Session 1: Basic Arithmetic Basic Maths Session 1: Basic Arithmetic [NB class register and give out handouts] START 12.40pm: Good afternoon and welcome to the Basic Maths sessions [introductions]. Remember that these sessions are just meant to refresh your basic maths skills and not turn you into mathematicians. These maths skills are needed for some of your Study Modules, e.g. Basic Statistics and Basic Epidemiology, as well as the more advanced statistics and epidemiology modules. If once you’ve looked through the handouts you think you’d be better off going through the exercises at home or that you don’t really need to be at these classes then please feel free to leave. I have a few slides – 1 or 2 for each section. So I’ll go through the first couple of slides and then if you have a go at section 1 (fractions) on the exercises sheets we’ll come round and help you and then you should have a bit of time to check over at least a few of the answers (which you have in your handouts) before I move on to going through the overheads about section 2 (decimals) and continue like that – quite fast-paced I’m afraid – so best to spend some time working through the exercises and further exercises at home too! Please keep an eye on your email and in reception since the groups and rooms may vary slightly for the other 3 sessions.

Intended learning objectives At the end of this session you should be able to: add, subtract, multiply and divide fractions convert fractions to decimals and decimals to fractions, multiply and divide decimals by multiples of 10 change a fraction to a percentage, find a percentage of a number, increase/decrease a number by a percentage round whole numbers and decimals use ratio and dividing in proportion understand the order of arithmetic operations and use some basic calculator functions Aims and Objectives of Session 1 At the end of this session you should have some basic knowledge of the following key concepts in arithmetic: fractions, decimals, percentages, rounding, ratio and proportion and order of operations. You should also have some understanding of the use of calculators in arithmetic and some indication of topics in your Term 1 modules which use these techniques.   In particular you should be able to: 1) add, subtract, multiply and divide fractions 2) convert fractions to decimals and decimals to fractions, multiply and divide decimals by multiples of 10 3) change a fraction to a percentage, find a percentage of a number, increase/decrease a number by a percentage 4) round whole numbers and decimals 5) use ratio and dividing in proportion 6) understand the order of arithmetic operations and use some basic calculator functions

§ 1. Fractions (simplifying and expanding) If we multiply (or divide) BOTH numerator and denominator by the same number we don’t change the value of the expression ‘simplifying’ or ‘cancelling down’ (÷ top and bottom by 3) ‘expanding’ (x top and bottom by 2) Definition: A fraction is a portion of a whole It’s a common concept that we use in everyday life without really thinking it’s maths! E.g. We often say things like “would you like half of this cake?” We call the top number the numerator and the bottom number the denominator. So that we don’t have big numbers that we can’t deal with easily in our heads, we try to ‘simplify’ our fractions by a technique called ‘cancellation’ or ‘cancelling down a fraction’. We can use this technique if we can divide both the numerator and the denominator by the same number. E.g. 6/9 = 2/3 by dividing the top and bottom number both by 3. If we do the reverse of cancellation by multiplying both the numerator and the denominator by the same number then we call this ‘expanding a fraction’. E.g. 3/4 = 6/8 by multiplying the top and bottom number both by 2.

§ 1. Fractions (multiplying and dividing) Multiplying fractions multiply numerators multiply denominators simplify Dividing fractions invert second (to get the ‘reciprocal’) and multiply by first To multiply fractions together we multiply the top numbers together and multiply the bottom numbers together. See the example on the left-hand-side of this slide (where we cancel at the end by dividing through by 2). To divide one fraction by another we invert the one we are dividing by (i.e. turn it upside-down) and then multiply by it. This ‘upside-down’ fraction is called the ‘reciprocal’. See the example on the right-hand-side of this slide (where we turn 2/5 upside-down to get 5/2 and then multiply this by the first fraction of 1/9).

§ 1. Fractions (adding and subtracting) Find ‘common denominator’ Expand both fractions Add or subtract numerators To add and subtract fractions we first need to rewrite the fractions so that they have the same denominator – called the ‘common denominator’. Let us consider 2/5 + 1/7. We try to find the smallest number that can be divided by both the denominators (known as the ‘lowest common denominator’). Often it is not possible to find a common denominator that is smaller than the product of the two denominators. This is the case in our example where the common denominator is 5x7 = 35. So then we expand both fractions – in the first fraction (2/5) we need to multiply the denominator by 7 to get 35 and so we also need to multiply the numerator by 7 so that the fraction retains the same value (see slide). Then adding the two numerators we get 14+5 = 19 .

§ 1. Fractions (rewriting) You may need to rewrite any mixed fractions as improper fractions BEFORE performing these operations ‘mixed fraction’ to ‘improper fraction’ ‘improper fraction’ to ‘mixed fraction’ When, adding, subtracting, multiplying or dividing fractions involving mixed fractions, first rewrite the mixed fractions as improper fractions. A ‘mixed fraction’ is a whole number and a fraction. E.g. 4½ (“four and a half”). A fraction in which the numerator is bigger than the denominator (so it is ’top-heavy’) is called ‘improper’. E.g. 7/3 - This can also be written as a mixed fraction (as shown on the slide). You might wish to convert an improper fraction to a mixed fraction since the latter is generally conceptually easier. .

§ 2. Decimals Converting decimals to fractions: Convert fractions to decimals by dividing numerator by denominator: Multiplying a decimal by a multiple of 10: 3.27x10 = 32.7, 3.27x100 = 327, 3.27x1000 = 3270 Dividing a decimal by a multiple of 10: 43.1÷10 = 4.31, 43.1÷100 = 0.431, 43.1÷1000 = 0.0431 Decimals are fractions like tenths, hundredths, thousandths etc. which have denominators of 10 and multiples of 10 such as 100, 1000 etc. So we can convert a decimal into a fraction by taking the number before the decimal point as the whole number and the numbers after the decimal point as a fraction with the denominator as 10 for one decimal place, 100 for two decimal places, 1000 for three decimal places etc. E.g. We can convert a fraction to a decimal by dividing the numerator by the denominator. E.g. 1/8 = 1÷8 = 0.125. To multiply a decimal by 10 we move the decimal point one place to the right. To multiply by 100 we move the point two places to the right etc. E.g. 3.27x10 = 32.7, 3.27x100 = 327, 3.27x1000 = 3270 To divide a decimal by 10 we move the decimal point one place to the left. To divide by 100 we move the point two places to the left etc. E.g. 43.1÷10 = 4.31, 43.1÷100 = 0.431, 43.1÷1000 = 0.0431 BREAK FOR EXERCISES 12.50 – 1.05pm zero after decimal point is to keep place value

§ 2. Fractions/Decimals in Pictures 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 1/2 1/2 1/3 1/3 1/3 1/4 1/4 1/4 1/4 1/6 1/6 1/6 1/6 1/6 1/6 This slide shows a pictorial representation of the conversion between fractions and decimals The coloured blocks represent fractions of width 1/1, 1/2, 1/3, 1/4, 1/6 and 1/8, the decimal measurements are on the top and bottom scales By reading off the decimal values for the proportions above we can see that 1/4 =0.25 and similarly that 5/6=0.833. If you were to try the second example on your calculator you will find that 5/6 =0.83333333333333 etc., which has been rounded to 0.833 in the example. This will be covered in more detail in section 4. 1/10 1/10 1/10 1/10 1/10 1/10 1/10 1/10 1/10 1/10 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

§ 3. Percentages Percentages are fractions out of 100 and can be written as decimals Converting a fraction to a percentage: Percentage of a number: Increase/decrease a number by a percentage: e.g. To increase 5 by 60% RESTART 1.05pm: Percentages are fractions out of 100 and can also be written as decimals. E.g. To convert a fraction to a percentage we multiply the fraction by 100 (i.e. multiply the numerator by 100) and cancel down. To find a percentage of a number write the percentage as a fraction and multiply this by the number. To increase or decrease a number by a percentage work out that percentage of that number and then add or subtract it from that number. E.g. To increase 5 by 60% first work out 60% of 5: (Note: we divide numerator and denominator by 10 (cancel zeros) and then divide numerator and denominator by 5 (or by 2)). Then add 3 onto 5 which gives us 8. Try to cancel down fractions as early as possible in a calculation (so method shown on slide may be seen as preferable to 30/10=3 then 5+3=8). If we had to decrease 5 by 60% we would subtract 3 from 5 to give us 2. [or do 5x160% for 60% increase or 5x40% for 60% decrease]. or need to have 1 as denominator

§ 4. Rounding 7284 rounded to the nearest ten is 7280 Thousands Hundreds Tens Units 7 2 8 4 7284 rounded to the nearest ten is 7280 7284 rounded to the nearest hundred is 7300 Rounding decimals is similar e.g. 3.854 rounded to 1 decimal place is 3.9 To round up we think of the number in hundreds, tens and units columns etc. E.g. 7284 is 7 Thousands, 2 hundreds, 8 tens and 4 units. This can be written as Th H T U 7 2 8 4 Rounding 7284 to the nearest ten means looking at the tens column and the column after this (units column). If the number in the units column is 5 or more then we round up the number in the tens column. In this example the number in the units column is 4, which is less than 5, so we leave the tens column alone. In either case we replace the number in the units column with a zero. E.g. So 7284 rounded to the nearest ten is 7280. Rounding 7284 to the nearest hundred means looking at the hundreds and the column next to this (tens column). Since the number in the tens column is 8 (which is bigger than 5) we round up the number in the hundreds column from 2 to 3. We then put zeros in the tens and units columns to keep the right place value. E.g. So 7284 rounded to the nearest hundred is 7300. Rounding decimals is similar but then the columns become tenths, hundredths, etc. after the decimal point. Also we don’t need to fill in the zeros after the number as it keeps its place value without them. E.g. 3.854 rounded to 1 decimal place is 3.9 (by looking at the numbers in the 1st decimal place column and the 2nd decimal place column). Note: important to round up at end of calculation – not earlier on since then get rounding errors carried thorough/building up in a calculation e.g. in question 4 of exercises.

§ 5. Ratios and proportion A and B are in the ratio 1:2 1 + 2 = 3 15ml in ratio 1:2 A 5 B 10 In this example we want to set the amount of a substance (A) to be quantity 1 and then double this for the amount of the other substance (B) that is required (quantity of 2). So total quantity is 1+2 = 3. So we imagine the test tube in 3 segments. A third of the total test tube contains substance A and 2 thirds of the test tube contains substance B. If the total volume of the test tube is 15 ml then a third of 15ml will tell us how much of substance A we require (5 ml). Similarly, 2 thirds of 15ml will be the amount of substance B we require (10 ml). We write ratios as whole numbers and in the simplest terms. We do this by cancelling down or expanding, similar to what we did for fractions. To keep the fraction the same – whatever we did to the top number we also did to the bottom number. With ratios, whatever we do to the left number we do to the right number. Do not confuse ratios with fractions though since the ratio 2:3 is not the same as 2/3 etc.

§ 6. Order of operations Do calculation from left to right obeying ordering: Brackets (innermost 1st) Exponents Multiplication and Division Addition and Subtraction This is the order in which we do the various operations in calculations, working from left to right. Exponents will be explained in the session 3 – basically they are powers e.g. 2^3, or e.g. ½ of 16. Important to take note of the order of calculation or else can get different answers – e.g. in question 1 of exercises. [British: BODMAS (B = brackets, O = of/exponents) American: PEMDAS (P = parentheses, E = exponents)] BREAK FOR EXERCISES 1.15 – 1.30pm

§ 7. Applied problems 30ml of drug solution consists of two thirds drug A (costing 10p per ml), a sixth of drug B (costing 50p per ml) and rest of volume made up with water (no cost) How much does the whole solution cost? A B Total cost = How much water is required? RESTART 1.30PM 30ml of drug solution made up of two thirds drug A, a sixth of drug B, and topped up with water. Drug A costs 10 pence per millilitre while drug B costs 50 pence per millilitre, and we assume the water is free. How much does whole solution cost? A (since 100p = £1) (note cancellation of units of volume) B Total cost = How much water is needed? Using working from 1st question: …or if asked this 2nd question before other one: or

§ 8. Topics in Term 1 modules using basic maths skills EPIDEMIOLOGY: Incidence Point prevalence (proportion) Risk ratio Rate ratio Odds ratio Mortality rates …etc STATISTICS: Frequency Relative (percentage) frequency Arithmetic mean Standard deviation Confidence intervals p-values …etc [see list on slide] Calculator use – e.g. AC and DEL buttons and memory function example below. On CASIO fx-85GT to use memory to calculate (43x37)+(181-64) key in: 43 x 37 SHIFT RCL M+ 181 – 64 + RCL M+ = 1708 (SHIFT key is in top left corner. Note: When pressing SHIFT RCL the calculator uses the memory function STO). Could also just key in the brackets for this one!...but if it was a more complex calculation it might make more sense to use the memory function. Will usually already be in MODE 1 but if not MODE key is top right. See ‘Basic Statistics for PHP’ or Statistics for EPH’ or ‘Statistics with Computing’ notes. Remember to try to finish these and the further exercises off at home for extra practice. The further exercises have styles of questions not covered in the notes – so worth having a go at them.

Intended learning objectives (achieved?) You should be able to: add, subtract, multiply and divide fractions convert fractions to decimals and decimals to fractions, multiply and divide decimals by multiples of 10 change a fraction to a percentage, find a percentage of a number, increase/decrease a number by a percentage round whole numbers and decimals use ratio and dividing in proportion understand the order of arithmetic operations and use some basic calculator functions Aims and Objectives of Session 1 At the end of this session you should have some basic knowledge of the following key concepts in arithmetic: fractions, decimals, percentages, rounding, ratio and proportion and order of operations. You should also have some understanding of the use of calculators in arithmetic and some indication of topics in your Term 1 modules which use these techniques.   In particular you should be able to: 1) add, subtract, multiply and divide fractions 2) convert fractions to decimals and decimals to fractions, multiply and divide decimals by multiples of 10 3) change a fraction to a percentage, find a percentage of a number, increase/decrease a number by a percentage 4) round whole numbers and decimals 5) use ratio and dividing in proportion 6) understand the order of arithmetic operations and use some basic calculator functions

Key messages If we multiply or divide BOTH numerator and denominator by the same number then the value of the fraction stays the same To multiply fractions together we multiply the top numbers together and _______ the bottom numbers together To divide one fraction by another we invert the one we are dividing by and then _______ by it To add and subtract fractions we first need to rewrite the fractions so that they have the same ____________ Percentages are fractions out of ____ N.B. For next session: http://www.lshtm.ac.uk/edu/studyskills.html (subheading ‘Maths and Numeracy Skills’) multiply multiply ‘simplifying’ or ‘cancelling down’ – e.g. can divide top and bottom numbers of fraction both by 2 (or by any other number) without changing value of the fraction ‘expanding’ – same rule applies in reverse, e.g. can multiply top and bottom numbers of fraction both by 2 (or by any other number) without changing value of the fraction To multiply fractions together we multiply the top numbers together and multiply the bottom numbers together To divide one fraction by another we invert the one we are dividing by (i.e. turn it upside-down) and then multiply by it - The inverted fraction is called the ‘reciprocal’ of the original fraction To add and subtract fractions we first need to rewrite the fractions so that they have the same denominator – called the ‘common denominator’ Percentages are fractions out of 100 – so 9% is 9/100 (‘9 hundredths’) BREAK FOR EXERCISES 1.45 – 1.55pm denominator 100 16