Part 1. Fundamentals of Mathematics

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. Percentages vs. percentage points
• “We produced only 28% more faults than the industry
average”
25A percentage is a portion of the whole (where the whole isn’t necessarily 100)
A percentage point is a unit of measurement that is calculated as a portion of 100

26. Percentages vs. percentage points
• “We produced only 28% more faults than the industry
average”
• Actually, it was 28 percentage points higher
• To calculate the percentage difference, you have to divide 28
by 35
• 80% more faults than the national average
26A percentage is a portion of the whole (where the whole isn’t necessarily 100)
A percentage point is a unit of measurement that is calculated as a portion of 100

27. Examples of the conflation of percentages and percentage points
Nationwide has upped the cost of its fixed-rate deals by up to
0.86%, and state-owned Northern Rock has raised its five-year
fixed rates by 0.2%, both with effect from tomorrow*
27*“Buyers face hike in mortgage rates as inflation fears mount” The Guardian, 11th June 2009,
**“Lenders rush to raise fixed-rate mortgages “ The Times 12th June 2009
On Wednesday, Times Online revealed that Nationwide Building
Society, Britain's third biggest lender, was putting up rates by up to
0.86 percentage points today, the biggest hike in mortgage rates
for months. A five-year fix has jumped from 4.78 per cent to 5.64
per cent**

28. Agenda
1. Looking at proportions
2. Basic algebra
3. Compound Annual Growth Rates (CAGR)
4. Back of the Envelope Calculations (BotEC)
28

29. 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
29

30. 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?
?
30

31. 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?
?
31
€
PW =1+ (0.5 × PW )
€
PW − 0.5PW =1
€
0.5PW =1
€
PW = £2

32. Algebra
10.If you buy a computer for £680, how much VAT have you
paid?
32
Note: £680 includes the VAT, therefore we ask two questions:
10a. What is the price before VAT gets added? 10b. What was the VAT?

33. Algebra
10.If you buy a computer for £680, how much VAT have you
paid?
33
€
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

34. Algebra
11.If you buy a computer for £5,000, how much VAT have you
paid?
34

35. 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
35

36. Algebra
12. A jacket costs £185.00 inc vat. What is the cost excluding
vat?
36http://www.thehogman.co.uk/www.thehogman.co.uk/info.php?p=26&pno=0

37. Algebra
12. A jacket costs £185.00 inc vat. What is the cost excluding
vat?
37http://www.thehogman.co.uk/www.thehogman.co.uk/info.php?p=26&pno=0

38. Agenda
1. Looking at proportions
2. Basic algebra
3. Compound Annual Growth Rates (CAGR)
4. Back of the Envelope Calculations (BotEC)
38

39. CAGR: Average Growth
• A company boasts that they’ve achieved average growth of
25% in the last two years. Are you impressed?
39
Year Value Percentage
Return
2004 £1,000
2005 £2,000 + 100%
2006 £1,000 - 50%
Average growth:
%25
2
%50%100
=
-
But growth = 0

40. CAGR: Average Growth
• A company boasts that they’ve achieved average growth of
25% in the last two years. Are you impressed?
40
Year Value Percentage
Return
2004 £1,000
2005 £2,000 + 100%
2006 £1,000 - 50%
Average growth:
%25
2
%50%100
=
-
But growth = 0

41. CAGR: Formula
• Compound Annual Growth Rate (CAGR) is given by:
1
1
-÷÷
ø
ö
çç
è
æ
=
÷
ø
ö
ç
è
æ
n
ValueBeginning
ValueEnd
CAGR
41

42. 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?
42
€
=
1,404,000
845,200
"
#
$
%
&
'
1
4
"
#
$
%
&
'
−1
€
=13.5%

43. 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?
43
€
=
1,404,000
845,200
"
#
$
%
&
'
1
4
"
#
$
%
&
'
−1
€
=13.5%

44. 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?
44
€
=
38.2
27.8
"
#
$
%
&
'
1
3
"
#
$
%
&
'
−1
€
=11.1%

45. 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?
45
€
=
38.2
27.8
"
#
$
%
&
'
1
3
"
#
$
%
&
'
−1
€
=11.1%

46. Agenda
1. Looking at proportions
2. Basic algebra
3. Compound Annual Growth Rates (CAGR)
4. Back of the Envelope Calculations (BotEC)
46

47. 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
47

48. 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
48

49. 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
49

50. 50

51. 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
51

52. 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
52

53. 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
53

54. 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)
54

55. • This presentation forms part of a free, online course
on analytics
• http://econ.anthonyjevans.com/courses/analytics/
55