Introduction and simple interest

As an easy example of simple interest, consider how much you will get by investing R1 000 for 1 year with a bank that pays you 5% simple interest. At the end of the year, you will get an interest of:

So, with an “opening balance" of R1 000 at the start of the year, your “closing balance" at the end of the year will therefore be:

We sometimes call the opening balance in financial calculations the Principal , which is abbreviated as $P$ (R1 000 in the example). The interest rate is usually labelled $i$ (5% in the example), and the interest amount (in Rand terms) is labelled $I$ (R50 in the example).

So we can see that:

and

This is how you calculate simple interest. It is not a complicated formula, which is just as well because you are going to see a lot of it!

Not just one

You might be wondering to yourself:

1. how much interest will you be paid if you only leave the money in the account for 3 months, or
2. what if you leave it there for 3 years?

It is actually quite simple - which is why they call it Simple Interest .

1. Three months is 1/4 of a year, so you would only get 1/4 of a full year's interest, which is: $1/4\times (P\times i)$ . The closing balance would therefore be:
   $$\begin{array}{ccc}\hfill \mathrm{Closing\; Balance}& =& P+1/4\times (P\times i)\hfill \\ & =& P(1+(1/4\left)i\right)\hfill \end{array}$$
2. For 3 years, you would get three years' worth of interest, being: $3\times (P\times i)$ . The closing balance at the end of the three year period would be:
   $$\begin{array}{ccc}\hfill \mathrm{Closing\; Balance}& =& P+3\times (P\times i)\hfill \\ & =& P\times (1+(3\left)i\right)\hfill \end{array}$$

If you look carefully at the similarities between the two answers above, we can generalise the result. If you invest your money ( $P$ ) in an account which pays a rate of interest ( $i$ ) for a period of time ( $n$ years), then, using the symbol $A$ for the Closing Balance:

As we have seen, this works when $n$ is a fraction of a year and also when $n$ covers several years.

Other applications of the simple interest formula

Many items become less valuable as they are used and age. For example, you pay less for a second hand car than a new car of the same model. The older a car is the less you pay for it. The reduction in value with time can be due purely to wear and tear from usage but also to the development of new technology that makes the item obsolete, for example, new computers that are released force down the value of older models. The term we use to descrive the decrease in value of items with time is depreciation .

Depreciation, like interest can be calculated on an annual basis and is often done with a rate or percentage change per year. It is like ”negative” interest. The simplest way to do depreciation is to assume a constant rate per year, which we will call simple depreciation. There are more complicated models for depreciation but we won't deal with them here.

Simple interest

1. An amount of R3 500 is invested in a savings account which pays simple interest at a rate of 7,5% per annum. Calculate the balance accumulated by the end of 2 years.
   
2. Calculate the simple interest for the following problems.
   1. A loan of R300 at a rate of 8% for l year.
   2. An investment of R225 at a rate of 12,5% for 6 years.
3. I made a deposit of R5 000 in the bank for my 5 year old son's 21st birthday. I have given him the amount of R 18 000 on his birthday. At what rate was the money invested, if simple interest was calculated ?
   
4. Bongani buys a dining room table costing R 8 500 on Hire Purchase. He is charged simple interest at 17,5% per annum over 3 years.
   1. How much will Bongani pay in total ?
   2. How much interest does he pay ?
   3. What is his monthly installment ?