# KS4. Focus on Compound Interest

The method we learn for compound interest is used whenever we have an amount that is increasing by a fixed percentage more than once.

The typical example of this is money held at the bank in an interest bearing account, but it can also be used for questions considering a loan borrowed from the bank at a fixed rate of interest, or the increase in population of an animal over several generations and in other circumstances.

The formulae we use for compound interest, which must be memorised, are:

$F = P(1+ \frac{R}{100})^n$, $I = F - P$ and $I_{p} = I/P$

where F = final amount, P = principal amount, R = rate of interest, n = number of time periods, I = Interest earned and Ip = percentage interest earned.

Let’s analyse the parts of the formula.

So, the right hand side of the equation starts with just P. This is the amount of money deposited in the bank at the start, called the principal amount.

Then we multiply this principal amount, which reflects the impact of the interest. We multiply it first by $1 + \frac{R}{100}$ . The 1 represents 100%. Multiplying by this would leave the principal amount unchanged. The $R/100$ represents the amount by which the principal increases due to the interest. The sum of these two amounts gives the total amount held at the end of the period. Note that if we are given an example where the principal amount is reducing over time, then instead of R, we will use -R.

But because there is more than one period, we need to multiply by this $1 + \frac{R}{100}$ more than one time. This is done using the power n in the formula, representing the number of times we multiply by this multiplier to get to our final amount.

As with all formulae in mathematics, we will typically be told the value of some of our variable amounts (i.e. F, P, R, n or I) and asked to find the value of an unknown variable amount (i.e. F, P, R, n or I).

Examples

1.) £1,500 is invested at 5% p.a. compound interest. What will the investment be worth after 4 years?

2.) A sum of money invested for 5 years at a rate of 5% interest, compounded yearly, grows to $2,500. What was the initial sum invested? 3.) €100 is invested subject to compound interest at a rate of 8% per annum. Find the value of the investment after a period of 3 years. 4.) The value of a new computer system depreciates by 30% per year. If it cost £1,200 new, what will it be worth in two years time? 5.) Dasha spent £2,000 on her credit card, which charges interest on a monthly basis. She noticed after a year that the amount she owed had doubled to 40%. What interest rate is her credit card charging? Exercise Note: For questions (1b), “calculated as a flat rate” means that the interest is not compounded, but just charged once at the end of the two years. 9.) Masha puts £1,500 into her bank account. She notices that it pays interest yearly. After 4 years she gets an email from her bank account telling her she has already earned £300 in interest. What percentage interest does the bank account pay? 10.) Sasha buys a brand new PS5 for$500. After two years his mother notices that his grades in mathematics have deteriorated and that he spends most of his time in the evening playing on his PS5. She tells him that he must sell it to buy more maths textbooks. He advertises on Facebook but finds that he can only get \$200 for it. At what percentage rate has the PS5 depreciated?