Should you ever find yourself stuck with a mathematics question on a television quiz show, you will probably wish you had remembered how many even prime numbers there are between 1 and 100 for the sake of R1 000 000. And who does not want to be a millionaire, right?
Welcome to the Grade 10 Finance Chapter, where we apply maths skills to everyday financial situations that you are likely to face both now and along your journey to purchasing your first private jet.
If you master the techniques in this chapter, you will grasp the concept of compound interest, and how it can ruin your fortunes if you have credit card debt, or make you millions if you successfully invest your hard-earned money. You will also understand the effects of fluctuating exchange rates, and its impact on your spending power during your overseas holidays!
Before we begin this chapter it is worth noting that the vast majority of countries use a decimal currency system. This simply means that countries use a currency system that works with powers of ten, for example in South Africa we have 100 (10 squared) cents in a rand. In America there are 100 cents in a dollar. Another way of saying this is that the country has one basic unit of currency and a sub-unit which is a power of 10 of the major unit. This means that, if we ignore the effect of exchange rates, we can essentially substitute rands for dollars or rands for pounds.
If you had R1 000, you could either keep it in your wallet, or deposit it in a bank account. If it stayed in your wallet, you could spend it any time you wanted. If the bank looked after it for you, then they could spend it, with the plan of making profit from it. The bank usually “pays" you to deposit it into an account, as a way of encouraging you to bank it with them, This payment is like a reward, which provides you with a reason to leave it with the bank for a while, rather than keeping the money in your wallet.
We call this reward "interest".
If you deposit money into a bank account, you are effectively lending money to the bank - and you can expect to receive interest in return. Similarly, if you borrow money from a bank (or from a department store, or a car dealership, for example) then you can expect to have to pay interest on the loan. That is the price of borrowing money.
The concept is simple, yet it is core to the world of finance. Accountants, actuaries and bankers, for example, could spend their entire working career dealing with the effects of interest on financial matters.
In this chapter you will be introduced to the concept of financial mathematics - and given the tools to cope with even advanced concepts and problems.
Interest
The concepts in this chapter are simple - we are just looking at the same idea, but from many different angles. The best way to learn from this chapter is to do the examples yourself, as you work your way through. Do not just take our word for it!
Simple interest is where you earn interest on the initial amount that you invested, but not interest on 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!
You might be wondering to yourself:
how much interest will you be paid if you only leave the money in the account for 3 months, or
what if you leave it there for 3 years?
It is actually quite simple - which is why they call it Simple Interest.
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×(P×i). The closing balance would therefore be:
For 3 years, you would get three years' worth of interest, being: 3×(P×i). The closing balance at the end of the three year period would be:
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.
Annual Rates means Yearly rates. and p.a.(per annum) = per year
If I deposit R1 000 into a special bank account which pays a Simple Interest of 7% for 3 years, how much will I get back at the end of this term?
opening balance, 
interest rate, i=7%
period of time, 
We are required to find the closing balance (A).
We know from Equation 4.7 that:
Solve the problem :
The closing balance after 3 years of saving R1 000 at an interest rate of 7% is R1 210.
If I deposit R30 000 into a special bank account which pays a Simple Interest of 7.5%, for how many years must I invest this amount to generate R45 000?
opening balance, 
interest rate, i=7,5%
closing balance, 
We are required to find the number of years.
We know from Equation 4.7 that:
Solve the problem :
The period is 6 years and 8 months for R30 000 to generate R45 000 at a simple interest rate of 7,5%. If we were asked for the nearest whole number of years, we would have to invest the money for 7 years.
Troy is keen to buy an additional hard drive for his laptop advertised for R 2 500 on the internet. There is an option of paying a 10% deposit then making 24 monthly payments using a hire-purchase agreement where interest is calculated at 7,5% p.a. simple interest. Calculate what Troy's monthly payments will be.
A new opening balance is required, as the 10% deposit is paid in cash.
10% of R 2 500 = R250
new opening balance, 
interest rate, i=7,5%
period of time, 
We are required to find the closing balance (A) and then the monthly payments.
We know from Equation 4.7 that:
Solve the problem :
Troy's monthly payments = R 107,81
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.
Seven years ago, Tjad's drum kit cost him R12 500. It has now been valued at R2 300. What rate of simple depreciation does this represent ?
opening balance, 
period of time, 
closing balance, 
We are required to find the interest rate(i).
We know from Equation 4.7 that:
Therefore, for depreciation the formula will change to:
Solve the problem :
Therefore the rate of depreciation is 11,66%
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.
Click here for the solution
Calculate the simple interest for the following problems.
A loan of R300 at a rate of 8% for l year.
An investment of R225 at a rate of 12,5% for 6 years.
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 ?
Click here for the solution
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.
How much will Bongani pay in total ?
How much interest does he pay ?
What is his monthly installment ?
To explain the concept of compound interest, the following example is discussed:
I deposit R1 000 into a special bank account which pays a Simple Interest of 7%. What if I empty the bank account after a year, and then take the principal and the interest and invest it back into the same account again. Then I take it all out at the end of the second year, and then put it all back in again? And then I take it all out at the end of 3 years?
opening balance, 
interest rate, i=7%
period of time, 
at a time, for 3 years
We are required to find the closing balance at the end of three years.
We know that:
Determine the closing balance at the end of the first year :
After the first year, we withdraw all the money and re-deposit it. The opening balance for the second year is therefore 
, because this is the balance after the first year.
After the second year, we withdraw all the money and re-deposit it. The opening balance for the third year is therefore 
, because this is the balance after the first year.
The closing balance after withdrawing all the money and re-depositing each year for 3 years of saving R1 000 at an interest rate of 7% is R1 225,04.
In the two worked examples using simple interest (Exercise 1.: Simple Interest and Exercise 5.: Using Simple Interest to lead to the concept Compound Interest ), we have basically the same problem because P=R1 000, i=7% and n=3 years for both problems. Except in the second situation, we end up with R1 225,04 which is more than R1 210 from the first example. What has changed?
In the first example I earned R70 interest each year - the same in the first, second and third year. But in the second situation, when I took the money out and then re-invested it, I was actually earning interest in the second year on my interest (R70) from the first year. (And interest on the interest on my interest in the third year!)
This more realistically reflects what happens in the real world, and is known as Compound Interest. It is this concept which underlies just about everything we do - so we will look at it more closely next.