Chapter 2: Basic Sums

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Chapter 2 Basic Sums
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Objectives

• Deal with basic algebra
• Combine expressions involving powers
• Recognise and use basic functions
• Construct graphs of equations
• Perform frequency counts

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Remember this?

• Brackets
• Exponentiation
• Division
• Multiplication
• Addition
• Subtraction

When you are doing calculations, it is important
to do things in the right order. Most people
remember this as BEDMAS
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For Example -
20 6 x 3 We multiply first 20 18 38
(20 6) x 3 We calculate brackets first 26 x 3
78
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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

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For Example (2)
So if a person saves 10 of their income, we
could write this as 0.1 I where I is the
persons 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.
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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

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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 a36
a9 If the two are divided, then you subtract the
powers a6 /a3 a6-3 a3
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Powers (3)
There are a few special cases, for example a½ is
the square root of a Because a½½ 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
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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, cant do the algebra

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Graphs (2)
Have 2 axes, often labelled X and Y
Y
.
Where they cross, X and Y are both zero, called
the origin
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X
Any point can be uniquely identified by the X and
Y values
So this point is labelled (8,10)
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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

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Constant
Y
Yk
k
X
A constant has the same value, whatever the value
of X
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Linear Function
Y
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Y 4 X
Intercept
4
10
X
A linear function changes proportionally to the X
value It has an equation of the form Y a bX
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Linear (2)
Y
100
Y 100 2.5X
50
20
X
When the value of b in the equation is
negative The graph looks like this
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Quadratic Function
A quadratic has one bend, either like this
or like this
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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
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Drawing a Quadratic (2)
Now we can plot the X and Y values, either by
hand or using the spreadsheet software
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Roots of a Quadratic
A root is where the function crosses the X-axis
(if it does)
Here we can see the roots are at X2 and X6
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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
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Roots of a Quadratic (3)
If you find the idea of finding brackets
difficult, you can always use the formula
For Y X2 8X 12 Remember YaX2bXc So a
1 b -8 and c 12 So we have
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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
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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
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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-
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Frequency Counts (2)
If we have this data, where 1 Yes and 2 No
By counting up, we get
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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
• Dont just sit back and ignore them