Business Mathematics | Module 2

SCATTER PLOTS AND LINES OF BEST FIT

(SO 2: AC 2.1-5)
One way to find out whether two variables are related is to draw a graph of a scatter plot, showing a dot
for each piece of data. e.g. using measurements of lung capacity of a group of smokers, compared to
the number of cigarettes they smoke a day: each dot represents a different person's lung capacity and
his/her number of cigarettes smoked per day.
There is a steady decline in lung capacity as people smoke more cigarettes a day. This could be
approximately a linear relationship, as is shown by the dotted straight line which is drawn through as
many of the dots as possible. There are usually a couple of dots which fall outside the line.

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Sometimes the scatter plot seems to indicate a quadratic relationship (i.e. the dotted line is curved in
the shape of a parabola.) The scatter plot below shows such a relationship.

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PROBABILITY

(S0 3: AC 1-6)

PROBABILITY CALCULATIONS

The probability of something occurring refers to the likelihood of it happening. There is a scale ranging
from 0 (cannot happen) to 1 (will definitely happen.) All other probabilities lie in between these two.

Impossible Unlikely Equal chance Likely Certain

0 1/4 1/2 3/4 1

0% 25% 50% 75% 100%

e.g. Probability that an object will fall when dropped = 1 or 100%

Probability that an object will rise when dropped = 0.

Probability that an coin will land on Heads when dropped = 1/2 or 50%

Suppose a 6-sided die is thrown. The probability that it lands on a 3 is the fraction 1/6. This is calculated
by asking how many ways it could have fallen: it could have fallen any one of 6 ways. Another way of
saying the same thing is that the odds on it landing on a 3 are 1 to 5: there is only one way it could land
on a 3, but 5 ways it could land on other numbers.

In theory, one would expect that one sixth of the throws of a die would show a 3, but in practice this is
not usually the case. One can throw a die seven times and perhaps there will be two 3's, or maybe none.
However, if one continues to repeat the experiment many times, the relative frequency of 3's compared
to other numbers comes close to 1/6.

Calculating Probability

Suppose 5 coloured balls are placed in a bag. If you put in your hand and draw out a ball at random, you

are more likely to get a red ball than green. We calculate the probability by putting the number of red

balls over the total number of balls: in this case, the probability of a red ball is 3/5. In the same way, the

probability of getting a green ball is 2/5.

Calculating Probability using addition law
Suppose 1 blue and 4 yellow balls are added to the bag. If you put in your hand and draw out a ball at
random, what is the probability that you would get a red or yellow?
There are now 10 balls altogether, of which 3 are red and 4 are yellow. We simply add the two
probabilities: 3/10 + 4/10 = 7/10.

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The addition law is used when we see the word 'or.'

Calculating Probability using multiplication law
If you throw a die three times, what is the probability that you would get three 6's? It is a small chance.
In this case, the probability is 1/6 x 1/6 x1/6 = 1/216 because we are looking at the chance of getting a 6
AND another 6 AND another 6.
The multiplication law is used when we see the word 'and.' This only works when the events are
independent, that is, they have no effect on each other.
The following terms are used when describing probability experiments:
A trial is to take a turn (e.g. at throwing a die)
An event is a general occurrence one observes, e.g. obtaining an even number when throwing a die,
An outcome is a specific fact about what happened, e.g. the die fell on a 3.
Random events are events which have an equal likelihood of occurring: e.g. getting an even number
when throwing a die, or getting heads or tails when spinning an unbiased coin.
Non-random events are when there is not the same likelihood, e.g. when a biased coin is used, or
drawing a picture card from a pack of playing cards (there are 12 picture cards in a pack of 52, so if a
card is drawn at random from a pack, there is a chance of 12/52 that it will be a picture card, but a
chance of 40/52 that it won't be a picture card.)
Bias must be avoided when setting up experiments or holding competitions. e.g. in a competition if
employees of the sponsoring company are allowed to enter, there may be more of them than other
competitors, so it will cause a biased result.

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TREE DIAGRAMS

A tree diagram is useful to calculate probabilities that are
dependent. Suppose a bag contains 5 marbles: 3 red and 2
blue. If you select a marble, we can draw a tree diagram with
branches to show the probabilities of selecting red and blue.

If we select a second marble without divided
replacement, the next stage is shown by extra
branches. Remember there are only 4
marbles left once the first marble has been
selected, so the second stage probabilities are
by 4.

Suppose we want to look at possible weather patterns over 3 days. We are looking at Sunshine (S) or
Rain (R). It could happen that the first day is sunny, followed by 2 days of rain. This would be written as
SRR. All the possible patterns are shown in the tree diagram below. There are 8 different possible
patterns.

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FINANCIAL TRANSACTIONS

If someone wishes to make money by engaging in business, there are various financial transactions which
may occur.

INCOME AND EXPENDITURE

Income is the amount of money or its equivalent received during a period
of time in exchange for labour or services, from the sale of goods or
property, or as profit from financial investments.
Expenditure is money that is spent.

COSTS
For example, if someone wishes to buy products for re-sale, they must make sure that they have enough
money to cover the costs of transporting the goods, rental of the selling venue, registration of their
business, hire of assistants and other such factors, in addition to the cost of the actual items they intend to
buy for re-sale.

PRICES
The costs of production must always be built into a price. For
example, it would be no use selling cakes which cover only the cost
of the ingredients: there are other factors, such as labour, cost of
electricity, expected profit and so on which must be built into the
price.

REVENUE
This refers to money which comes in. For example, the money
collected by the sales-person at the shop is the revenue. It has not
yet had any deductions made to cover the costs of selling the
product.

COST PRICE
The amount which the person pays for the goods he/she intends to sell is called the COST PRICE. Some
discount houses or markets sell goods in bulk for a reduced price. If someone buys from them, he/she can
sell the goods individually at a higher price and be able to make some money. The cost price is usually the

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price of one item: e.g. if one dozen cans of cold drink cost R36.00 at a discount house, then the cost price
of one cold drink is R3.00.

SELLING PRICE
Selling Price. The term 'selling price' is defined as the price at which a good or service is sold by the seller to
the buyer. It is generally expressed as currency units. In other words, it is a market value or agreed
exchange value that enables a buyer to purchase goods or services.
In the above example, the selling price is the price for which the person decides to sell the goods. It is the
price of one item. e.g. if the person decides to sell a can of cold drink for R4.50, then the selling price is
R4.50.

PROFIT
Profit is financial gain. It is the amount earned minus the amount spent. In the case of selling goods:
Formula to calculate profit is:
Profit = selling price minus cost price

In the above example, the profit on one can of cold drink is R4.50
minus R3.00 = R1.50

LOSS
If the operation runs at a loss, it is not successful. This can occur
for a number of reasons: the seller could be asking prices that are
too high, the costs may be too high (e.g. transport,) the items
being sold may lose their value over time (e.g. food can go rotten),
there could be competition, etc.
Formula to calculate Loss is:
Loss = cost price minus selling price

In the above example, if the seller had bought 10 dozen cans of cold drink and 7 dozen cans were stolen
from his bakkie, he will only have 3 dozen left to sell. So the cost price was R360.00 and the selling price
was 36 x R4.50 = R162.00 In this case, he made an overall loss of R360 minus R162 = R198.00

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ESTIMATION AND APPROXIMATION

Estimates
In finance, an estimate is a rough idea of what we expect a cost will be. It is a rough calculation, often using
rounded-off numbers. It is not always possible to predict the exact cost in advance because of variables
such as time taken, therefore we use an estimate.

Approximation
An approximation is an inexact result which is sufficient for the purpose. For example, when budgeting for
an event, one does not need to know the exact cents in advance if the amount is likely to be thousands of
rand.

Difference between Estimation and approximation:
Estimation is making an educated guess and approximation is finding a result precise enough for a
specific purpose.

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CHAPTER 12 - BUDGETS

PERSONAL BUDGETS (SO 1 AC 1.1-4)

What is a Budget ?

A budget is an estimation of revenue and expenses over a specified future period of time and is usually
compiled and re-evaluated on a periodic basis. Budgets can be made for a person, a group of people, a
business, a government, or just about anything else that makes and spends money.

Why is a budget important?

In today's world there are very few people who take the time to create a personal budget. Some people do
not see the value in doing so; other people simply have no desire to restrict their spending habits. With this
in mind, it should surprise no one that the number of personal bankruptcies has reached an all-time high.
People have reached a point in our society where they buy on impulse with no thoughts to the
consequences. In order to reverse this trend people need to become more responsible with their patterns
of spending. One of the best tools to help an individual accomplish this behaviour is the personal budget.

Budgetary Steps

 Create a list of all of your monthly income. If you have any sources of income that are received
annually then simply divide this number by 12. It is important to list all sources including child
support grants, side jobs, etc. This figure will set the cap on your total budget.

 Create a list of all your monthly expenses. If an expense occurs less frequently, simply prorate (split)
it to fit a monthly format. Be sure to include such expenses as housing, food, transportation,
utilities, entertainment, etc.

 It is wise to track your spending for a full month during this stage of budgetary planning. Save your
receipts and each evening write down your expenses for the day. This is the best way to gain an
accurate reflection of actual expenses.

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Example of a simple budget:

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 Determine if your income covers all of your current expenses. If the answer is no, then
expenses need to be reduced.

 It is a basic rule that one should not pay more than 30% of one's income on rent or home loan.

 Adjust expenses. This can be done in a variety of ways. Depending on the amount of the shortfall, it
may be a simple matter of reducing some discretionary spending, such as entertainment, or food
(i.e. the number of times you eat out in a given month.) If the deficit is larger, then it may be a
matter of downsizing your vehicle or your living arrangements. If your income covers all of your
expenses then this is still a good opportunity to trim down some your spending habits. This can help
free up extra money for a variety of reasons ranging from education to a vehicle, your own home,
or the future needs of your children.

 Add new categories if necessary. Three areas that
are often overlooked are 1) debt reduction 2)
retirement savings and 3) emergency savings. An
emergency fund will ensure that there is an
adequate amount available to cover an unforeseen
event should it arise (e.g. the car breaks down).
This will prevent the use of credit which can
quickly break a personal budget.

 Once you have adjusted your budget using the above steps, keep a monthly record such as the one
below. It shows the budgeted amount for each category as well as the actual expenses. Variance is
the difference between the two: actual minus budgeted. If the variance of income is positive, it
means you have earned more than you expected to.

 If the variance of expenses is positive, it means you have spent over your budget. It is useful to
calculate the percentage variance for each category, as in the table below. (Remember: to calculate
the percentage, divide by the amount and multiply by 100% as shown in the first row).

March 2014 Budget Actual Variance Variance as % of budget

PROJE Income R12 500 R 12 12 200 – 12 500 = – 300 (– 300 ÷ 12 500) x 100% = – 2,4%
CTED Wages R11 500 200 0
INCO
R11 500 0

ME Commission R1 000 R700 – 300 – 30%
(SO 1 R12 000 R12 100 +100 + 0,83%
AC 1.2) Expenses

To Rent R4 000 R4 000 0 0
create Groceries R3 000 R3 300 +300 +10%

an Other R5 000 R4 800 – 200 –4%
accur Savings R500 R100 – 400 – 80%
ate

budget, one needs to know what one's income will be over the forthcoming period. This is not always

possible, as wage increases are not always known in advance however, in some cases, companies give a

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small regular increase and this can be built into the budget. Another source of income may be dividends
from shares or investments. These can also be built into the projected income.

NATIONAL BUDGETS (SO 1 AC 1.3-4)

The pie chart below shows how government apportioned the more than R350 billion to be spent in the
2020 tax year.

What is the National Budget?

Every year in February, Government publishes details of how much money it will spent over the next
twelve months. This is the Budget. The Budget explains how much money is going to be spent on public
services such as health, education, and defence. Government also announces in the Budget what taxes it
will impose to pay for these services. Government provides services for everyone, not just a special few.
Because of this, Government is changing the way it provides these services to make them fair and to help
the poor. This means making changes to how Government spends its money.

Tito Mboweni’s budget speech for 2020 is available for reading at the following link:

https://www.gov.za/BudgetSpeech2020

Some Key statements from the 2019 budget speech:

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Economic Context

What are the conditions under which we must implement our plan to win?

In 2020, global economic growth is expected to strengthen to 3.3 per cent. Global inflation remains
contained. Global monetary policy is supportive, and we are benefiting from demand for emerging market
assets.

Asia (excluding Japan) is expected to grow by 5.8 per cent in 2020. The Coronavirus is a source of
uncertainty to this forecast.

With growth of 3.5 per cent, sub-Saharan Africa is forecast to be the second-fastest growing region
in the world.

Against this backdrop we forecast that the South African economy will grow by 0.9 per cent and inflation
will average 4.5 per cent in 2020.

Over the next eighteen months, the economy should get a number of jump starts.

These include, amongst others:
1. The fruits of the reform agenda led by the President
2. Lower inflation
3. The interest rate reduction earlier this year
4. The recent gains in platinum group metals prices
5. The impending change to the electricity regulatory framework
6. The tax proposals we are setting out today

Persistent electricity problems will, however, hold back growth. Over the next three years, we expect
growth to average just over 1 per cent.

Therefore, a stable supply of electricity will be our number one task.

Towards an economic strategy

Last year, the Government embraced the ideas contained in the document Towards an Economic Strategy
for South Africa. This is our plan, and it contains the basic and fundamental pillars of our approach:

1. Strengthening the macroeconomic framework to deliver certainty, transparency and lower borrowing
costs
2. Focusing spending on education, health and social development
3. Modernising “network industries” and restructuring our state-owned enterprises
4. Opening markets to trade with the rest of the continent

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5. Implementing a re-imagined industrial strategy
6. Lowering the cost of doing business
7. Focusing on job-creating sectors, such as agriculture and tourism

Underpinning all of this is the need for an efficient and capable state. We must also leverage the private
sector as far as possible.

Today, we report on our progress. “We are moving forward!”

Prudent Fiscal Policy
Outline of the Budget for 2020/21

A sound macroeconomic framework always lays the foundation for growth. Budgets are complex, but the
numbers are simple. The numbers show that we have work to do.
For 2020/21, revenue is projected to be R1.58 trillion, or 29.2 per cent of GDP.

Expenditure is projected at R1.95 trillion, or 36 per cent of GDP.

This means a consolidated budget deficit of R370.5 billion, or 6.8 per cent of GDP in 2020/21.

2020 Budget Speech Gross national debt is projected to be R3.56 trillion, or 65.6 per cent of GDP by the
end of 2020/21.

Tax Adjustments

To support growth, we propose no major tax increases. Indeed, there is some real personal income tax
relief. This Budget means that a teacher who earns on average R460 000 a year, will see their taxes
reduced by nearly R3 400 a year.

Hard-working tax payers, who earn on average R265 000 a year, will see their income tax reduced by over
R1 500 a year.

Our income tax system is progressive, and the adjustments reflect this. Someone earning R10 000 a month
will pay 10 per cent less in tax. Someone earning R100 000 a month will pay about 1.5 per cent less.

We are also proposing broadening the corporate income tax base. This additional revenue will be used to
reduce the corporate tax rate in the near future to help our businesses grow.

Start-ups will ignite the economy. The tax system supports them in a number of ways, including the
preferential small business tax regime, the VAT registration threshold and the turnover tax. We will review
these to improve their effectiveness while at the same time reducing the scope for fraud and abuse.

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To support the property market, the threshold for transfer duties is adjusted. Property costing R1 million
or less will no longer be subject to transfer duty.

There will be a renewed focus on illicit and criminal activity, including non-compliance by some
religious public benefit organisations. Religious bodies must operate within the strict boundaries of the law
if they are to enjoy tax exempt status. The annual tax free savings account contribution limit rises to R36
000.

We have increased excise duties to keep pace with inflation. From today:

 A 340ml can of beer or cider will cost only an extra 8c
 A 750ml bottle of wine will cost an extra 14c
 A 750ml bottle of sparkling wine an extra 61c
 A bottle of 750 ml spirits, including whisky, gin or vodka, will rise by R2.89
 A packet of 20 cigarettes will be an extra 74c
 A 25 gram of piped tobacco will cost 40c more
 A 23 gram cigar will cost an extra R6.73

I am again happy to report that there is no increase in the price of sorghum beer.

In line with Department of Health policy, we will start taxing heated tobacco products, for example hubbly
bubbly. The rate will be set at 75 per cent of the rate of cigarettes. Electronic cigarettes, or so-called vapes,
will be taxed from
2021.

To adjust for inflation, the fuel levy goes up by 25 cents per litre, of which 16 cents is for the general fuel
levy and 9 cents is for the Road Accident Fund levy.

Despite this increase, the liabilities of the RAF are forecast to exceed R600 billion by 2022/23. We
need to take urgent steps to reduce this risk to the fiscus and bring about a more equitable way of sharing
these costs. One option is to introduce compulsory third-party insurance.

The carbon tax and other measures will help green the economy, and will bring in R1.75 billion over the
next few months. This will be complemented by more focussed spending on climate change mitigation. We
remain extremely concerned about plastic bags throughout the length and breadth of our country. In this
regard, we have increased the plastic bag levy to 25 cents.

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Youth Employment

8.2 million young people between the ages of 15 and 34 are not in education, employment or training.
Government is committed to helping them. Raising skills and improving the matching of young people and
jobs is an important focus of the Presidential Youth Employment Intervention.

To date, Jobs Fund projects have created more than 175 000 permanent jobs, and helped 21 000 people
into internships and created 59 900 short term jobs. Of these, 65 per cent went to youth.

As the President announced, we will reprioritise resources to raise spending on this critical area. We will
start work immediately! I will provide more details in the 2020 Medium Term Budget Policy Statement.

We intend to make this intervention a resounding success.
4.1.3 Social grants

We are a caring society. We are a caring government. More than 18 million people receive a grant, which is
a lifeline for many.

Grants reduce inequality and protect the most vulnerable in society. I am happy to announce that grants
are adjusted as follows:

1. R80 increase for the old age, disability and care dependency grants to R1860 per month
2. R80 increase in the war veterans grant to R1880
3. R40 increase for the foster care grant to R1040 per month
4. The child support grant will increase by R20 to R445 per month

Changing the way we provide social grants has generated about R1 billion per annum in efficiency savings,
which will be partly used to raise the daily subsidy per child.

REGIONAL BUDGETS (SO 1 AC 1.3-4)

Departments which are run by the regional government (such as education, health, transport for the
provinces) receive allocations from central government according to specific needs and the availability of
funds. Municipal councils also have regional budgets which take care of the localized needs within a city,
such as refuse collection, water and electricity supply, etc. These are financed by rates and taxes paid by
land-owners.

Extra reading on the Budget review - National treasury RSA
http://www.treasury.gov.za/documents/National%20Budget/2020/review/FullBR.pdf

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INTEREST

Interest is the cost of borrowing money. If someone wants to borrow money from a bank or credit
institution, they will have to pay interest on it. It is usually lent at a fixed interest rate.

e.g. if a bank charges 13% interest per annum on a particular loan it means that 13% of the loan (per year)
will have to be paid back to the bank by the borrower, in addition to the original amount. The longer one
takes to pay back, the more interest will be added on.

If one is investing money in a bank, then the bank will pay the investor interest. Here, in effect, the bank is
borrowing money from the investor, so if the bank offers 11% per annum interest on an investment, it
means that the bank will pay the investor 11% of the amount invested (per year.)

There are two types of interest: Simple Interest and Compound Interest.

SIMPLE INTEREST

Simple interest is interest calculated on the original amount or Present Value. If someone invests R300 at
12% simple interest for 5 years, then the interest earned after one year is (12/100) x 300 = R36. It will be
the same each year, so for 5 years the amount will be multiplied by 5.
The formula for simple interest is:
Simple Interest = P x i X n
Where P = present value
i = interest rate / 100
n= number of years

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Example of Simple Interest :

We invest R4500 at 7% interest for 3 years at simple interest. How much interest is earned?

P = 4500.

i = 7/100

n=3

Substitute into the above formula:

Simple Interest = 4500 x 0,07 x 3

= R945

COMPOUND INTEREST WITH ANNUAL COMPOUNDING

Compound Interest increases with each time period, because interest is earned not only on the original
amount but also on the interest. Thus interest earns interest.

The formula for calculating Compound Interest is:

F=P(1+i) n
Where F = Future value
P = present value
i = interest rate / 100
n= number of years

Example:

We invest R4500 at 6,5% interest for 3 years at compound interest. What is the final amount?

P = 4500.

i = 6,5/100

n=3

Substitute into the above formula:

F = 4500(1+0,065)3

= R5435-77

NOTE: if we are asked to calculate the interest earned, it must be obtained by subtracting the principal
from the future value. In this example, interest = 5435-77 - 4500 = R935-77.

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COMPOUND INTEREST WITH FREQUENT COMPOUNDING

Very often interest is compounded more often than just once a year. This means that n, the number of
times compounded, will be higher, but it also means that “i”, the interest rate paid over one compounding
period, will be less.
The rule is: To find n, multiply by the number of periods in one year (e.g. for quarterly compounding, we
would use number of years x 4)
To find i, divide the rate per annum by the number of periods in one year (for quarterly compounding,
divide by 4)

Example:

Find the final amount if we invest R4500 at 6,5% interest for 3 years at compound interest, COMPOUNDED
MONTHLY. We use the compound interest formula: Note the change in i and n:

F=P(1+i) n

P (i.e. the present value) = 4500.

i (i.e. interest rate as a percentage) = 0,065/12 since it is compounded monthly

n(i.e. number of time periods) = 3 x 12, since it is compounded every month

Substitute into the above formula:

F (i.e. future value) = 4500(1+0,065/12)^3x12 or 4500 1+ 0,065 3 × 12
12

= R5466-02

You will have to use a scientific calculator to find (1,0054166)^36, as you
cannot multiply the bracket by itself 36 times manually! - Use a scientific
calculator on the computer by selecting ‘view’ on a standard computer
calculator, then select ‘scientific.’

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NOMINAL AND EFFECTIVE INTEREST

A nominal interest rate is the quoted rate, the value of in the compound interest formula.

The effective interest rate is the actual rate which is applied when compounding is done more frequently
than once a year. It is the rate which has been adjusted by including the number of compounding periods.
e.g. if an effective rate is 7% p.a., it means that a nominal rate of some other amount has been adjusted to
take into account the fact that it is applied every month or every quarter, say, and it is applied the same as
a rate of 7% once a year.

Example: Find the effective interest rate if the nominal rate is 7% per annum, and compounding is done
monthly.

Let i = effective rate for 1 year.
Then P(1 + i)1 = P(1 + 0,07/12)12
So 1 + i = (1,0058333...) 12

1 + i = 1,0723

So i = 0,0723 thus i = 7,23%

COMMISSION

Commission is a means of rewarding people for work done, particularly
in selling goods. A certain percentage of the cost of the sale is given to
the person who completes the sale. This can range from estate agents
to people selling informally.
There are three basic models, as seen in IT companies, for example: a
basic salary only, commission only and a combination of the two.

Trevor Green, MD of Electron Highway, IT service and solution providers,
pays no commission to sales staff, preferring to give them a flat salary.
His rationale is that his salespeople are selling the Electron Highway
brand and their success is thus difficult to determine. They are motivated,
rather, by means of various incentive schemes.

At Advanced Chips and Upgrades, salespeople have a choice: a basic salary plus 10% commission or no
basic and 40% commission. The second option, with its unusually high percentage, says Halperin,
encourages individual responsibility for sales and is a powerful motivator.

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The third model involves a combination of basic and commission. Rather than have a set percentage, many
companies prefer to establish, together with the salesperson, a desired personal annual income. That
figure is then split into a percentage basic salary and a percentage commission. If the company works
according to, for example, a 60-40 split, 60% of the figure is paid as a basic and remains constant, while
40% depends on commission and is thus variable. This method helps to establish earning potential and to
set targets. It also determines the percentage commission that the salesperson makes on each deal.

Companies, such as Xerox Channels Group (SA) (manufacturers and suppliers of the Xerox range of copiers,
printers and multifunctional devices) and The Internet Solution (an Internet service provider), both of
which work according to a 60-40 split, agree that this is effective.

INVESTMENTS

If money is kept in a safe place without being invested, it cannot earn interest. To invest money is to put it
in a financial scheme which enables it to earn interest, with the expectation of receiving a profit. There is
always an element of risk when investing money. Some schemes are safer than others: these usually offer
less interest.

BANK SAVINGS ACCOUNTS

Most people have some form of bank savings account.
Because of their simplicity and familiarity, a large portion of
the public’s savings is tied up in bank investments of one
type or another.

There are different types of bank investments; two
examples are given below.

CALL ACCOUNTS

Almost every bank offers some type of call account. With these accounts the investor’s money is on call,
which means it can be withdrawn at very short notice (usually not more than 24 hours). Interest rates
fluctuate while the investor’s money is invested.

FIXED DEPOSITS

With fixed deposits, the investor’s money is tied up for a pre-specified term, usually ranging from one
month to five years. Although interest rates are usually fixed at a pre-determined rate for the full term,
some banks are now offering linked deposits, where the interest rate can increase during the term, but has
a guaranteed minimum. Interest rates on these deposits are generally higher than those on call deposits
and the longer the term the higher the interest rate.

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STOKVELS

A stokvel is a savings scheme by group of people who save a regular
amount each month, say R100.00. Each person then gets a chance to
have the month’s collected cash. Minimum investments into stokvel
investments are very low – in fact, they are at the discretion of the
people forming the stokvel. However, an important element of the
stokvel model is trust. Members have to be fairly confident that the
other members will keep up their monthly payments, especially after
having received the cash.

South Africa has more than half a million stokvels. A nationwide survey by research company African
Response found there are 11.4-million stokvel members in South Africa. Stokvels are defined as group
savings schemes providing for mutual financial assistance, as well as social and entertainment needs.

The most popular type of stokvel is for savings with 47% of respondents making use of this. It is followed
by a burial society stokvel, of which 41% of respondents are members. Grocery stokvels, which buy in bulk
from various retailers at the end of the year, make up 20% of stokvel membership. Investment stokvels
make up 5% of the total stokvel market.

The survey found that the average number of members per stokvel was 27.

INFLATION

Inflation is the increase in the cost of goods and services over time. If the inflation rate is 10%, it means
that an item which used to cost R60 will cost R60 + 10% of 60, i.e. R66 in a year's time, and a year after that
the same item will cost R66 + 10% of 66, i.e. R72-60. Because inflation builds up on itself in this way,
calculations must be done using the compound interest formula, with “i” = inflation rate.

Example: A 2 litre bottle of Coke costs R 11 now. If annual inflation is 9%, what will a 2 litre bottle of Coke
cost in 3 years' time?

Use the formula

FV = P(1+i) n

Where Fv = Future value

P = present value

i = inflation rate / 100

n= number of years

Fv = 11(1+0,09)3

= R14-25

The cost will be R14-25 in 3 years' time.

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APPRECIATION AND DEPRECIATION

APPRECIATION

The value of certain items increases with time. Such items are often bought as an investment. One such
example is property, another is art-works by great masters. The value of appreciation is calculated using
the normal compound interest formula, with i = appreciation rate.
Example: An art work currently worth R4 500 appreciates at a rate of at 6,5%. How much will it be worth
after 4 years?
Use the formula
Fv= P(1+i) n
Where Fv = Future value
P = present value
i = rate of appreciation / 100
n= number of years
Fv = 4500(1+0,065)4

= R 5789-10

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DEPRECIATION

The value of many items decreases with time. One example is motor vehicles. A car which costs R120000
brand new will only be worth a fraction of that in a year’s time. The percentage by which its value reduces
is called the depreciation rate. The formula used is very like the normal compound interest formula, except
that there is a minus instead of a plus.
Use the formula
Fv = P(1-i) n
where Fv = Future value
P = present value
i = rate of depreciation / 100
n= number of years

Example: A car costs R120 000 new. What will it be worth after 4 years, if the depreciation rate is 11%?

Fv = 120000(1- 0,11)4
= R75 290-69

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THE NATIONAL ECONOMY

BUDGET CALCULATIONS

In 2013, the national budget was R1,06 trillion. Of this, the largest amount was allocated to education:
R232,5 billion. What percentage is that?
(232 500 000 000 / 1 060 000 000 000) x 100% = 22%

TAX

Tax is money that must be paid to the government (SARS is
the South African Revenue Service). It is used to pay for
budgeted categories such as education, health, defence,
infrastructure, etc.

Examples of amounts an individual may receive, and from
which the taxable income is determined, include:

 Remuneration (income from employment), such as, salaries, wages, bonuses, overtime pay, taxable
(fringe) benefits, allowances and certain lump sum benefits

 Profits from a business or trade
 Investment income, such as interest and foreign dividends
 Annuities
 Pension income

Who is it for?

You are liable to pay income tax if you earn more than R78 150.00 in the 2020 year of assessment, and are
younger than 65 years of age. If you are 65 years of age or older, the tax threshold (i.e. the amount above
which income tax becomes payable) increases to R121 000. For taxpayers aged 75 years and older, this
threshold is R135 300.

Step one: You must register for income tax

If you earn a taxable income which is above the tax threshold (see above), you must register as a taxpayer
with SARS. To register for income tax, you must go to the nearest SARS branch where the staff will register
you on their system. You must take along the following supporting documents.

- Proof of ID

- Proof of address (the list of what is an approved proof of address is on the SARS website)

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- Proof of bank details

You must register for income tax at SARS within 60 days of becoming liable for tax. Once you have been
registered and given a tax number, then you may register for e-filing, which will allow you to submit your
tax returns on-line.

Step two: You must submit a return.

If you are registered for income tax, you will be required to submit an annual income tax return to SARS.
These days e-filing is the most widely used means of submitting a tax return.

Tax Calculations

PAYE, or Pay-As-You-Earn, is tax which is deducted each month from employees’ salaries. The total annual
tax payable is calculated from tax tables and divided by 12 to find the monthly deduction. Each financial
year, SARS provides new tax tables. The table below is for the tax year from March 2014 to February 2017.

Rebates are amounts that are subtracted from the total tax at the end of the calculation. Older people
have additional rebates. There is a tax threshold according to age. People earning below the threshold are

not liable for tax.
Example. Calculate the monthly tax deduction for the following:
a. A person under the age of 65 who earns R8 000 per month.
Annual income: 12 x 8 000 = R96 000.
This is less than R195 850 so raw tax = 18% of 96 000 = R17 280
Less primary rebate: 17 280 – 14 067 = R3 213

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Monthly tax payable = 43213 ÷ 12 = R267.75

b. A person aged 70 who earns R20 000 per month.

Annual income: 12 x 20 000 = R240 000.
This is between 195 851 and 305 850, so raw tax = 35 253 + 26% of (240 000 – 195 851) = R 44149
Less primary and secondary rebates: R44149 – (14 067 + 7 713) = R 22 369
Monthly tax payable = R22 369 ÷ 12 = R1 864.08

PRODUCTIVITY

Productivity is the amount of output produced relative to the amount of resources (time and money) that
go into the production.
Productivity in economics refers to measures of output from production processes, per unit of input.
Labour productivity, for example, is typically measured as a ratio of output per labour-hour, an input.
Productivity may be thought of as a measure of the technical or efficiency of production. As such
quantitative measures of input, and sometimes output, are emphasized.
Economic activity can be identified with production and consumption. Production is a process of combining
various immaterial and material inputs of production so as to produce tools for consumption.
Economic growth is a production increase achieved by an economic community. It is usually expressed as
an annual growth percentage depicting (real) growth of the national product. Economic growth is created
by two factors: an increase in production input and an increase in productivity.

The formula of total productivity is normally written as follows:

Total productivity = Output quantity / Input quantity

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CONCLUSION TO BUSINESS MATHEMATICS

By completing this module you will now have an insightful use of Basic Business Mathematics and the
ability to become a self-managing person in all aspects of present and future life including in the personal
daily operations in become a participating citizen within society .

In completion of this module you will be expected to complete a series off assessments that will solidify
your understanding as well as your competency in what has been covered within the module you will also
be expected to complete a workplace assessment.

You will be tested on the following:
 Measure, estimate, and calculate physical quantities in practical situations relevant to the adult in
life or the workplace.
 Explore, describe and represent, interpret and justify geometrical relationships and conjectures to
solve problems in two and three dimensional geometrical situations
 Convert numbers between the decimal number system and binary number system
 Work with numbers in different ways to express size/magnitude
 Demonstrate the effect of error in calculations
 Pose questions, collect and organise data
 Represent and interpret data using various techniques to investigate real life and work problems
 Use random events to explore and apply probability concepts in simple life and work related
situations
 Use mathematics to plan and control personal and/or household budgets and income and
expenditure
 Use simple and compound interest to make sense of and define a variety of situations including
investments, stokvels, inflation, appreciation and depreciation
 Investigate various aspects of financial transactions including costs, prices, revenue, cost price,
selling price, loss and profit

Therefore it is safe to say that any staff member within your organisation from warehousing staff, drivers,
entry level clerks all the way up to Directors/Owners of any business organisation need to have a
fundamental understanding of Mathematics in order to ensure that your day to day running’s of your
organisation run at a profit and is a success.

A simple mistake can cost you or an organisation millions.

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