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Part 1. Fundamentals of Mathematics

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- 1. Numeracy Skills Part 1. Fundamentals of Mathematics Anthony J. Evans Associate Professor of Economics, ESCP Europe www.anthonyjevans.com London, February 2015 (cc) Anthony J. Evans 2015 | http://creativecommons.org/licenses/by-nc-sa/3.0/
- 2. Description • Fundamentals of Mathematics is usually a pre-term course that provides a basis for the numerical literacy required by other courses on an MBA programme • This course is intended to be a short refresher for students wishing to gain general confidence with numbers, and will provide an opportunity to practice the types of numeracy tests used in graduate recruitment • I will assume that you have little or no mathematical training so basic terminology and methods will be explained 2
- 3. Agenda 1. Looking at proportions 2. Basic algebra 3. Compound Annual Growth Rates (CAGR) 4. Back of the Envelope Calculations (BotEC) 3
- 4. Proportions 25 100 25% 0.25= = Fractions A quotient of numbers Percentages “Percent” means “per 100” Decimal Relating to powers of 10 • There are three equivalent ways to express a proportion 4
- 5. Percentages Use Dynamic % Formula Finding the proportion of a given fixed sizeStatic % Finding the proportional change between two values measured over different time periods 5 € Part = % ×Whole € %Δ = Absolute Δ Original value
- 6. Percentages • 20 is half of 40 We can write this in different ways: 20 is 50% of 40 20 is 1/2 of 40 20 is 0.5 of 40 6 € Part = % ×Whole
- 7. 1. What’s 20% of 48,200? Percentages 7
- 8. 1. What’s 20% of 48,200? Percentages 8 € Part = % ×Whole € = 0.2 × 48,200 € = 9,640
- 9. Percentages 2. How much would you save? Was £20 Now 10% off 9
- 10. Percentages 2. How much would you save? Was £20 Now 10% off 10 € Part = % ×Whole € = 0.10 × 20 € = £2
- 11. Percentages 3. Which product is cheaper? Was £12 Now 40% off Was £18 Now 50% off 11
- 12. Percentages 3. Which product is cheaper? Was £12 Now 40% off Was £18 Now 50% off 12 = 12 - (0.40) * 12 = 12 - 4.80 = £7.20 = 18 - (0.50) * 18 = 18 - 9 = £9.00 As before… = (0.60) * 12 = £7.20 = (0.50) * 18 = £9.00 What’s the new %?
- 13. Percentages 4. Suppose the profits of a certain company go from £365 000 in January to £425 000 in February. What is the % increase in their profits? 13
- 14. Percentages 4. Suppose the profits of a certain company go from £365 000 in January to £425 000 in February. What is the % increase in their profits? 14 € %Δ = Absolute Δ Original value € = 425,000 − 365,000 365,000 € = 0.164 € =16.4%
- 15. Percentages 5. The number of first year students at a certain university studying Law was 127 in 1996 and 114 in 1997. What was the % decrease? 15
- 16. Percentages 5. The number of first year students at a certain university studying Law was 127 in 1996 and 114 in 1997. What was the % decrease? 16 € %Δ = Absolute Δ Original value € = 127 −114 127 € =10.2%
- 17. Percentages 6. An antique jug is now worth 25% more than when it was first bought. The original price was £40. How much is it worth now? 17
- 18. Percentages 6. An antique jug is now worth 25% more than when it was first bought. The original price was £40. How much is it worth now? 18 € %Δ = Absolute Δ Original value € %Δ × Original value = Absolute Δ € 0.25 × 40 = Absolute Δ € =10 Rearrange the formula… € ∴New value = 40 +10 = 50
- 19. Percentages 6. An antique jug is now worth 25% more than when it was first bought. The original price was £40. How much is it worth now? 19 € =1.25 × 40 What’s the new %? Calculate 125% of £40 € = 50
- 20. Percentages 7. The price of a certain model of car goes up by 8%. It used to cost £7,800. What does it cost now? 20
- 21. Percentages 7. The price of a certain model of car goes up by 8%. It used to cost £7,800. What does it cost now? 21 Rearrange the formula… What’s the new %? € =1.08 × 7,800 Calculate 108% of £7,800 € = £8,424 € 0.08 × 7,800 = Absolute Δ € = 624 € ∴New value = 7800 + 624 = 8,424
- 22. Percentages 7. The price of a certain model of car goes up by 8%. It used to cost £7,800. What does it cost now? Note: here’s the algebra… New value = Original value + Absolute change New value = 7,800 + (0.08 x 7,800) New value = 7,800 x (1 + 0.08) New value = 7,800 x 1.08 = £8424 22
- 23. Percentages 8. The price of a certain model of car goes down by 8%. It used to cost £7,800. What does it cost now? 23
- 24. Percentages 8. The price of a certain model of car goes down by 8%. It used to cost £7,800. What does it cost now? 24 What’s the new %? € = 0.92 × 7,800 Calculate 92% of £7,800 € = £7,176
- 25. Agenda 1. Looking at proportions 2. Basic algebra 3. Compound Annual Growth Rates (CAGR) 4. Back of the Envelope Calculations (BotEC) 25
- 26. Algebra • Algebra can be useful since it denotes numbers as symbols (e.g. a, b, c etc) – This helps us to find general rules and arithmetic laws – It allows us to recognise unknown numbers – It allows functional relationships 26
- 27. Algebra 9. The price of a widget is £1 plus half of the total price. How much would you have to pay to buy one? ? 27
- 28. Algebra 9. The price of a widget is £1 plus half of the total price. How much would you have to pay to buy one? ? 28 € PW =1+ (0.5 × PW ) € PW − 0.5PW =1 € 0.5PW =1 € PW = £2
- 29. Algebra 10.If you buy a computer for £680, how much VAT have you paid? 29 Note: £680 includes the VAT, therefore we ask two questions: 10a. What is the price before VAT gets added? 10b. What was the VAT?
- 30. Algebra 10.If you buy a computer for £680, how much VAT have you paid? 30 € 680 = RRP + 0.175RRP € 680 = RRP(1+ 0.175) € 680 1.175 = RRP € £578.82 = RRP Note: £680 includes the VAT, therefore we ask two questions: 10a. What is the price before VAT gets added? 10b. What was the VAT? € = 0.175 × 578.72 € = £101.28
- 31. Algebra 11.If you buy a computer for £5,000, how much VAT have you paid? 31
- 32. Algebra 11.If you buy a computer for £5,000, how much VAT have you paid? Need to calculate the original price: • 5,000/(1+0.175) = P • P = £4255.32 Solution: 0.175 * 4255.32 • VAT = £744.68 32
- 33. Algebra 12. A jacket costs £185.00 inc vat. What is the cost excluding vat? 33http://www.thehogman.co.uk/www.thehogman.co.uk/info.php?p=26&pno=0
- 34. Algebra 12. A jacket costs £185.00 inc vat. What is the cost excluding vat? 34http://www.thehogman.co.uk/www.thehogman.co.uk/info.php?p=26&pno=0
- 35. Agenda 1. Looking at proportions 2. Basic algebra 3. Compound Annual Growth Rates (CAGR) 4. Back of the Envelope Calculations (BotEC) 35
- 36. CAGR: Average Growth • A company boasts that they’ve achieved average growth of 25% in the last two years. Are you impressed? 36 Year Value Percentage Return 2004 £1,000 2005 £2,000 + 100% 2006 £1,000 - 50% Average growth: %25 2 %50%100 = − But growth = 0
- 37. CAGR: Average Growth • A company boasts that they’ve achieved average growth of 25% in the last two years. Are you impressed? 37 Year Value Percentage Return 2004 £1,000 2005 £2,000 + 100% 2006 £1,000 - 50% Average growth: %25 2 %50%100 = − But growth = 0
- 38. CAGR: Formula • Compound Annual Growth Rate (CAGR) is given by: 1 1 −⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ n ValueBeginning ValueEnd CAGR 38
- 39. CAGR: Example • In 2002, 845.2m units were shipped globally. Unit shipments are expected to reach 1.404bn in 2006 • What is the CAGR for the global smartcard market? 39 € = 1,404,000 845,200 " # $ % & ' 1 4 " # $ % & ' −1 € =13.5%
- 40. CAGR: Example • In 2002, 845.2m units were shipped globally. Unit shipments are expected to reach 1.404bn in 2006 • What is the CAGR for the global smartcard market? 40 € = 1,404,000 845,200 " # $ % & ' 1 4 " # $ % & ' −1 € =13.5%
- 41. CAGR: Example • TowerGroup estimates that the financial services industry’s global IT spending on outsourcing services will grow from $27.8bn in 2003 to $38.2bn in 2006 • What’s the CAGR? 41 € = 38.2 27.8 " # $ % & ' 1 3 " # $ % & ' −1 € =11.1%
- 42. CAGR: Example • TowerGroup estimates that the financial services industry’s global IT spending on outsourcing services will grow from $27.8bn in 2003 to $38.2bn in 2006 • What’s the CAGR? 42 € = 38.2 27.8 " # $ % & ' 1 3 " # $ % & ' −1 € =11.1%
- 43. Agenda 1. Looking at proportions 2. Basic algebra 3. Compound Annual Growth Rates (CAGR) 4. Back of the Envelope Calculations (BotEC) 43
- 44. BotEC • “Back of the Envelope” means rough calculations • Tests analytic abilities • Requires logical thought process and ease with numbers • Somewhere between a guess and a proof – Demonstrate a structured thought process to arrive at a numerical answer • Many alternative ways to proceed • Have to use assumptions, and therefore justify them 44
- 45. BotEC: TV Sets • You are consulting an advertising agency who wish to launch a major television advert campaign in the US, and they ask you to estimate the potential market size • Proxy: How Many TV Sets in the US? • Two variables – # Households 100m 100m – # TVs per household 2 2.4 • Total # TV Sets 200m 240m 45
- 46. BotEC: TV Sets • You are consulting an advertising agency who wish to launch a major television advert campaign in the US, and they ask you to estimate the potential market size • Proxy: How Many TV Sets in the US? • Two variables – # Households 100m 100m – # TVs per household 2 2.4 • Total # TV Sets 200m 240m 46
- 47. 47
- 48. BotEC: UK Ringtone Market • A French media company is intending on breaking into the UK ringtone market (in 2006), but require an estimate of the size of the market • Split the question up – UK Population 60m – Mobile phone penetration rate 80% – Users who download ringtones 1/3 – Annual av. no. of ring tones 18 – Price per ringtone 1 • Market Size £288m 48
- 49. BotEC: UK Ringtone Market • A French media company is intending on breaking into the UK ringtone market (in 2006), but require an estimate of the size of the market • Split the question up – UK Population 60m – Mobile phone penetration rate 80% – Users who download ringtones 1/3 – Annual av. no. of ring tones 18 – Price per ringtone 1 • Market Size £288m 49
- 50. BotEC: ESCP Europe • A Middle-East consortium wish to enter the market for European business education. They realise that the most important resource in a business school is the quality of the faculty, and they have identified ESCP Europe as being especially world-class (in particular the London campus) • They have hired your team to provide a ball park estimate of the current market value of ESCP Europe 50
- 51. BotEC: Useful Figures USA • Population: 300m (US Census Bureau estimate, 2006) • GNI per capita: US $43,740 (World Bank, 2006) UK • Population: 60.2 million (National Statistics, 2005) • GNI per capita: US $37,600 (World Bank, 2006) India • Population: 1.1 billion (UN, 2005) • GNI per capita: US $720 (World Bank, 2006) China • Population: 1.3 billion (UN, 2005) • GNI per capita: US $1,740 (World Bank, 2006) 51
- 52. • This presentation forms part of a free, online course on analytics • http://econ.anthonyjevans.com/courses/analytics/ 52

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