Essential Mathematics
for
Economics
and
Business
Second Edition
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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,


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
form
or by any
means,
electronic,
mechanical,
photocopying,
recording,
scanning
or
otherwise, except under
the
terms

Paul Patton have asserted their rights under
the
Copyright, Designs
and
Patents
Act
1988.
to be
identified
as the
authors
of
this work.
Other
Wiley
Editorial
Offices
John
Wiley
&
Sons, Inc.,
605
Third
Avenue,
New
York.
NY
10158-0012,
USA
WILEY-VCH

&
Sons (Canada) Ltd,
22
Worcester
Road,
Rexdale.
Ontario
M9W
1L1, Canada
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

each
one
used
for
paper
production.
To my
parents,
Lily
and
Richard T.B.
To my
mother
and the
memory
of my
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

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

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

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

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

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

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

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

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

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

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

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

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

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
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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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
•

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)

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

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:

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

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.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

(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

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

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

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

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
=

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

+
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
+ 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

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

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