Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN © Cengage Chapter 2: Basic Sums

Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN © 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 © Cengage 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

Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN © Cengage For Example x 3 We multiply first = 38 (20 + 6) x 3 We calculate brackets first 26 x 3 = 78

Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN © 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 © 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 © 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 = 2 3

Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN © Cengage Powers (2) You can extend the idea of powers to algebra, So a x a x a x a = a 4 When a number raised to a power is multiplied by the same number raised to a power You add the powers: a 3 x a 6 = a 3+6 = a 9 If the two are divided, then you subtract the powers: a 6 /a 3 = a 6-3 = a 3

Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN © Cengage Powers (3) There are a few special cases, for example: a ½ is the square root of a Because a ½+½ = a 1 = a Using the same logic, a ¼ is the fourth root of a Think of a 3 /a 3 You get a 3-3 = a 0 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 © 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 © Cengage Graphs (2) Have 2 axes, often labelled X and Y X Y Where they cross, X and Y are both zero, called the origin 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 © Cengage 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 © Cengage Constant X Y k Y=k A constant has the same value, whatever the value of X

Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN © Cengage Linear Function A linear function changes proportionally to the X value It has an equation of the form Y = a + bX X Y Y = 4 + X Intercept

Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN © Cengage Linear (2) When the value of b in the equation is negative The graph looks like this X Y Y = 100 – 2.5X

Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN © Cengage Quadratic Function A quadratic has one bend, either like this or like this

Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN © 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

Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN © Cengage Drawing a Quadratic (2) Now we can plot the X and Y values, either by hand or using the spreadsheet software

Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN © Cengage 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 X=2 and X=6

Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN © Cengage Roots of a Quadratic (2) The roots can also be found algebraically The standard function is Y = aX 2 + 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 = X 2 – 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 © Cengage Roots of a Quadratic (3) If you find the idea of finding brackets difficult, you can always use the formula For Y = X 2 – 8X + 12 Remember Y=aX 2 +bX+c So a = 1; b = -8; and c = 12 So we have :

Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN © 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 Y = 43, so 10Y = 40, and Y =10

Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN © 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 © 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 © Cengage Frequency Counts (2) If we have this data, where 1 = Yes and 2 = No By counting up, we get: AnswerCodeFrequency Yes112 No218

Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN © 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