essential mathematics for economics and business

Essential Mathematics
for
Economics
and
Business
Second Edition
This page intentionally left blank
Essential Mathematics
for
Economics
and
Business
Teresa
Bradley
Limerick
Institute
of
Technology
Paul Patton
Limerick
Senior
College
Second
Edition Revised
by
Teresa Bradley
JOHN
WILEY
&
SONS,

LTD
Second
Edition
Copyright
f
2002
by
John
Wiley
&
Sons Ltd,
Baffins
Lane, Chichester,
West
Sussex PO19 1UD, England
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customer service enquiries):
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our
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First edition printed
in
1998, 1999
All
Rights Reserved.
No
part
of
this publication
may be
reproduced,
stored
in a
retrieval system,
or
transmitted,
in any
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1988
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90
Tottenham Court Road, London,
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permission
in
writing
of the
Publisher.
Teresa
Bradley
and

Paul Patton have asserted their rights under
the
Copyright, Designs
and
Patents
Act
1988.
to be
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authors
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British
Library
Cataloguing
in
Publication
Data
A
catalogue record
for
this
book
is
available
from
the
British Library
ISBN
0-470-84466-3
Typeset
in
9.5/1 l.5pt Times

by
C.K.M. Typesetting, Salisbury, Wiltshire
Printed
and
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Great
Britain
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Biddies Ltd, Guildford
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each
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Richard T.B.
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father
P.P.
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Introduction
Introduction
to the first
edition
1
Mathematical preliminaries
1.1
Arithmetic operations
1.2
Fractions
1.3

Solving equations
1.4
Currency conversions
1.5
Simple inequalities
1.6
Calculating percentages
1.7
Using
the
calculator
1.8
Introducing Excel
The
straight line
and
applications
2.1
The
straight line
Mathematical modelling
Applications: demand, supply,
cost,
revenue
More mathematics
on the
straight
line
Translations
of

linear
functions
Elasticity
of
demand,
supply
and
income
Budget
and
cost constraints
Excel
for
linear
functions
Summary
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
xi
xiii
1
o
4
7

9
14
17
19
20
29
30
47
51
66
71
72
80
81
86
Simultaneous
equations
3.1
Solving simultaneous linear equations
3.2
Equilibrium
and
break-even
ebooksdownloadrace.blogspot.in
viii
CONTENTS
3.3
Consumer
and
producer surplus

114
3.4
The
national income model
and the
IS-LM model
118
3.5
Excel
for
simultaneous linear equations
123
3.6
Summary
127
Appendix
129
4
Non-linear functions
and
applications
131
4.1
Quadratic, cubic
and
other polynomial functions
132
4.2
Exponential functions
151

4.3
Logarithmic functions
165
4.4
Hyperbolic functions
of the
form a/(bx
+ c) 177
4.5
Excel
for
non-linear functions
183
4.6
Summary
185
5
Financial mathematics
188
5.1
Arithmetic
and
geometric
sequences
and
series
189
5.2
Simple
interest,

compound
interest
and
annual
percentage
rates
195
5.3
Depreciation
205
5.4
Net
present value
and
internal rate
of
return
207
5.5
Annuities, debt repayments, sinking
funds
213
5.6
The
relationship between interest rates
and the
price
of
bonds
224

5.7
Excel
for
financial mathematics
227
5.8
Summary
229
Appendix
231
6
Differentiation
and
applications
234
6.1
Slope
of a
curve
and
differentiation
235
6.2
Applications
of
differentiation,
marginal functions, average functions
245
6.3
Optimisation

for
functions
of one
variable
260
6.4
Economic applications
of
maximum
and
minimum points
276
6.5
Curvature
and
other applications
291
6.6
Further differentiation
and
applications
306
6.7
Elasticity
and the
derivative
319
6.8
Summary
328

7
Functions
of
several variables
332
7.1
Partial differentiation
332
7.2
Applications
of
partial
differentiation
349
7.3
Unconstrained optimisation
369
7.4
Constrained optimisation
and
Lagrange multipliers
378
7.5
Summary
390
8
Integration
and
applications
394

8.1
Integration
as the
reverse
of
differentiation
394
8.2
The
power rule
for
integration
396
8.3
Integration
of the
natural exponential
function
401
8.4
Integration
by
algebraic substitution
402
8.5
The
definite
integral
and the
area under

a
curve
407
ebooksdownloadrace.blogspot.in
CONTENTS
ix
8.6
Consumer
and
producer surplus
414
8.7
First-order
differential
equations
and
applications
422
8.8
Differential
equations
for
limited
and
unlimited growth
433
8.9
Summary
438
9

Linear algebra
and
applications
441
9.1
Linear programming
441
9.2
Matrices
452
9.3
Solution
of
equations: elimination methods
462
9.4
Determinants
468
9.5
The
inverse matrix
and
input/output analysis
481
9.6
Excel
for
linear algebra
495
9.7

Summary
498
10
Difference equations
503
10.1
Introduction
to
difference
equations
503
10.2
Solution
of
difference
equations (first-order)
505
10.3
Applications
of
difference
equations (first-order)
517
10.4
Summary
526
Solutions
to
progress exercises
528

Bibliography
621
List
of
worked
examples
622
Index
628
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Introduction
This book
is
intended
for
students
who are
studying mathematics
as a
subject
on
economics,
business
or
management courses.
The
text assumes minimal mathematical background
but
demonstrates
the

usefulness
and
relevance
of
basic mathematics
in
economics
and
business.
To
assist students, particularly those
who are
apprehensive about mathematics,
the
mathematical
methods
are set out and
explained
step
by
step
then
illustrated
in
worked
examples. Economic
and
business applications
of
these methods

follow
immediately, again
illustrated
in
worked examples.
It is
essential that students attempt
the
progress exercises
at
the
end of
each section
in
order
to
consolidate
and
retain
the
ideas
and
methods introduced
and to
test
and
enhance understanding. Detailed solutions
are
given
at the

back
of the
text.
In
this edition several sections within chapters have been rewritten
in a
clearer, more
accessible style,
for
example,
the
straight line, logs,
differentiation,
optimisation
and
integration.
Chapters
4 and 8
were
reorganised,
introducing
e
x
earlier, therefore including
it
as an
integral part
of the
rules
for

indices, logs, integration
and
applications.
New
topics
have been introduced: currency conversions
in
Chapter
1;
annuities, debt repayment, sinking
funds
in
Chapter
5;
integration
by
substitution (for functions
of
linear
functions)
in
Chapter
8;
elimination methods (Gaussian
and
Gauss-Jordan)
applied
to the
solution
of

systems
of
equations
and the
inverse matrix
in
Chapter
9. A
short section
on the use of
Excel
in
linear
algebra
is
also included
in
Chapter
9.
In an
attempt
to
produce
a
slimmer volume, some
of the
less frequently used sections
(on
the
mathematics,

but
more especially
on
applications) have been moved
to the web
site
which
has
been
developed
to
accompany this book
at
www.wiley.co.uk/bradley2ed.
This
material
is
referenced
by JHH
within
the
text,
so
that
it may be
retrieved
quickly
and
easily
from

the web
site
as
required. Also available
on the web
site:
•
Integration
by
substitution
and
parts with progress exercises
and
solutions (Chapter
8).
•
PowerPoint slides: these slides present introductory material
and a
resume
of
certain
topics
from
selected
chapters.
xii
ESSENTIAL MATHEMATICS
FOR
ECONOMICS
AND

BUSINESS
•
Instructors manual: this
is
available,
via
password,
to
instructors
who
adopt
the
book
as
the
main course text. This
is an
updated version
of the
manual that
was
formerly
available
in
hardcopy
form.
Details
of
content
are

given
in the
introduction
to the first
edition
of
the
text.
Acknowledgements
I
thank
all
those whose help
was
invaluable
in
producing
the
second edition: Bart Mooney,
Gabriel Daly, Chris Naughton.
I
thank those
who
reviewed
the first
edition
and
made
very
helpful

suggestions
for the
second edition; these suggestions were adopted whenever possible:
Dr
Reza Arabsheibani,
University
of
Wales;
Dr
Donal Hurley, University College Cork;
Professor
John
O'Donoghue,
Dr
Eugene
Gath
and Dr
Michael
Hayes,
University
of
Limerick;
J.
Colin
Glass,
University
of
Ulster; Hillary Lamaison, Brunei University;
Dr
Alexander Lee, University

of
Western Sydney; Helen
Chadda,
Limerick Institute
of
Technology; Anca Porojan, University
of
Bradford; R.J.
Stroeker,
Erasmus University:
Dr
Stephen Weissenhofer, University
of
Western Sydney.
I
thank John McGuinness,
who
made suggestions
for the
design
of the
cover.
And I
thank Steve Hardman, publishing editor
and
Sarah Booth
of
John Wiley
&
Sons Ltd, Chichester.

Teresa Bradley
A
web
site
accompanies this book:
www.wiley.co.uk/bradley2ed
Many
students
who
pursue
the
study
of
economics
and
business studies
are
surprised
and
perturbed when they discover that mathematics
is a
core subject
on first-year
courses.
However,
a
certain
level
of
skill

and
understanding
of the
basic mathematical methods
is
required
if the
study
of
economics
or
business
is to be
pursued beyond
the
descriptive
level.
Students should
be
reassured
that
it is not
necessary
to
become
a
mathematician
to use
mathematical
techniques

and
methods
effectively.
Learning
to use
mathematics could
be
compared
to
learning
to
drive.
In
either case,
the
quote
from
the
Chinese philosopher
Lao Tse is
appropriate:
You
read
and you
forget;
you
see and you
remember;
you
do and you

learn.
At the
outset
the
learner-driver
is
presented
with
a
bewildering
set of
rules
and
tasks, some
of
which must
be
performed simultaneously, some sequentially.
There
are
sound,
sensible
reasons
for
each
of
these rules,
as
learners
will

discover
on
their
first
outing
on a
public road.
Mastering driving skills
and
gaining
a
sense
of how to
control
the car
only comes
about
by
following
closely
the
routines demonstrated
by the
instructor, then practising them over
and
over
again, sometimes patiently, sometimes not!
In the
end,
the new

driver will
be
able
to
handle
a car
easily
and
effortlessly,
as if it
were second nature. With these newly acquired
skills
life
is
enhanced
with
previously unavailable choices.
The new
driver (with
a
car!)
can
xiv
ESSENTIAL MATHEMATICS
FOR
ECONOMICS
AND
BUSINESS
choose where
to go, who to go

with, what route
to
take, what time
is
convenient, etc.
And so it
is
with
maths.
Worked Examples
At
an
introductory
level,
you
will
not
become proficient
at
using mathematics
by
simply
'reading'
a
mathematics book, cover
to
cover.
A
better approach
is to

read just
one
section
at
a
time, then
put pen to
paper
and
follow
the
methods which
are
demonstrated
in the
Worked
Examples,
line
by
line.
An
index
of
Worked Examples
is
provided
at the end of the
book
for
easy

reference.
Graphs
Can
you
imagine someone
who has
never seen
a car
before, attempting
to
understand what
the
controls, gears, steering, etc. look like
from
a
verbal description? Understanding would
be
enhanced enormously
by the
provision
of
some well-labelled sketches
and
diagrams.
Visualising
mathematical functions/equations
is not
easy, especially
for
beginners. When

the
function
is
graphed, much
of the
vagueness
and
abstractness
is
removed.
In
fact, many
of
the
properties
of the
function
are
revealed.
In
Worked Examples, whenever
appropriate,
graphs
are
plotted over
an
interval
by
calculating
a

table
of
points, then drawing
the
graph.
Unfortunately,
in
many economic applications, this process
is
often
a
very time-consuming
exercise
and
errors
are all too
easily
made
in
calculations.
In
this
text,
the use of
spreadsheets
has
been introduced
to
expedite
the

process.
Use of
Spreadsheets
(Excel
or
Others)
The
text
refers
to
Excel, since
it is
available
in
Microsoft
Office,
but in
fact,
any
spreadsheet
serves
the
same purpose. Learning Excel
is not
difficult
and
outline steps
are
given
in the

text.
However,
no
attempt
is
made
to
teach Excel,
so
beginners
are
advised
to
refer
to
introductory
texts
on the
subject.
The use of
Excel
is
discussed
at the
ends
of
Chapters
1 to 5,
with
appropriate Worked Examples. Progress Exercises

in
which Excel would
be
particularly
useful
are set at the
ends
of
Chapters
2 and 3. In the
remainder
of the
text, Excel
may be
used
whenever
the
student chooses,
for
graphs, tables, etc. With
practice,
the
student should
discover
that
a
spreadsheet
is an
invaluable tool: Tables
of

points
are
calculated
and
good
accurate
graphs
are
drawn
in
minutes.
©
Remember:
the
text
can be
used
in its
entirety without touching
on the
Excel sections, but,
when
used,
Excel
will
prove
invaluable
for
graph plotting
and in

certain calculations.
Progress
Exercises
and
Test
Exercises
The
Progress Exercises provide
the
vitally
important practice
on the
mathematics
and on the
economic/business
applications covered
in the
preceding section. Answers
to all
problems,
INTRODUCTION
TO THE
FIRST EDITION
xv
with
outline solutions
to
some
of the
longer ones,

are
given
at the end of the
text.
However,
the
Progress
Exercises
alone
are not
sufficient.
In
more realistic applications (not
to
mention
exam
papers)
the
student
is
required
to
draw
on a
variety
of
mathematical techniques.
Questions based
on the
entire chapter

are set in the
Test Exercises
at the end of
each chapter.
Answers/solutions
to the
Test
Exercises
are
given
in the
Lecturer's Manual only.
You
Are Not
Expected
to be a
Mathematician!
It
is
important
to
point
out
that
this text
is
introductory, hence
the
approach
adopted

is not
mathematically rigorous.
The
mathematical functions used
in
Worked Examples
and in
applications
are
those which
are
normally encountered
at
this
level.
However,
the
student
is
alerted
to the
unusual
and
exceptional cases.
For
example,
the
student
is
made aware

of the
fact
that
equations
do not
always have
solutions,
that
functions
and/or
their derivatives
may
not
exist
at
every point
and
this gives rise
to
problems
in
graph sketching
and
optimisation,
etc. This situation
is
similar
to
that faced
by the

vast
majority
of car
drivers. Many
excellent
drivers have only
a
vague
idea
about
how the
engine works, but, nonetheless quickly
recognise when normal conditions break down.
So
they consult manuals
or
seek advice
from
the
experts.
For
reference,
a
short Bibliography
of
mathematical
texts
is
given
at the end

of
the
book.
Instructor's Manual
This
manual
by
Teresa
Bradley
is
available
from
the
publisher
to
instructors
who
adopt
this
book
as the
main course text.
For
each chapter
in the
text
the
manual contains:
•
Introduction: general comments

on the
contents
of
each chapter
•
Details
of
PowerPoint presentations
on
disk,
if
applicable
•
Reference
to
mathematical software: Calmat; Maple
•
General comments
on
areas
of
difficulty
experienced
by
students
•
Answers/solutions
to the
Test
Exercises

•
Additional problems, with answers/solutions
•
Worked Examples
on
applications
of
translations
are
given
in
Chapter
2
•
Sample
Examination
Papers:
stage
1
papers
examine material from
the
text
at a
basic
level:
stage
II
papers
at a

more
advanced
level
A
disk with Powerpoint presentations
on
selected topics
from
the
text
Transparency masters
on
selected figures from
the
text
may be
printed
from
the
PowerPoint
presentations.
Mathematics
is a
hierarchical subject.
The
core topics
in
most foundation mathematics
courses (linear functions, non-linear functions, differentiation
and

integration)
are
covered
in
XVI
ESSENTIAL
MATHEMATICS
FOR
ECONOMICS
AND
BUSINESS
Chapters
1, 2, 3, 4, 6 and 8.
These
provide
the
basis
for the
usual
economic
applications.
(Note:
an
introductory course
may be
pursued
by
covering
the
earlier

sections
within these
chapters.) Some
flexibility is
possible
in
deciding when
to
introduce further material
based
on
these core areas, such
as
linear algebra,
difference
equations, etc.
The
prerequisite
chapters
are
indicated
by the
chart below.
Chapter
1
Preliminaries
Chapter
2
Straight line
Chapter

9
Linear
programming
matrices
determinants
Chapter
3
Simultaneous
equations
Chapter
10
Difference
equations
Chapter
4
Quadratics:
indices
logs:
a/(bx
+ c)
Chapter
5
Financial maths
Chapter
6
Differentiation
Chapter
7
Partial differentiation
Chapter

8
Integration
For
example,
for
those
who
wish
to
continue
with
more mathematics
and
applications
of
the
basic
simultaneous
linear equations
in
Chapter
3, go
straight
to
Chapter
9. The
financial
maths
in
Chapter

5
requires
a
knowledge
of
indices
and
logs covered
in
Chapter
4.
Difference
equations
(Chapter
10)
could
be
introduced
on
completion
of the
section
on
indices
in
Chapter
4. The
material
in
Chapter

7
follows
immediately
from
Chapter
6.
However,
many
introductory
mathematics courses
may
omit this chapter completely, unless required
for
related
subjects
on the
course.
INTRODUCTION
TO THE
FIRST EDITION
xvii
Acknowledgements
We
would like
to
thank
a
number
of
friends

and
colleagues:
Chris Naughton
and
Dave O'Neill
for the
painstaking hours spent reviewing various chapters
of
the
text
and who
contributed many invaluable ideas
and
suggestions.
Dr Jan
Podivinsky
(Southampton
University), Hilary
Lamaison
(Brunei University),
Julian
Wells (Southbank University),
Dr
Matthias Lutz (Sussex University), Lars Wahlgren
(Lund
University), Mark Baimbridge (University
of
Bradford), Professor Hassan Molana
(University
of

Dundee)
and any
anonymous reviewers
who
offered
useful
suggestions
for
improvements,
many
of
which
we
have incorporated into this text.
Alan Barry
for his
technical
support
and
advice
at all
stages
throughout
the
project.
Orla
Gavigan
for her
patience
and

assistance
in the final
preparation
of the
manuscript.
Steve
Hardman, publishing editor
and his
assistant
See
Hanson,
and
Mary Seddon,
marketing
manager
and the
production team
at
John Wiley.
Finally,
thanks
to
Harry
and Joe and our
families
for
their support.
We
bear
full

responsibility
for all
errors
and
omissions.
T.B.
and
P.P.
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Mathematical
Preliminaries
At the end of
this chapter
you
should
be
able
to:
•
Perform basic arithmetic operations
and
simplify
algebraic expressions
•
Perform basic arithmetic operations with fractions
•
Solve equations
in one
unknown, including equations involving fractions
•

Understand
the
meaning
of no
solution
and
infinitely
many solutions
•
Currency conversions
•
Solve simple inequalities
•
Calculate percentages
In
addition,
you
will
be
introduced
to the
calculator
and a
spreadsheet.
Some
mathematical
preliminaries
Brackets
in
mathematics

are
used
for
grouping
and
clarity.
Brackets
may
also
be
used
to
indicate multiplication.
Brackets
are
used
in
functions
to
declare
the
independent variable (see later).
Powers:
positive whole numbers such
as 2
3
,
which means
2 x 2 x 2 = 8:
(anything)

3
=
(anything)
x
(anything)
x
(anything)
(x)
3
= x x x x x
5
Note
2
ESSENTIAL MATHEMATICS
FOR
ECONOMICS
AND
BUSINESS
Brackets:
(A)(B)
or A x B or AB
all
indicate
A
multiplied
by B.
Variables
and
letters: When
we

don't
know
the
value
of a
quantity,
we
give that quantity
a
symbol, such
as x. We may
then make
general
statements
about
the
unknown quantity,
.x, for
example 'For
the
next
15
weeks,
if I
save
£x per
week
I
shall have
£4000

to
spend
on a
holiday'. This statement
may be
expressed
as a
mathematical equation:
15
x
weekly savings
=
4000
15
x x =
4000
or
15.x
=
4000
Now
that
the
statement
has
been
reduced
to a
mathematical
equation,

we may
solve
the
equation
for the
unknown,
.x:
15*
=
4000
15.x
4000
divide
both sides
of the
equation
by 15
15 15
x
=
266.67
Square roots:
the
square root
of a
number
is the
reverse
of
squaring:

(2)
2
= 4 -+ x/4 = 2
(2.5)
2
=
6.25
->
v/6^25
= 2.5
1.1
Arithmetic
Operations
G
Addition
and
subtraction
Adding:
If all the
signs
are the
same, simply
add all the
terms
and
give
the
answer with
the
common overall sign.

Subtracting: When subtracting
any two
numbers
or two
similar terms, give
the
answer with
the
sign
of the
largest number
or
term.
If
terms
are
identical,
for
example
all
.x-terms,
all
.xy-terms,
all
x
2
-terms, then they
may be
added
or

subtracted
as
shown
in the
following
examples:
Add/subtract
with
numbers, mostly
5
+ 8 + 3 = 16
similarity
-
5
+ 8 + 3 + v = 16 + y
similarity
-
The
j-term
is
different,
so it
cannot
be
added
to the
others
Add/subtract
with
variable terms

5x
+
8.x
+
3.x
=
16.x
(i)
5.x + 8.x + 3.x + v =
16.x
+ y
(ii) 5.xv
+
8.xv
+
3.xv
+ v =
16.xv
+ y
MA
THEMATICAL
PRELIM
IN
A
RIES
7– 10 = -3
similarity
7
-
10

-
10*
= -3 -
10.x similarity
-
The
.x-terrn
is
different,
so it
cannot
be
subtracted from
the
others
The
y-term
is
different,
so it
cannot
be
added
to the
others
(i)
7A – 10x = – 3x
(ii)
7x
2

–
10.x
2
=
-3x
2
7x
2
–
10,x
2
– 10* = – 3X
2
– 10*
The
x-term
is
different,
so it
cannot
be
subtracted from
the
others
WORKED
EXAMPLE
1.1
ADDITION
AND
SUBTRACTION

For
each
of the
following, illustrate
the
rules
for
addition
and
subtraction:
(a)
2 +3+ 2.5 = (2 +3+
2.5)
= 7.5
(b)
2.x + 3.x +
2.5.x
= (2 + 3 +
2.5)*
=
7.5.x
(c)
–3xy
–
2.2xy
– 6xy
=(-3-2.2-
6)xy
=
–11.2xy

(d)
8* +
6xy
– 12* + 6 +
2xy
= 8* - 12* +
6xy
+
2xy
+ 6 = -4x +
8*y
+ 6
(e)
3.x
2
+ 4* + 7 –
2.x
2
–
8.x
+ 2 =
3.x
2
– 2*
2
+
4.x
–
8.x
+ 7 + 2 = x

2
-
4.x
+ 9
Multiplying
and
dividing
Multiplying
or
dividing
two
quantities with like signs gives
an
answer with
a
positive sign.
Multiplying
or
dividing
two
quantities with
different
signs gives
an
answer with
a
negative
sign.
WORKED
EXAMPLE

1.2
MULTIPLICATION
AND
DIVISION
Each
(a)
(b)
(c)
(d)
(e)
(0
v
(g)
(h)
of the
following examples illustrate
the
rules
for
multiplication.
5x7
-5 x
5
x -
-5 x
7/5 =
(–7)
(–7);
!/(
= 35

-7 = 35
7
= -35
7= –35
: 1.4
(-5)
= 1.4
5
1
4
J 1
.^T
5)
=
–1.4
©
Remember
It is
very useful
to
remember that
a
minus sign
is a
—
1,
so —5 is the
same
as
—

1 x 5
ESSENTIAL
MATHEMATICS
FOR
ECONOMICS
AND
BUSINESS
(i)
(j)
00
(1)
(m)
(n)
(o)
5(7)
= 35
(–5)(–7)
= 35
(–5)y
= –5y
(-x)(-y)=xy
Remember
= x(x + 2) + 4(x + 2)
= x
2
+ 2x + 4x + 8
= x
2
+ 6x + 8
0

x
(any real number)
= 0
0
-r-
(any real number)
= 0
But
you
cannot
divide
by 0
= x(x +
y)+y(x
= xx +
xy+yx
+ yy
= x
2
+ 2xy + y
2
The
following
identities
are
important:
multiply
each term inside
the
bracket

by the
term
outside
the
bracket
multiply
the
second bracket
by x,
then
multiply
the
second bracket
by
(+4)
and
add,
multiply
each bracket
by the
term outside
it
add or
subtract similar terms, such
as
2x
+ 4x = 6x
multiply
the
second bracket

by x and
then
by y
add
the
similar terms:
xy + yx = 2xy
1.
(x + y)
2
= x
2
+ 2xy + y
2
2.
(
x
-
y
f=
x
2
-2xy
+ y
2
3.
(x +
y)(x-y)=x
2
-y

2
©
Remember: Brackets
are
used
for
grouping terms
together
in
maths
for:
(i)
Clarity
(ii) Indicating
the
order
in
which
a
series
of
operations should
be
carried
out
1.2
Fractions
Terminology:
fraction
=

numerator
denominator
3 is
called
the
numerator
7
is
called
the
denominator
1.2.1
Add/subtract
fractions:
method
The
method
for
adding
or
subtracting fractions
is:
Step
1:
Take
a
common denominator, that
is, a
number
or

term which
is
divisible
by the
denominator
of
each fraction
to be
added
or
subtracted.
A
safe
bet is to use the
product
of all the
individual
denominators
as the
common denominator.
MATHEMATICAL
PRELIMINARIES
5
Step
2: For
each fraction, divide each denominator into
the
common denominator, then
multiply
the

answer
by the
numerator.
Step
3:
Simplify
your answer
if
possible.
WORKED
EXAMPLE
1.3
ADD AND
SUBTRACT FRACTIONS
Each
of the
following
illustrates
the
rules
for
addition
and
subtraction
of
fractions.
Numerical
example
1 2_4
7

+
3~5
Step
1: The
common denominator
is
(7)(3)(5)
1
2 4
Step
2:-
+

Step
3:
2(7)(5)-4(7)(3)
15
+ 70- 84 1
105
105
I ?
Step
1: The
common denominator
is
(7)(3)
1
2
=
1(3)+

2(7)
73
(7)(3)
_3+14_
17
21
~2l
Step
3:
Same example,
but
with
variables
x 2x 4x
7
+
l
5~
x(3)(5)+2x(7)(5)-4x(7)(3)
105
x
105
1
2
1(x)
x + 4 ' x (x +
4)(x)
x
+ 2x + 8
x

2
+ 4x
4x
1.2.2 Multiplying fractions
In
multiplication, write
out the
fractions, multiply
the
numbers across
the top
lines
and
multiply
the
numbers
across
the
bottom lines.
Note:
Write whole numbers
as
fractions
by
putting them over
1.
Terminology:
RHS
means right-hand side
and LHS

means
left-hand
side.
ESSENTIAL MATHEMATICS
FOR
ECONOMICS
AND
BUSINESS
WORKED
EXAMPLE
1.4
MULTIPLYING FRACTIONS
Ca)
/2W5\
(2)(5)_10
1
' ' VW
(3)(7)
21
,u. /
2W7\
(-2M7)
-14
(b)
I -T
3/V5/
(3)(5)
15
(C
)

3
x
2
(
3
-}(
2
-}-VM-
6
l
l
-
(C)
3x
5-(\)(5)-(\)(5)-5'
l
5
The
same
rules apply
for
fractions involving variables,
.x, v,
etc.
(d)
, 3\
(x
+
3)
3(.x

+ 3)
3.x
+
jc/
(.x-5)
.x(.x-5) .x
2
-5.x
1.2.3
Dividing
fractions
General
rule:
Dividing
by a
fraction
is the
same
as
multiplying
by the
fraction inverted
WORKED
EXAMPLE
1.5
DIVISION
WITH
FRACTIONS
The
following examples illustrate

how
division with fractions
operates.
3J
/2\/ll\
22
/5\
V"/
5\ 3s 15
"
(b)
5
f
4 5 4 20 , 2
v
/ — ^v — v — — r»
~ 3~1 3~3 3
(
c
\ \"/ — \"/
—
_
x
1
—
_
(0
°-e~38~24