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/

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

Agenda

1.   Looking at proportions 2.  Basic algebra 3.  Compound Annual Growth Rates (CAGR) 4.  Back of the Envelope Calculations (BotEC)

3

Proportions

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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

Percentages

Use

Dynamic %

Formula

Finding the proportion of a given fixed size Static %

Finding the proportional change between two values measured over different time periods

5

€

Part =% ×Whole

€

%Δ =Absolute Δ

Original value

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

1.  What’s 20% of 48,200?

Percentages

7

1.  What’s 20% of 48,200?

Percentages

8

€

Part =% ×Whole

€

= 0.2 × 48,200

€

= 9,640

Percentages

2. How much would you save?

Was £20

Now 10% off

9

Percentages

2. How much would you save?

Was £20

Now 10% off

10

€

Part =% ×Whole

€

= 0.10 × 20

€

= £2

Percentages

3. Which product is cheaper?

Was £12

Now 40% off

Was £18

Now 50% off

11

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 %?

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?

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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%

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

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 −114127

€

=10.2%

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?

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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

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

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

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 × 7,800Calculate 108% of £7,800

€

= £8,424

€

0.08 × 7,800 = Absolute Δ

€

= 624

€

∴New value = 7800 + 624 = 8,424

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

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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

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?

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What’s the new %?

€

= 0.92 × 7,800Calculate 92% of £7,800

€

= £7,176

Agenda

1.  Looking at proportions 2.   Basic algebra 3.  Compound Annual Growth Rates (CAGR) 4.  Back of the Envelope Calculations (BotEC)

25

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

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

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? ?

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€

PW =1+ (0.5 × PW )

€

PW − 0.5PW =1

€

0.5PW =1

€

PW = £2

Algebra

10.If you buy a computer for £680, how much VAT have you paid?

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Note: £680 includes the VAT, therefore we ask two questions:

10a. What is the price before VAT gets added? 10b. What was the VAT?

Algebra

10.If you buy a computer for £680, how much VAT have you paid?

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€

680 = RRP + 0.175RRP

€

680 = RRP(1+ 0.175)

€

6801.175

= 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

Algebra

11.If you buy a computer for £5,000, how much VAT have you paid?

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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

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Algebra

12. A jacket costs £185.00 inc vat. What is the cost excluding vat?

33 http://www.thehogman.co.uk/www.thehogman.co.uk/info.php?p=26&pno=0

Algebra

12. A jacket costs £185.00 inc vat. What is the cost excluding vat?

34 http://www.thehogman.co.uk/www.thehogman.co.uk/info.php?p=26&pno=0

Agenda

1.  Looking at proportions 2.  Basic algebra 3.   Compound Annual Growth Rates (CAGR) 4.  Back of the Envelope Calculations (BotEC)

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CAGR: Average Growth

•  A company boasts that they’ve achieved average growth of 25% in the last two years. Are you impressed?

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Year Value Percentage Return

2004 £1,000

2005 £2,000 + 100%

2006 £1,000

- 50%

Average growth:

%252

%50%100=

−

But growth = 0

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:

%252

%50%100=

−

But growth = 0

CAGR: Formula

•  Compound Annual Growth Rate (CAGR) is given by:

1

1

−⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎟⎠

⎞⎜⎝

⎛n

ValueBeginningValueEndCAGR

38

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

€

=1,404,000845,200

"

# $

%

& '

14"

# $

%

& '

−1

€

=13.5%

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

€

=1,404,000845,200

"

# $

%

& '

14"

# $

%

& '

−1

€

=13.5%

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

€

=38.227.8"

# $

%

& '

13"

# $

%

& '

−1

€

=11.1%

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

€

=38.227.8"

# $

%

& '

13"

# $

%

& '

−1

€

=11.1%

Agenda

1.  Looking at proportions 2.  Basic algebra 3.  Compound Annual Growth Rates (CAGR) 4.   Back of the Envelope Calculations (BotEC)

43

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

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

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

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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

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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

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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

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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)

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•  This presentation forms part of a free, online course on analytics

•  http://econ.anthonyjevans.com/courses/analytics/

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