Transcript Chapter 2

Chapter 2: Basic Sums
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Cengage
Objectives
•
•
•
•
•
Deal with basic algebra
Combine expressions involving powers
Recognise and use basic functions
Construct graphs of equations
Perform frequency counts
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Cengage
Remember this?
When you are doing
calculations, it is
important to do things
in the right order.
Most people remember
this as BEDMAS
Brackets
Exponentiation
Division
Multiplication
Addition
Subtraction
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Cengage
For Example -
20 + 6 x 3
We multiply first
20 + 18
= 38
(20 + 6) x 3
We calculate brackets first
26 x 3
= 78
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Cengage
What is Algebra?
• It uses letters instead of numbers
• It is a way of generalising a calculation
• That is, it creates a formula which can be used
over and over again for similar sums,
• Or it can be used to explain to others what to
do
• We need to be able to manipulate algebraic
expression to help understand relationships
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Cengage
For Example (2)
So if a person saves 10% of their income, we could write this as:
0.1 * I
where I is the person’s income
This could be extended by letting s represent the proportion saved,
Then the amount saved is:
sI
Now we have a (very simple ) formula which shows the amount
saved for anyone
This is a very simple example, but we use algebra over and
over again both to generalise and to write down formulae.
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Cengage
Working with Powers
• When trying to work something out, we often find that we
are multiplying the same number over and over again, for
example:
• This happens when we try to work out interest and return
on sums of money
• It also happens when we are looking at probability
• Powers are a sort of short-hand, instead of writing out the
individual items or terms,
• So 2 x 2 x 2 is
• = 23
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Cengage
Powers (2)
You can extend the idea of powers to algebra,
So
a x a x a x a = a4
When a number raised to a power is multiplied by the same
number raised to a power
You add the powers:
a3 x a6 = a3+6 = a9
If the two are divided, then you subtract the powers:
a6 /a3 = a6-3 = a3
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Cengage
Powers (3)
There are a few special cases, for example:
a½ is the square root of a
Because
a½+½ = a1 = a
Using the same logic, a¼ is the fourth root of a
Think of a3 /a3
You get
a3-3 = a0
Which must be 1
So anything raised to the power zero is equal to 1
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Cengage
Graphs
• Provide a visual representation of a function
• Illustrate standard shapes
• These can be used to make comparisons to
“real world” situations
• Can be used to help explain a situation to
others who, maybe, can’t do the algebra
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Cengage
Graphs (2)
Have 2 axes, often
labelled X and Y
Y
Where they cross, X and
Y are both zero, called
the origin
10
.
8
Any point can be uniquely
identified by the X and Y
values
So this point is
labelled (8,10)
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Cengage
X
A Few Functions
• Constant
– Something which stays the same
– Used for things like fixed costs
• Linear
– Surprisingly powerful function
– Works well even if “real” situation is not quite linear
• Quadratic
– Often used for cost curves
– Arise when try to solve problems algebraically
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Cengage
Constant
Y
Y=k
k
X
A constant has the same value, whatever the value of X
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Cengage
Linear Function
Y
14
Y=4+X
Intercept
4
10
A linear function changes proportionally to the X value
It has an equation of the form Y = a + bX
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Cengage
X
Linear (2)
Y
100
Y = 100 – 2.5X
50
20
When the value of b in the equation is negative
The graph looks like this
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Cengage
X
Quadratic Function
A quadratic has one bend,
either like this
or like this
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Cengage
Drawing a Quadratic
The easiest way is to
use a spreadsheet
Put a series of X
values into a column
then calculate the
parts of the function
Finally add across
the rows to get the Y
value
Y=-2x2+5x+120
=b19+c19+d19
=-2*a19*a19
=5*a19
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Cengage
Drawing a Quadratic (2)
Now we can plot the X and Y values, either by hand or using the
spreadsheet software
140
120
100
80
60
40
20
0
-20
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
-40
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Cengage
9
10
Roots of a Quadratic
A root is where the
function crosses the
X-axis
(if it does)
140
120
100
Here we can see
the roots are at
X=2 and X=6
80
60
40
20
0
0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15
-20
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Cengage
Roots of a Quadratic (2)
The roots can also be found algebraically
The standard function is
Y = aX2 + bX + c
Either by breaking the function into 2 brackets,
(X + p)(X + q)
So that p times q = c, and p + q = b
For example:
if Y = X2 – 8X + 12
This can be broken down to:
(X – 6)(X – 2) = 0 for roots
So either (X – 6) = 0 and X = 6
Or
(X – 2) = 0 and X = 2
These are the 2 roots
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Cengage
Roots of a Quadratic (3)
If you find the idea of finding brackets difficult, you can always
use the formula
 b  (b 2  4ac)
2a
For Y = X2 – 8X + 12
Remember Y=aX2+bX+c
So a = 1; b = -8; and c = 12
So we have :
 (8)  (8) 2  4(1)(12)
2(1)
8  64  48 8  16 8  4


2
2
2
12
4
or  6 or 2
2
2
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Cengage
Simultaneous Equations
We often find a situation where two equations must both be true
These are called simultaneous equations
For example:
2X + 5Y = 26 – equation 1
X + 10Y = 43 – equation 2
We want to find the vales of X and Y for which they are both true.
To do this we must make the coefficients of one of the variables equal on
both equations,
Here we would multiply the first equation by 2; 4X + 10Y = 52
then subtract one from the other, to get
3X
= 9,
so X = 3
now substitute this value into one equation
3 + 10Y = 43, so 10Y = 40, and Y =10
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Cengage
Simultaneous Equations (2)
You may only have used simultaneous equations for
“maths” exercises at school,
but they will be particularly useful when we look at
Linear Programming
If you have a module in Economics, you will also find
yourself using simultaneous equations to find things like
Market Equilibrium
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Cengage
Frequency Counts
Finally we will look at simple counting.
For example, with a set of questionnaire results, there are
only a few different answers,
so we can count up how many of each
These are called frequency counts
Such tables make it much easier to understand the data
For example:-
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Cengage
Frequency Counts (2)
If we have this data, where 1 = Yes and 2 = No
1
2
1
2
1
2
2
1
2
2
2
2
2
1
2
1
2
1
2
1
2
1
1
1
2
2
2
1
2
2
By counting up, we get:
Answer
Code
Frequency
Yes
1
12
No
2
18
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Cengage
Conclusions
• These slides cover very basic materials
• They should remind you of things you have done in the
past
• None are difficult in themselves
• They do form the basis of much of what will be covered in
the course
• If you are not comfortable with these topics, ask someone
for help now
• Don’t just sit back and ignore them
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Cengage