their original worth?

EDDIE SAYS
James assets depreciate by 10% per year, so we need to calculate: 1 - 0.1 = 0.9 £5000 × 0.90? Now we can play around with the powers until we reach an answer below £2500 (which is half of £5000). 5000 × 0.907 = £2391.48 (to 2 d.p.) That's not long James to lose half your assets! He needs to do something NOW to resolve this issue. Great work completing this activity! You can now calculate compound interest, which assumes that the percentage increase or decrease is accumulating over time, in problems involving interest and depreciation.
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