Compound-Interest Aqa Higher

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

Compound interest formula

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

  • Final balance = where

    • P is the original amount,

    • r is the % increase,

    • and n is the number of years

  • Note that  is the same value as the multiplier

    • e.g. 1.15 for 15% interest

• This formula is not given in the exam

How do I solve reverse compound interest problems?

• You could be told the final balance after compound interest has been applied, and need to find the original amount

  • This could be referred to as a “reverse compound interest” problem

• For example if:

  • The final balance is £432

  • After 20% interest has been applied each year

  • For 3 years

• Using the same method as above, this can be written as an equation:

  • where <img alt=”P” data-mathml=”<math ><semantics><mi>P</mi><annotation encoding=”application/vnd.wiris.mtweb-params+json”>{“fontFamily”:”Times New Roman”,”fontSize”:”18″,”autoformat”:true,”toolbar”:”<toolbar ref=’general’><tab ref=’general’><removeItem ref=’setColor’/><removeItem ref=’bold’/><removeItem ref=’italic’/><removeItem ref=’autoItalic’/><removeItem ref=’setUnicode’/><removeItem ref=’mtext’ /><removeItem ref=’rtl’/><removeItem ref=’forceLigature’/><removeItem ref=’setFontFamily’ /><removeItem ref=’setFontSize’/></tab></toolbar>”}</annotation></semantics></math>” height=”22″ role=”math” src=”data:image/svg+xml;charset=utf8,%3Csvg%20xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F2000%2Fsvg%22%20xmlns%3Awrs%3D%22http%3A%2F%2Fwww.wiris.com%2Fxml%2Fmathml-extension%22%20height%3D%2222%22%20width%3D%2212%22%20wrs%3Abasel