2.2 Compound interest

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Other applications of compound growth

South Africa's population is increasing by 2,5% per year. If the current population is 43 million, how many more people will there be in South Africa in two years' time ?

• initial value (opening balance), $P=43\phantom{\rule{3.33333pt}{0ex}}000\phantom{\rule{3.33333pt}{0ex}}000$
• period of time, $n=2\phantom{\rule{3.33333pt}{0ex}}\mathrm{year}$
• rate of increase, $i=2,5%\phantom{\rule{3.33333pt}{0ex}}\mathrm{per}\mathrm{year}$

We are required to find the closing balance( $A$ ).

1. We know from [link] that:

$A=P{\left(1+i\right)}^{n}$
2. $\begin{array}{ccc}\hfill \mathrm{A}& =& P{\left(1+i\right)}^{n}\hfill \\ & =& 43\phantom{\rule{3.33333pt}{0ex}}000\phantom{\rule{3.33333pt}{0ex}}000{\left(1+0,025\right)}^{2}\hfill \\ & =& 45\phantom{\rule{3.33333pt}{0ex}}176\phantom{\rule{3.33333pt}{0ex}}875\hfill \end{array}$
3. There will be $45\phantom{\rule{3.33333pt}{0ex}}176\phantom{\rule{3.33333pt}{0ex}}875-43\phantom{\rule{3.33333pt}{0ex}}000\phantom{\rule{3.33333pt}{0ex}}000=2\phantom{\rule{3.33333pt}{0ex}}176\phantom{\rule{3.33333pt}{0ex}}875$ more people in 2 years' time

A swimming pool is being treated for a build-up of algae. Initially, $50{m}^{2}$ of the pool is covered by algae. With each day of treatment, the algae reduces by 5%. What area is covered by algae after 30 days of treatment ?

• opening balance, $P=50{\mathrm{m}}^{2}$
• period of time, $n=30\phantom{\rule{3.33333pt}{0ex}}\mathrm{days}$
• rate of increase, $i=5%\phantom{\rule{3.33333pt}{0ex}}\mathrm{per}\mathrm{day}$

We are required to find the closing balance( $A$ ).

1. We know from [link] that:

$\mathrm{A}=\mathrm{P}{\left(1+\mathrm{i}\right)}^{\mathrm{n}}$

But this is compound decrease so we can use the formula:

$\mathrm{A}=\mathrm{P}{\left(1-\mathrm{i}\right)}^{\mathrm{n}}$
2. $\begin{array}{ccc}\hfill \mathrm{A}& =& P{\left(1-i\right)}^{n}\hfill \\ & =& 50{\left(1-0,05\right)}^{30}\hfill \\ & =& 10,73{m}^{2}\hfill \end{array}$
3. Therefore the area still covered with algae is $10,73{m}^{2}$

Compound interest

1. An amount of R3 500 is invested in a savings account which pays compound interest at a rate of 7,5% per annum. Calculate the balance accumulated by the end of 2 years.
2. If the average rate of inflation for the past few years was 7,3% and your water and electricity account is R 1 425 on average, what would you expect to pay in 6 years time ?
3. Shrek wants to invest some money at 11% per annum compound interest. How much money (to the nearest rand) should he invest if he wants to reach a sum of R 100 000 in five year's time ?

Summary

• A foreign exchange rate is the price of one currency in terms of another.
• There are two types of interest: simple and compound.
• The following table summarises the key definitions that are used in both simple and compound interest.  $P$ Principal (the amount of money at the starting point of the calculation) $A$ Closing balance (the amount of money at the ending point of the calculation) $i$ interest rate, normally the effective rate per annum $n$ period for which the investment is made
• For simple interest we use:
$\mathrm{A}=P\left(1+i·n\right)$
• For compound interest we use:
$\mathrm{A}=P{\left(1+i\right)}^{n}$
Always keep the interest and the time period in the same units of time (e.g. both in years, or both in months etc.).

The following three videos provide a summary of how to calculate interest. Take note that although the examples are done using dollars, we can use the fact that dollars are a decimal currency and so are interchangeable (ignoring the exchange rate) with rands. This is what is done in the subtitles.

Note in this video that at the very end the rule of 72 is mentioned. You will not be using this rule, but will rather be using trial and error to solve the problem posed.

End of chapter exercises

1. You are going on holiday to Europe. Your hotel will cost 200 euros per night. How much will you need in Rands to cover your hotel bill, if the exchange rate is 1 euro = R9,20?
2. Calculate how much you will earn if you invested R500 for 1 year at the following interest rates:
1. 6,85% simple interest.
2. 4,00% compound interest.
3. Bianca has R1 450 to invest for 3 years. Bank A offers a savings account which pays simple interest at a rate of 11% per annum, whereas Bank B offers a savings account paying compound interest at a rate of 10,5% per annum. Which account would leave Bianca with the highest accumulated balance at the end of the 3 year period?
4. How much simple interest is payable on a loan of R2 000 for a year, if the interest rate is 10%?
5. How much compound interest is payable on a loan of R2 000 for a year, if the interest rate is 10%?
6. Discuss:
1. Which type of interest would you like to use if you are the borrower?
2. Which type of interest would you like to use if you were the banker?
7. Calculate the compound interest for the following problems.
1. A R2 000 loan for 2 years at 5%.
2. A R1 500 investment for 3 years at 6%.
3. An R800 loan for l year at 16%.
8. If the exchange rate for 100 Yen = R 6,2287 and 1 Australian Doller (AUD) = R 5,1094 , determine the exchange rate between the Australian Dollar and the Japanese Yen.
9. Bonnie bought a stove for R 3 750. After 3 years she had finished paying for it and the R 956,25 interest that was charged for hire-purchase. Determine the rate of simple interest that was charged.

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