Business Mathematics | Module 2

59365 – Freight Forwarding & Customs Compliance

Business Mathematics

Compiled by:

Global Maritime Legal Solutions

A World Class Knowledge Centre
This Module is suitable for training towards the FIATA Diploma, TETA, QCTO Qualifications and similar such
courses.

Learner Guide compiled by:

Mark Goodger

Edited date – May 2020.
# Page layout by Charlotte Francis
Copyright © 2020 edition. Date of revision: May 2020.
Global Maritime Legal Solutions (Pty) Ltd.

No part of this book may be reproduced in any form or by any means without written
permission from the publisher.

WARNING AGAINST PLAGIARISM

ASSIGNMENTS ARE INDIVIDUAL TASKS AND NOT GROUP ACTIVITIES. (UNLESS
EXPLICITLY INDICATED AS GROUP ACTIVITIES)
Copying of text from other learners or from other sources (for instance the study guide,
prescribed material or directly from the internet) is not allowed – only brief quotations are
allowed and then only if indicated as such.
You should reformulate existing text and use your own words to explain what you have read.
It is not acceptable to retype existing text and just acknowledge the source in a footnote –
you should be able to relate the idea or concept, without repeating the original author to
the letter.
The aim of the assignments is not the reproduction of existing material, but to ascertain
whether you have the ability to integrate existing texts, add your own interpretation and/or
critique of the texts and offer a creative solution to existing problems.
Be warned: students who submit copied text will obtain a mark of zero for the assignment
and disciplinary steps may be taken by the Faculty and/or University. It is also unacceptable
to do somebody else’s work, to lend your work to them or to make your work available to
them to copy – be careful and do not make your work available to anyone!

2

Learner Guide – Business Mathematics

Credits 16

This Integrated Module will compose of the following Unit Standards:
Unit Standard 9013 Level 3 Credits 4 - Describe, apply, analyse and calculate shape and motion in 2-and
3-dimensional space in different contexts
Unit Standard 9010 Level 3 Credits 2 - Demonstrate an understanding of the use of different number
bases and measurement units and an awareness of error in the context
Unit Standard 9012 Level 3 Credits 5 - Investigating life and work related problems using data and
probabilities
Unit Standard 7456 Level 3 Credits 5 - Use mathematics to investigate and monitor the financial aspects
of personal, business and national issues

Overall Objectives of this Module

The purpose of this module is designed provide credits towards the mathematical literacy requirements of
the NQF at level 3. The essential purposes of the mathematical literacy requirements are that, as the learner
progresses with confidence through the levels, the learner will grow in:

 An insightful use of mathematics in the management of the needs of everyday living to become a self-
managing person

 An understanding of mathematical applications that provides insight into the learner`s present and
future occupational experiences and so develop into a contributing worker.

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 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

Unit Standard 9013 Work with numbers in different ways to express size/magnitude

 Demonstrate the effect of error in calculations

Level 3 Credits 4 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

3

TABLE OF CONTENTS

Introduction to Business Mathematics ………………………………………………………………………….... 7

CHAPTER 1 – SCALES..…….………………………………………………….…………………….………………………. 9
1.1 – Measuring tools and scales
1.2 – Reading scales

CHAPTER 2 – UNITS OF MEASUREMENT AND CONVERSIONS…………………………..……………… 13

2.1 – Length and distance
2.2 – Mass
2.3 – Conversion according to size
2.4 – Conversion from SI to Imperial System
2.5 – Volume
2.6 – Temperature
2.7 – Speed

CHAPTER 3 – ESTIMATION …….…………………………………………………..…………………………………….. 16
3.1 – Rounding Off

CHAPTER 4 – MAPS AND SCALE PLANS …………………………………………………………………………..... 18
4.1 – Longitude and Latitude

CHAPTER 5 – SHAPES ……………………………………………………………………………………………………….. 20
5.1 – Pythagoras Theorem
5.2 – Calculation of Perimeter and Area
5.2.1 – Rectangle
5.2.2 – Parallelogram
5.2.3 – Triangle
5.2.4 – Trapezium
5.2.5 – Circle
5.3 – Volume and Surface area of Solids
5.4 – Volume of Prisms including Cylinders
5.4.1 – Net of Prisms
5.4.1 – Surface Area of Prisms

CHAPTER 6 – TRANSFORMATIONS, SYMMETRY AND TESSELLATIONS …………………….…….... 32
6.1 – Translation
6.2 – Rotation
6.3 – Reflection
6.4 – Symmetry
6.5 – Tessellations

CHAPTER 7 – BINARY NUMBERS ……………………………………………………………………………….…… 37
7.1 – Why use Binary numbers

4

7.2 – Addition and Subtraction of Binary numbers

CHAPTER 8 – SCIENTIFIC NOTATION ………………………………………………………………………………. 40
8.1 – Writing a number in scientific notation
8.2 – Converting from scientific notation to ordinary numbers
CHAPTER 9 – RATIONAL AND IRRATIONAL NUMBERS ………………………………………………….…. 42
9.1 – Rational numbers explained
9.2 – Irrational numbers explained
9.3 – Working with irrational numbers
9.4 – Conversion of recurring decimals to fractions

CHAPTER 10 – DATA COLLECTION …………………………………………………………………………….………45
10.1 – Dichotomous
10.2 – Likert
10.3 – Discrete and continuous variables
10.4 – Methods of representing data
10.4.1 – Tables
10.4.2 – Pie Charts
10.4.3 – Bar Graphs
10.4.4 – Stem and Leaf
10.5 – Measures of central tendency and spread
10.5.1 – Mean
10.5.2 – Median
10.5.3 – Mode
10.5.4 – Range
10.5.5 – Quartiles
10.5.6 – Inter- quartile range
10.6 – Scatter plots and lines best fit
10.7 – Probability
10.7.1 – Probability calculations
10.7.2 – Tree Diagrams

CHAPTER 11 – FINANCIAL TRANSACTIONS ………………………………………………………….…………… 56
11.1.1 – Income and expenditure
11.1.2 – Costs
11.1.3 – Prices
11.1.4 – Revenue
11.1.5 – Cost Price
11.1.6 – Selling Price
11.1.7 – Profit
11.1.8 – Loss
11.2 – Estimation and approximation
11.2.1 – Estimation
11.2.2 – Approximation
11.2.3 – Difference between estimation and approximation

CHAPTER 12 – Budgets …………………………………………………………………………………………………… 59

5

12.1 - Personal budgets
12.1.1 – What is a budget
12.1.2 – Why is a Budget important
12.1.3 – Budgetary steps
12.2 – Projected income
12.3 – National Budgets
12.3.1 – What is the National Budget
12.4 – Regional Budgets
CHAPTER 13 – Interest ……………………………………………………………………………………………….…… 67
13.1 – Simple Interest
13.2 – Compound Interest with Annual Compounding
13.3 – Compound Interest with frequent Compounding
13.4 Nominal and Effective interest
13.5 – Commission
13.6 – Investments
13.6.1 – Call Accounts
13.6.2 – Fixed Deposits
13.6.3 – Stokvels
13.6.4 – Inflation
13.7 – Appreciation and Depreciation
13.8.1 The National Economy
13.8.2 – Budget Calculations
13.8.3 – Tax and Tax Calculations
13.8.4 – Productivity
CONCLUSION OF BUSINESS MATHEMATICS ………………………………………………………………..…… 78
ALIGMENT MATRIX …………………………………………………………………………………………………….…….. 79

6

INTRODUCTION TO BUSINESS MATHEMATICS

Mathematics plays an integral part in the successful management of any business. In this module you
will learn about and understand the vital importance of understanding, and being able to carry out basic
mathematics, within the workplace and the day-to-day running’s of a business.

You will be surprised that, not only in a business environment, but also in your personal capacity, you
are unknowingly, yet constantly solving and evaluating mathematical problems. How, you may ask?
When you left for work this morning you already estimated the time you needed to leave, so that you
could arrive at your place of work on time, whether you left on foot, by public transport, or travelled in
a vehicle. Right there, you used estimation, distance, time and speed.

When you go to the shops and purchase goods, you are calculating costs as you go, so that you do not
have that embarrassing moment at the till where you swiped for more items than what you have money
for – hopefully you have gone as far as to put a budget in place, so that you make your money last, as
well as save some of it for unforeseen circumstances.

When you sit on a chair, or on any object, you have automatically calculated if the object will hold the
weight of your body, and you also sit down you gradually decline, until you feel the chair below you.
Your eyes have automatically seen the angle and made and automatic calculation so that you do not
just flop down .

These are just a few simple examples of how you are automatically calculating in the everyday world all
the time.

In this module we will provide you with the tools to understand and articulate, to accurately solve
problems in the work place, as well as in your personal environment.

In Global Trade almost every factor encompasses mathematics from salaries, to invoicing, dealing with
percentage mark-ups, profit and loss, areas of warehouses, in order to see how much stock can be kept
on your premises, to packing a container ensuring that weight distribution gets taken into account so
that the goods do not get damaged, estimated times of arrival, financial budgets, and forecasts, and
conversions of currencies and temperature to name a few.

Business ownership requires more than skill in creating a product, or the talent at providing a service.
Overseeing the finances of your company is key to survival and success. Understanding basic business
maths is necessary for profitable operations and accurate record keeping. Knowing how to add, subtract,
multiply, divide, round and use percentages and fractions is the minimum you will need to price your
product and meet your budget. If math is not your strength, partner with someone who can take over
that role or hire a trusted employee to help your operation stay in the black and grow responsibly.

7

Calculate Production Costs
Before you formally establish your business, you must estimate the cost to manufacture or acquire your
product or perform your service. Adding all expenses associated with making or buying items helps you
realize if you can be competitive with other companies and profitable enough to sustain your business
and make a reasonable income. In addition to the standard costs of production, such as materials and
machinery, add accompanying expenses, such as shipping, labour, interest on debt, storage and
marketing. The basis to your business plan is an accurate representation of how much you will spend on
each item.

Determine Product Pricing

To ensure you can operate your business and produce enough cash flow to invest into your enterprise,
you must charge enough for your product to be profitable. Mark-up is the difference between your
merchandise cost and the selling price, giving you gross profit. If your operations require a large mark-
up, such as 70 percent, you may not be competitive in your industry if other companies sell the same
items for less. Once you have determined your mark-up, one way to calculate the retail price is to divide
using percentage or decimals.

Measure Business Profits

If you want to determine the net profit for a certain time period, you will need to subtract returns, costs
to produce an item and operating expenses from your total amount of sales, or gross revenue, during
that time. Discounts on products, depreciation on equipment and taxes also must be calculated and
subtracted from revenue. To arrive at your net profit, add any interest you earned from credit extended
to customers, which is reflected as a percent of the amount each person owes. Your net profit helps you
understand if you are charging enough for your product and selling an adequate volume to continue to
operate your business or even expand.

Analyse Financial Health

To analyse the overall financial health of your business, you will need to project revenue and expenses
for the future. It is important to understand the impact to your accounting records when you change a
number to reflect an increase or decrease in future sales. Estimating how much an employee affects
revenue will indicate if you can afford to add to your staff and if the profits realized will be worth the
expense. If a competitor starts selling a cheaper product, you may need to calculate the number by
which your volume must increase if you reduce prices.

You may need to know if you can afford to expand your operations to improve sales. Using basic
business math to understand how these types of actions impact your overall finances is imperative
before taking your business to the next level.

Let us take a more insightful look at what is mentioned above to increase your understanding.

8

CHAPTER 1 - SCALES AND CONVERSIONS

MEASURING TOOLS AND SCALES

(SO 1 AC 1.1 – 1.5)

Quantity Tool Unit Example

Temperature Thermometer 0Celsius

Length Ruler, mm, cm, m

Distance measuring
Speed
tape/ Vernier

callipers/

micrometre

screw

Odometer km

Speedometer Km per hour or Speedometer
metres per second

Odometer

Capacity Measuring ml, litres
cylinder

9

Mass Scale g, kg

Time Clock/ watch sec, min, hr

Reading scales
An important part of practical measurement is being able to read a scale properly.
There are ten divisions between each whole number in the diagram, so each small division is one tenth
of a cm (i.e. 0,1 cm.)
Example
To measure the length AB on the ruler below: A is at 0; B lies halfway between 0,6 and 0,7. Length AB =
0,65 cm.
For BC: First find the position of C. It lies between 1,5 and 1,6, but closer to 1,5. Estimate position of C:
1,53.
BC = 1,53 – 0,65 = 0,88 cm by subtraction

Example
What is the position of A on the scale below?

Solution: There are 8 spaces between 10,25 and 11,25. Thus each space represents one eighth of a
whole (0,125), and two spaces represent 0,25. Thus the position of A is 10,875.

10

CHAPTER 2 - UNITS OF MEASUREMENT

(SO 1 AC1.1 – 1.6)
The most commonly used system of units is the
Standard International system (S.I.) which uses a
decimal system of calculation. The prefixes milli-,
centi-, kilo and others are used to denote parts of
a whole.

 kilo means one thousand.
 milli means one thousandth,
 centi means one hundredth

This applies for all measurements: e.g. length (centimetres, millimetres, kilometres), mass (grams,
milligrams and kilograms) and liquid volume (litres, millilitres and kilolitres.)

1 mm = 0,001 m and 1000 mm = 1 m
1 mm = 0,1 cm and 10 mm = 1 cm.
1 cm = 0,01 m and 100 cm = 1 m
1 km = 1000 m
1 metric ton = 1 000 kg

Example:
Convert 3,06 g to mg.
Answer: Since there are 1000 mg in 1 g, we multiply by 1000.
Answer = 3 060 mg.
Area is given in square units. When converting area be careful to convert lengths first, before
multiplying them.
Example:
Find the area of a rectangle in cm2 if the length is 24 mm and breadth 30 mm.
Answer:
L= 2,4 cm and breadth = 3 cm
Area = 2,4 x 3 = 7,2cm2 (also written cm^2)

11

Example
1. Convert the following:
a. 35,6 kg to g
b. 0,92 g to mg
Answers:
a. 35 600 g
b. 920 mg
2. What is wrong with this calculation? A rectangle has length 3,8 m and breadth 3 m. So its area in cm2
is 3,8 x 3 x 100 = 1 140 cm2
Answer:
Length= 380 cm and breadth = 300 cm
area = 380 x 300 = 114 000 cm2

12

CONVERSION FROM IMPERIAL TO S.I. UNITS

Imperial measures are used in certain countries such as Britain (UK) and USA. We can convert from
Imperial to SI units using the conversions in the table below.
Length and distance:

Imperial 1 inch (or 1”) 1 foot (or 1’) = 12 inches 1 yard = 3 feet 1 mile

SI 2,54 cm 30,48 cm 91,44 cm 1,6 km

Example

1. Convert 4 foot 7 inches into metres.

4 ft = 4 x 30,48 cm = 121,92 cm

7 in = 7 x 2,54 cm = 17,78 cm

Answer = 121,92 + 17,78 = 139,7 cm = 1,40 metres.

2. If 1 inch = 2,54 cm, 1 foot = 30,48 cm, 1 yard = 91,44 cm:

What are the length, breadth and height (in centimetres,) of a box whose dimensions in the Imperial
system are: length = 2 feet; breadth = 11 inches; height = 2 yards?

Answer:

Length = 2 x 30,48 = 60,96 cm

breadth = 11 x 2,54 = 27,94 cm

height = 2 x 91,44 = 182,88 cm

Mass:

Imperial 1 pound (or lb) 1 ounce (or 1oz) 1 imperial ton or long ton

SI 454 g 28,6 g 1,1016 metric tons

Example:
4. A container has a mass of 6,2 long tons. Give this in kilograms.
Answer: 6,2 x 1,1016 x 1000 = 6 829,92 kg

a. Convert the mass shown on the scale to pounds.
b. Round up the pounds reading to the next highest half-pound.
Answer: a. 14.317 pounds
b. 14,5 pounds

13

CONVERSION ACCORDING TO SIZE

Different prefixes are used in the S.I. (Standard International) System to denote size. e.g. 'kilo-' always
refers to 10^3, or 1000, so 1 kilogram is 1000 grams; 1 kilometre is 1000 metres, and so on. Giga is a
very large quantity, being 1 000 000 000 times (think of the word Giant), whereas pico is extremely
small, being 0,000 000 000 0001 times. See table below:

Prefix Multiplied by
giga 10^9
mega 10^6
kilo 10^3
milli 10^-3
micro 10^-6
nano 10^-9
pico 10^-12

CONVERSION FROM SI TO IMPERIAL SYSTEM

Although most countries use the S.I. system, some countries still use the Imperial system, which
originated in the United Kingdom. If consignments go to one of these countries, it is often necessary to
convert S.I. measurements into Imperial. Multiply by the conversion factor given below.

Volume:

1 litre = 2,1 pints

1 litre = 0,264 US gallons

Temperature:
Degrees Fahrenheit are used in the Imperial System, degrees Celsius in the S.I.
The formula to convert from degrees Fahrenheit (F) to degrees Celsius (C) is:
C = 5(F – 32)/9
Example:
Convert 82o F into degrees Celsius.
Substitute F = 82 in the formula. Calculate the bracket first.
C = 5 (82 - 32) / 9 = 5 x 50 /9 = 27,80C
Speed:

14

We can use the diagram shown as a reminder that
Speed = distance / time,
Time = distance / speed, and
Distance = speed x time

Examples:
1. A vehicle travels 135 miles in 2 hours.
a. Calculate its speed in km/hour.
Answer: (135 x 1,6)/2 = 108 km/h
b. How many km can it cover in 5 hours, travelling at this speed?
Answer: Distance = 108 x 5 = 540 km.
2. A vehicle has an average speed of 70 miles per hour.
a. How long would it take to travel 253 km? Round off to the nearest quarter of an hour.
b. What is the estimated time of arrival if the vehicle left at 11h55?
Answer: a. 70 mph = 70 x 1,6 km/h = 112 km/h
Time = 253/ 112 = 2,25 hours or 2 ¼ hours.
b. 11.55 + 2.15 = 14h10

15

CHAPTER 3 - ESTIMATION

(SO 1 AC2 )

It is useful to know some common measurements. By comparing these with the size of the thing to be
estimated, we can arrive at a reasonable estimate.

Object Length of ruler Capacity of cup Teaspoon Arm span of adult

Measurement 30 cm 250 ml 5 ml 1m

Freight forwarding often requires estimates of the length of time to complete a job. We use ratio to do
this type of calculation. e.g. If it takes 3 hours to load 2 containers, how long will it take to load 5
containers?

First work out how long it takes to load 1 container: 3/2 = 1,5 hours.

Then multiply by the number of containers: 1,5 x 5 = 7,5 hours.

Example

If it takes 3 cranes to load 20 containers in a certain period of time, how many containers can 4 cranes
load in the same period of time?

Answer

First find how many containers one crane can load: 20/3 = 6,6 containers. Then multiply this by 4 to find
how many containers can be loaded: 4 x 6,6 = 26,6. That means 26 complete containers can be loaded
in that time.

Alternatively we can cross multiply and divide. Where we solve for x being the unknown value

20 = x

34

Step one: cross multiply the known factors: 20 x 4 = 80

Then divide by the last value to solve for x: 80 /3 = 26.6

Rounding off

 For rounding off to the nearest ten, look at the units digit. If it is 5 or more, the tens digit will
increase by one.



Example:
round off 36 to the nearest ten: it will be 40 because 6 is more than 5.

 For rounding off to the nearest hundred, look at the tens digit. If it is 5 or more, the tens digit
will increase by one.

16

Example:
round off 346 to the nearest hundred: it will be 300 because 4 is less than 5.

 For rounding off to the nearest thousand, look at the units digit. If it is 5 or more, the hundreds
digit will increase by one.

Example:
round off 3 629 to the nearest thousand: it will be 4 000 because 6 is more than 5.

 For rounding off to two decimal places, look at the third digit after the decimal point. If it is 5 or
more, the hundreds digit will increase by one.

Example:
round off 3,629 to two decimal places: it will be 3,63 because 9 is more than 5.
 Rounding up to the next half-measure: Here we look for the next half-number (multiple of 0,5)

Example:
a. round up 128,3 kg to the next highest half-kg. The answer is 128,5 kg
b. round up 128,8 kg to the next highest half-kg. The answer is 129,0 kg

 Never round off twice: it causes inaccuracies. (e.g. Round of 245 to the nearest hundred.

The answer is 200. if you first rounded off 245 to the nearest ten it would be 250, and then to
the nearest 100, you would get 300, which is wrong).
 Only round off in the last step of calculations. (e.g. to find 3,38 x 2,27 correct to 1 decimal place:
the correct answer is 7,7. If you first round off, you will get 3,4 x 2,3 which gives 7,8: this answer
is wrong.

17

CHAPTER 4 - MAPS AND SCALE PLANS

(SO 2 AC 2.1 - 7)
On a map, lines of longitude go from North to South, and they all meet at the North Pole and South Pole.
They appear vertical on maps, and they are labelled as so many degrees East or West of Greenwich,
which is 0o.
Lines of Latitude go from East to West and appear horizontal on maps. The Equator is in the centre, and
it is 0o.
These lines are marked as either North or South of the Equator.
The co-ordinates of a point are given with the latitude first, then longitude.

Example:

18

The point A represents a ship at sea. Its position is given by (37oS; 25oE)

A scale of 1: 300 means the actual length is 300 times larger than the scale length.

Example:
A room is rectangular, of length 9 m and width 4 m. A scale plan is to be drawn using a scale 1:140, what
are the dimensions of the room on the plan?
The length of the room would be 900 /140 = 6,4 cm on the scale drawing, and the width would be
400/140 = 2,9 cm.
Note: To find the actual size given the scale length, do it in reverse.

Example:
Find the actual length of a desk if its length on a scale drawing is 2cm, and the scale is 1: 50.
The answer is 2 x 50 = 100 cm or 1 m.

Example:
1. A map has a scale of 1: 300 000. The distance between two places on the map is 4,5 cm. What is the
actual distance between the places?
Answer:
1cm represents 300 000 cm = 3 000 m = 3 km
Distance = 4,5 x 3 = 13,5 km.
2. A building that is 20 m long is shown on a scale plan as 5 cm long.

a. What scale has been used for the plan?
b. How wide is the building, if it is shown as 3 cm wide on the plan?
c. A footpath is 32 m long. How long will it be on the plan?
Answer:
a. 5: 2000 = 1: 400
b. 5 cm represents 20 m, so 1 cm represents 20/5 = 4 m, and 3 cm represents
3 x 4 = 12 m.
c. 1: 400 = x: 3200; 32/4 = 8. The path will be 8 cm long on the plan.
________________________________________________________________________

19

CHAPTER 5 - SHAPES

PYTHAGORAS' THEOREM

(SO 1 AC 1.5; SO2: AC 2.4 – 2.7)

Pythagoras' theorem states that in a right- angled triangle,
the square of the hypotenuse (longest side) equals the sum
of the squares of the other two sides.

When a triangle has a right angle (90°) ...

... and squares are made on each of the three sides, ...

... then the biggest square has the exact same area as the other two squares put together!
It is called “Pythagoras Theorem” and can be written in one short equation:

Note:

C is the longest side
(Hypotenuse) of the triangle
A and b are the other two sides

- Let’s see if it really works
using an example.

Example:

Find the
length x in
the right-
angled
triangle
below

20

62 + 102 = x 2
36 + 100 = x2
136 = x2
x = √136 = 11,66 cm

If one of the shorter sides of a triangle must be found, subtraction must be used.

Example:
Find the length k in the right-angled triangle below

Example
1. Find he length marked x in the right-angled triangle shown.

Answer
x2 + 62 = 102
x2 = √(102 – 62)
x = 8 cm

2. Town B is 53 km due west of town A, and town C is 38 km due north of town A. Find the distance
between town B and town C.

Answer

BC2 = AB2 + AC2

21

= 532 + 382
BC = √(532 + 382) = √ 4253
BC = 65,2 km
3. Find the length of the sloping lines in the trapezium shown. Draw in extra lines where necessary.

Answer
In the right-angled triangles: let sloping line = r. Horizontal lines = 3 m, by subtraction.
r2 = 3 + 4
r = √(32 + 42)
r=5m

22

CALCULATION OF PERIMETER AND AREA

(SO 1 AC 4-6)
RECTANGLE
To find the perimeter of a rectangle, add the lengths of the rectangle's four sides. If you have only the
width and the height, then you can easily find all four sides (two sides are each equal to the height and
the other two sides are equal to the width).

Formulas to find the perimeter as well as the area of a
rectangle are:
PERIMETER= 2(L + B)
AREA = L X B

Example:
Find the perimeter and area of the rectangle below

Answer:
Perimeter = 2(16,6 + 7,8) = 48,8 m
Area = 16,6 x 7,8 = 129,48 m2

23

PARALLELOGRAM
The perimeter of a parallelogram is the distance around the outside of the parallelogram.
A parallelogram has four sides with opposite sides being congruent.
The formula for finding the perimeter is Side A + Side B + Side A + Side B. This could also be stated as
2*Side A + 2*Side B or 2*(Side A + Side B).

PERIMETER= sum of all sides

AREA = base x perpendicular height

Example:
Find the perimeter and area of the parallelogram shown.
Answer:
Perimeter = 12 + 8 + 12 + 8 = 40 m
Area = 12 x 7 = 84 m2

24

TRIANGLE

PERIMETER= sum of sides
AREA = ½ Base X Perpendicular height
A=½bxh

Example:
Find the perimeter and area of the triangle below.

Answer:
First find n and p using Pythagoras:
n2 = 32 + 42 = 9 + 16 = 25
n = 5 cm
p2 = 122 – 42 = 144 – 16 = 128
p = 11,3 cm
Perimeter = 3 + 11,3 + 12 + 5 = 31,3 cm
Area = ½ (14,3) x 4

2

= 28,6 cm

25

TRAPEZIUM

PERIMETER= sum of sides

AREA = ½ sum of parallel sides x perpendicular distance between them
A = ½ (a + b) x h

Example:
Find the perimeter and area of the trapezium below

Perimeter = 17 + 4 + 12+ 4 = 37 m
Area = ½ (17 + 12) x 3

= 43,5 m2

CIRCLE

CIRCUMFERENCE: =
AREA: =
Where π is approximately 3,14 and r = radius.

26

Example:
Find the circumference and area of the circle shown.

Circumference = 2 x 3,14 x 5 = 31,4 cm
Area = 3,14 x 5 x 5 = 78,5 cm2

Example
1. Find the perimeter and area of the parallelogram below.

AD2 = 22 + 3,82 First find AD:
AD = √18,44 = 4,3 cm
Perimeter = 2(7 + 4,3) Answer:
= 22,6 cm Area of each triangle = ½ (3,8 x 2) = 3,8 m2
Area = 7 x 3,8 Area of rectangle = 5 x 3,8 = 19 m2
= 26.6 cm2 Total area = 2 x 3,8 + 19 = 26,6 m2
2. Find the area of the shape below.

Example

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Find the shaded area in the following diagrams. Find two separate areas, then add or subtract as
required.

Answer:
Diameter = 16 cm, radius = 8 cm.
By subtraction, breadth of rectangle = 8 cm.
Area of rectangle = 16 x 8 = 128
Area of semi-circle = ½ (3,14 x 8 x 8)
= 100,48
Shaded area = 128 + 100,48 = 228,48 cm2
Answer:
Large circle:
3,14 x 4,5 x 4,5 = 63,585
Small circle:
3,14 x 2,5 x 2,5 = 19,625
Shaded area:
63,585 – 19,625 = 43,96 cm2

28

VOLUME AND SURFACE AREA OF SOLIDS

(SO 1: AC1.5; SO 2, AC 2.4-7)
VOLUME OF PRISMS (INCLUDING CYLINDERS)
A prism is an elongated shape in which the cross-section remains the same shape and size at all points
along the prism. The flat sides are known as faces. Between each face and the next are edges. At
each end of every edge is a corner or vertex.
The general formula for the volume of any prism is:

VOLUME OF PRISM = AREA OF CROSS-SECTION X HEIGHT

Note: volume always has cubic units, e.g. m3
A RECTANGULAR PRISM is a normal box. Since the area of the rectangular cross-section is L x B, the
volume of the prism is given by LxBxH.
A TRIANGULAR PRISM has a triangle for its cross-section. Find the area of the triangle then multiply this
by the height of the prism. (or length if it is on its side, as in the diagram below.)
Example:
find the volume of the triangular prism below.

Area of cross section = area of triangle
= (1/2) (6x8)

= 24 m2
Volume of triangular prism = 24 x 10 = 240 m3

29

A CYLINDER is a type of prism, whose cross-section is a circle. Therefore, the formula for the volume of a
cylinder is:

VOLUME OF CYLINDER = π r 2 h

Where r = radius of cylinder
h = height of cylinder.

Example:
Find the volume of the cylinder shown.

Vol = 3,14 (4)2(10)
= 502,4 cm2

NETS OF PRISMS

If a prism is cut open and all the sides are flattened out, the resulting flat shape is known as a net.

Prism rectangular prism triangular prism Cylinder

Net

30

SURFACE AREA OF PRISMS
Every prism can be opened out into a net consisting of: two identical end-pieces the shape of the cross-
section, and a rectangle whose one side is equal to the perimeter of the cross-section, and which other
side is the height of the prism (see diagram).

Thus the total surface area of any prism is given by the formula:
TSA of prism = 2 x (area of base) + (perimeter of base x H)

In particular, since the circumference of a circle is 2π r (2 pi r),
the formula for the surface area of a cylinder is:
SURFACE AREA OF CYLINDER = 2π r2 + 2πrh

Example:
Find the volume and total surface area of the prism shown in the diagram.

Answer:
First find h by subtraction:
h=7–4=3m
Find b by Pythagoras: b2 = 82 – 32 = 55, so b =

55 = 7,4 m
Area of cross section of prism = area of
trapezium = ½ (7 + 4) x 7,4 = m2

Volume of prism = 40,7 x 10 = 407 m3
Perimeter of trapezium = 7 + 8 + 4 + 7,4 = 26,4
Total surface area of prism =
(2 x area of cross-section) + (perimeter of cross-section x length of prism)
= (2 x 40,7) + (26,4 x 10) = 345,4 m2
________________________________________________________________________

31

CHAPTER 6 - TRANSFORMATIONS, SYMMETRY, TESSELLATIONS

(SO 2 AC2-4)
If a rigid shape is moved from one position to another, it undergoes a transformation. Three
transformations will be dealt with: translation, rotation and reflection.
TRANSLATION
In a translation transformation all the points in the object are moved in a straight line in the same
direction. The size, the shape and the orientation of the image are the same as that of the original
object. Same orientation means that the object and image are facing the same direction.
Example:

We describe a translation in terms of the number of units moved to the right or left and the number of
units moved up or down.

32

ROTATION
A rotation is a transformation in which the object is rotated about a fixed point. The direction of
rotation can be clockwise or anticlockwise.
The fixed point in which the rotation takes place is called the centre of rotation. The amount of rotation

made is called the angle of rotation.

Example:
The square is pivoted or fixed , and rotated in an anti- clockwise manner
through 55o to its new position.

REFLECTION
Reflection means obtaining a mirror image of a shape.
This requires a line of reflection, which acts as a mirror. If
the line of reflection is vertical, for example, then shapes
which were on the left switch over to the right, and vice
versa.
Example:
The shape on the left has been reflected or flipped in the
vertical line of reflection.

1. Describe the transformation that has occurred in each of the following cases. For each case, give the
details of what has happened in going from the first grid to the second.

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Answer:
a. Translation. 1 space to the right and 3 down.

Answer:
b. Rotation: clockwise, through 900

Answer:
c. Reflection about a vertical line of reflection.
2. Examine the diagrams below, then select the correct statement.

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a. The triangle has undergone rotation through 90 degrees in an anti-clockwise direction and the
rectangle has undergone reflection about the horizontal.

b. The triangle has undergone reflection about the vertical and the rectangle has undergone rotation
through 90 degrees in a clockwise direction.

c. The triangle has undergone reflection about the vertical and the rectangle has undergone reflection
about the horizontal.

d. Both items have undergone translation only

e. Both items have undergone reflection only

Answer: b

SYMMETRY

(SO 2 AC 1;2;5)

Mathematically, symmetry means that one shape becomes exactly like another when you move it in
some way: turn, flip or slide. For two objects to be symmetrical, they must be the same size and shape,
with one object having a different orientation from the first. There can also be symmetry in one object,
such as a face.
The line down the middle is called an axis of
symmetry.

Example: axis of

This drawing is symmetrical about a vertical
symmetry:

35

This drawing is symmetrical about a horizontal axis of symmetry:
Example
1. Of the capital letters A,B,C,D,E,K,V,T, which are symmetrical about the vertical?
Answer: A, V, T
2. Of the capital letters A,B,C,D,E,K,V,T, which are symmetrical about the horizontal?
B, C, D, E, K
TESSELLATIONS
(SO 2, AC 2.1-7)
Tessellations are patterns in which identical shapes are placed alongside each other to fill up a space.
There must be no gaps in between. Tiles are a typical example. Three examples of tessellations are given
below.

36

CHAPTER 7 - BINARY NUMBERS

(SO 1: AC 1.1-3)
The binary number system is a base-2 number system. This means it only has two numbers: 0 and 1. The
number system that we normally use is the decimal number system. It has 10 numbers: 0-9.
Why use binary numbers?

Binary numbers are very useful in electronics and computer systems. Digital electronics can easily work
with a sort of "on" or "off" system where "on" is a 1 and "off" is a zero. Often times the 1 is a "high"
voltage, while the 0 is a "low" voltage or ground.

Our decimal system uses ten digits, the numbers 0,1,2..., 9. We use a positional system to show
numbers bigger than 9.

Note: examine the pattern of numbers: 1000, 100, 10, 1, and 0,1. These could be written as powers of
10: 103, 102, 101, 100 and 10-1. (Note how the powers decrease in sequence.) We could fit this into a
table to illustrate how we write 365:

365 = 3 x 102+ 6x101 + 5x100. We normally don't write out the powers of ten, we just show by the
position of the digits what their meaning is.

Thousands Hundreds Tens Units
1 000 100 10 1
103 102 101 100

3 6 5

Binary numbers became useful with the invention of computers, where large amounts of information
needed to be stored in electronic circuits. A system using only 0s and 1s was developed, where 0 means
off and 1 means on.

37

The binary system works in a similar way to the decimal, the difference being that instead of ten digits
there are only two, thus instead of powers of 10, we use powers of 2. In the third row of the table below,
a number of 0s and 1s are shown.
This is how binary numbers are written. Each 0 or 1 is in a particular position, so it has a particular
meaning.
The number 100112 or 10011(binary) means 1x16 + 0x8 + 0x4 + 1x2 + 1x1 = 19 decimal.

24 23 22 21 20

16 8 4 2 1

10011

We can even include decimal numbers in this system. For example, the number 110,1 means 1x4 + 1x2
+ 0x1 + 1 x ½ = 6 ½ (see table below).

We add on another column to the right:

24 23 22 21 20 Point 2-1 2-2

16 8 4 2 1 ½¼

1 1 0 .1 0

Here is a table showing some numbers in both systems
Decimal Binary
11
2 10
3 11
4 100
8 1000
9 1001
14 1110
16 10000

38

Example:
a. Convert 100.012 to decimal.
Answer: 1x4 + 0x2 + 0x1 + 0 x ½ + 1 x ¼ = 4 ¼
b. Convert 23 to binary.
Answer: First split the number into 16s, 8s, 4s, and so on.
23 = 16 + 4 + 2 + 1
= 1 x 16 + 0 x 8 + 1 x 4 + 1 x 2 + 1 x 1 = 10111
Addition and subtraction of binary numbers is most easily done by first converting to decimal, doing the
calculation, and then converting the answer back to binary.

EXAMPLE: Find the answer in the binary system: 1101 - 1001
Convert to decimal:
1101 = 1x8 + 1x4 + 0x2 + 1x1 =13;
1001= 1x8 +0x4 +0x2 + 1x1 = 9
13 - 9 = 4, which is 100 binary

For more information on the binary system, visit http://www.mathsisfun.com/binary-number-
system.html

39

CHAPTER 8 - SCIENTIFIC NOTATION

(SO 2: AC 1-2)
Scientific notation is used to write very large or very small numbers in a convenient form. It is also used
for comparison of numbers.
Scientific notation requires ONE digit only before the comma. After that we write 10 to the power of
the number of places you have to move the comma.
NOTE:
* We can use ^3 to represent “to the power of” e.g. 2^3 = 2x2x2 = 8
* Anything to a negative power denotes a fraction, so 1/10 = 0,1 = 10^-1

Writing a number in scientific notation
Example (a): Write 6432 in scientific notation.
There must be one digit then a comma, so the first part will be 6,432. Now count up how many places
you must move the comma to get back to the original number. It is 3 (to the right.) This gives the power
of 10. Answer: 6432 = 6,432 x 10^3
(b) Numbers less than 1 will require a negative power.

40

Example: Write 0,0037 in scientific notation.
There must be one digit then a comma, so the first part will be 3,7
Now count up how many places you must move the comma to get back to the original number. It is 3
places (to the left.) Don't forget to put in the minus to show that the number is less than 1.
Answer: 0,0037 = 3,7 x 10^-3

Converting from scientific notation to ordinary numbers.
Here you must begin with the power of 10: this tells you how many places to move the comma.
Remember that if it's a negative power you must end up with a number less than 1, i.e. 0,...

Example (a): Convert 2,34 x 10^4 to an ordinary
number.
Look at the power of 10. It is 4, so there must be
4 digits after the 2. Put on as many 0's as
needed to make up 4 digits after the 2.
Answer: 2,34 x 10^4 = 23400
Example (b): Convert 3,14 x 10^-2 to an ordinary
number.
The power of 10 is -2. Because it's negative, it
means the answer will be 0,... (i.e. a number less
than 1) and the 2 means that you must move
the decimal point 2 places to the left.
Answer: 3,14 x 10^-2 = 0,0314

41

CHAPTER 9 - RATIONAL AND IRRATIONAL NUMBERS

(SO 3: AC 3.1-4)

Rational numbers are numbers which can be written as the RATIO of one integer to another. (Integers
are ...-3; -2; -1; 0; 1; 2; ...) A rational number can also be written as a decimal: either a terminating
decimal, e.g. 0,37894 or a recurring (repeating) decimal, like 3,2828....

Irrational numbers cannot be written in this form. Their decimal form goes on forever without ending or
repeating, e.g. 23, 7829184.... The dots at the end of a decimal tell us what kind of number it is.
Numbers like π and √ 2 or √ 8 are irrational, and we can only give an approximate value for them by
listing some of their decimals, thus an irrational number is a non-terminating and non-repeating decimal.

Example: Say whether the following numbers are rational or irrational, and give a reason.

(a) 3,295295...

(b) 62,90483

(c) √ 26

(d) √ 25

(e) 12,76243...

(f) 16

Answers
(a) is rational: repeating decimal
(b) is rational: terminating decimal
(c) is irrational: square root of a non-perfect square
(d) is rational: square root of a perfect square
(e) is irrational: non-terminating and non-repeating decimal.
(f) is rational: can be written as one integer over another, = 16/1

42

WORKING WITH IRRATIONAL NUMBERS

When working with the square root of a non-perfect square, such as √3, do not use a calculator unless
asked to do so. Leave the number in √ (surd) form. The reason is that a calculator cannot give a fully
accurate value of such a square root, as it is irrational.
Multiplication of surds:
If we multiply the square root of a number by itself, we get back to the number:
e.g. √3 x √3 = 3.
So 2√3 x 4√3 = 3x4x2= 24
Addition of surds:
The surd must remain as it is.
e.g. √3 + √3 = 2√3
So 6√7 + 5 √7 = 11√7

CONVERSION OF RECURRING DECIMALS TO FRACTIONS

Do the following experiment. On your calculator, divide any number by 99 or 990. Look at the first few
digits after the comma.

Since a recurring decimal is a rational number, it must be possible to write it as a fraction. There is a
special method for doing this. Follow the steps in the examples below.

Example 1: convert 4,62626... to a common fraction.
Put the given decimal =f.
f=4,62626...
Count up the number of digits that recur. If it is one, we multiply by 10, if it is two, multiply by 100. In
this case it is two: 6 and 2.
Multiply both sides by 100:
100f = 462,62626...
Then subtract to make the part after the decimal disappear.
100f-f = 462-4.
99f = 458
f = 458 /99
Check your answer by dividing 458 by 99 using a calculator.

Example 2: convert 4,386868... to a common fraction.

43

Put the given decimal =f.
f = 4,386868...
Count up the number of digits that recur. If it is one, we multiply by 10, if it is two, multiply by 100.
Multiply both sides by 100:
100f = 438,686868...
Then subtract:
100f-f =438,6868.. - 4,3868...
99f = 434,3
f = 434,3 /99
Multiply top and bottom to get rid of the decimal:
f = 4343 /990
Check your answer by dividing 4343 by 990 using a calculator.

44

CHAPTER 10 - DATA COLLECTION

What is data collection?
Data collection is the process of gathering and
measuring information on targeted variables in an
established system, which then enables one to
answer relevant questions and evaluate outcomes.

(SO 1: AC 1-4)
In statistics we deal with data, which means information that has been given to us. A population refers
to the widest group of items or people which the data describe. A sample is a set of items or people
taken from the entire population. A useful way to collect data is by a questionnaire. This is a set of
questions which can be given to members of a sample group. This helps us to know about the entire
population.

DICHOTOMOUS
A questionnaire which have only two categories or levels. For example, if we were looking at
gender, we would most probably categorize somebody as either "male" or "female". This is an
example of a dichotomous variable (and also a nominal variable).
LIKERT
A questionnaire in which there are five answers arranged in order from bad to good, makes use of a
Likert scale. e.g. What is your opinion of this policy: I agree strongly/ I agree/ neutral/ disagree/ disagree
strongly

45

DISCRETE AND CONTINUOUS VARIABLES (SO 1: AC 1-4)

Numerical data can be discrete or continuous.
Discrete data can only take certain values.
For Example,

- Number of boxes in a container.
- Number of containers per shipment
- Number of HAWBs per MAWB
Continuous data comes from measuring and can take any value within a given range.
For Example,
- Weight of a pallet.
- The time it will take for delivery of shipment.
- Temperature range at which the perishable goods would need to be kept at during transit.

46

METHODS OF REPRESENTING DATA (SO 1: AC 1-4)

In statistics, a collection of numerical facts (data) is made so that conclusions may be drawn. e.g. If
weather patterns are being examined, a large number of measurements of temperature and rainfall
must be made on a regular basis, and at various places. This is done by the weather bureau, and the
results are published in tables.

TABLES

Tables must be read according to the headings at the top of each column and also at the start of each
row. StatsSA publishes data on many facts in South Africa, e.g. Population in each province in different
years. Visit their web site http://www.statssa.gov.za/

PIE CHARTS

A pie chart is a useful means of displaying data. For of
example, the pie chart shows the relative proportion
different makes of vehicles which were found in a
company parking lot. There were not only five
makes of car: a category called OTHER was used for
those not mentioned by name.

In making a pie chart, all the facts must be collected for

each category. Suppose there were 12 Fords, and that

the total of all the cars was 48. Then the proportion of

cars which were Fords is 12/48 = 1/4. Thus One quarter of the pie chart is taken up by Ford. The key is

used to inform the reader which sector of the graph belongs to each category.

To draw a pie chart by hand, calculate the size of the angle for each sector (slice). The total angle at the
centre of the circle is 3600. Thus the angle for Ford is ¼ of 3600 = 900.

BAR GRAPHS

A bar graph has two axes, vertical and horizontal. They must both be clearly marked with type of data
and units where necessary.

In the example below, the height of the bars gives the percentage of people unemployed. The fact that
the tallest bar is labeled it tells us that Jan 2020 has the highest unemployment rate . By looking across

47

to the numbers on the vertical axis, we can see that Jan 2020 shows approximately 30.40%
unemployment rate.
If asked the range, subtract the lowest from the highest: in this case, 30.4 % – 21.50% = 8,9%.

South Africa Unemployment Rate in RSA

STEM-AND-LEAF
This display is a quick way of organizing several numbers. Left and Right are taken for two different
categories: e.g. Men on the left, women on the right showing their ages. The red number in the middle,
the stem, represents tens; the other numbers represent digits. In the case below, the highest number is
94, and it represents a woman. The next highest is also a woman, 84. Then comes a man, 83. There are
four people in the 70's, two men and two women

Men Woman

Key: 9 4 = 94 years

MEASURES OF CENTRAL TENDENCY AND
SPREAD

48

(SO 2: AC 2.1-5)

MEAN

The mean of several numbers is the average. It is calculated by adding the numbers and dividing by the
number of items.
e.g. Find the mean of 6; 6; 7; 3; 2; 1
Answer: (6+6+7+3+2+1)/6 = 4,17

MEDIAN
Arrange the numbers in order first. The number in the middle is the median (if there are an even
number of items, find the number halfway between the two middle numbers.)
e.g. Find the median of 3; 3; 6; 8; 9; 9
Answer: 7
e.g. Find the median of 3; 3; 7; 8; 9; 9
Answer: 7,5

MODE

The number which occurs most frequently. In the previous example both 3 and 9 were modes.

RANGE

Largest minus smallest. In the previous example, 9 minus 3= 6.

QUARTILES

A quartile is defined as a group of values and/or means that divide a data set into quarters, or groups of
four. To find the quartiles of a data set use the following steps: Order the data from least to greatest.
Find the median of the data set and divide the data set into halves.
Arrange the numbers in order first. The quartiles are quarter of the way along from either end of the
arranged numbers. Leave out the median when finding quartiles.
E.g. a For the numbers 1; 2; 3; 6; 7; 9, the quartiles are 2 and 7.
E.g. b For the numbers 1; 2; 3; 5; 6; 7; 9, the quartiles are 2 and 7.

49

INTER-QUARTILE RANGE
The interquartile range (IQR) is a measure of variability, based on dividing a data set into quartiles.
Quartiles divide a rank-ordered data set into four equal parts. The values that divide each part are called
the first, second, and third quartiles; and they are denoted by Q1, Q2, and Q3, respectively.
Upper quartile minus lower quartile: 7–2=5 in the above examples.

50