Compound Interest (Cambridge International IGCSE Mathematics)

Compound Interest

What is compound interest?

• Compound interest is where interest is calculated on the running total, not just the starting amount

  • This is different from simple interest where interest is only based on the starting amount

• e.g. $ 100 earns 10% interest each year, for 3 years

  • At the end of year 1, 10% of $ 100 is earned

    • The total balance will now be 100+10 = $ 110

  • At the end of year 2, 10% of $ 110 is earned

    • The balance will now be 110+11 = $ 121

  • At the end of year 3, 10% of $ 121 is earned

    • The balance will now be 121+12.1 = $ 133.10

How do I calculate compound interest?

• Compound interest increases an amount by a percentage, and then increases the new amount by the same percentage

  • This process repeats each time period (yearly or monthly etc)

• We can use a multiplier to carry out the percentage increase multiple times

  • To increase $ 300 by 5% once, we would find 300×1.05

  • To increase $ 300 by 5%, each year for 2 years, we would find (300×1.05)×1.05

    • This could be rewritten as 300×1.05²

  • To increase $ 300 by 5%, each year for 3 years, we would find ((300×1.05)×1.05)×1.05

    • This could be rewritten as 300×1.05³

• This can be extended to any number of periods that the interest is applied for

  • If $ 2000 is subject to 4% compound interest each year for 12 years
  • Find 2000×1.04¹²**, **which is $ 3202.06

• Note that this method calculates the** total balance** at the end of the period, not the interest earned

How do I calculate depreciation?

• A similar method can be used if something decreases in value by a percentage every year (e.g. a car)

• This is known as depreciation

• Change the multiplier to one which represents a percentage decrease

  • e.g. a decrease of 15% would be a multiplier of 0.85

• If a car worth $ 16 000 depreciates by 15% each year for 6 years

  • Its value will be 16 000 × 0.85⁶, which is $ 6034.39

Compound interest formula

• An alternative method is to use the following formula to calculate the final balance

  • Final balance = $P ( 1 + \frac{r}{100} )^{n }$ where

    • P is the original amount,
    • r is the % increase,
    • and n is the number of years

  • Note that $1+\frac{r}{100}$ is the same value as the multiplier

    • e.g. 1.15 for 15% interest

• This formula is not given in the exam

Exam Hint

• Double check if the question uses simple interest or compound interest
• The formula for compound interest is not given in the exam

Worked Example Jasmina invests $ 1200 in a savings account which pays compound interest at the rate of 4% per year for 7 years.

To the nearest dollar, what is her investment worth at the end of the 7 years?

Answer:

We want an increase of 4% per year, this is equivalent to a multiplier of 1.04, or 104% of the original amount

This multiplier is applied 7 times; $\times1.04\times1.04\times1.04\times1.04\times1.04\times1.04\times1.04=1.04^{7}$

Therefore the final value after 7 years will be

1200 \times 1.04^{7} = \ 1579.118135$

Round to the nearest dollar

\ 1579$

Alternate method Using the formula for the final amount $P ( 1 + \frac{r}{100} )^{n }$ *Substitute P is 1200, r = 4 and n = 7 into the formula *

$1200 ( 1 + \frac{4}{100} )^{7}$

\ 1579$