Visions 5 CST Vol 3 (English)

student book volume

The 3rd volume of the Secondary Cycle Two, Year Three Visions series, Cultural, Social and Technical option, was designed in the spirit of the Québec Education Program and addresses an update to the program.

Student Book, volume 3

• Vision 1 Supplement, including inequalities

• Vision 2 Supplement, including the law of cosines • Vision 5, a chapter on financial mathematics, including logarithms, simple interest and compound interest in different financial contexts

• Vision 6, a chapter on probabilities, including subjective probabilities, odds for, odds against and mathematical expectation

Dominique Boivin • Richard Cadieux • Claude Boivin • Antoine Ledoux Étienne Meyer • Dominic Paul • Nathalie Ricard • Vincent Roy

Cultural, Social and Technical

• Many

reproducible sheets and their answer keys (Support, Consolidation, Enrichment, Snapshot)

• LES including the logbooks and evaluation grids • Tests and their answer keys

Secondary Cycle Two, Year Three

• Teaching notes for each chapter, including detailed lesson plans

MATHEMATICS

Teaching Guide, volume 3

volume Cultural, Social and Technical

• Volume 1, Student Book • Volume 2, Student Book • Volume 3, Student Book • Volumes 1 and 2, Teaching Guide • Volume 3, Teaching Guide

3

The components of the Visions series:

MATHEMATICS Secondary Cycle Two, Year Three Dominique Boivin Richard Cadieux Claude Boivin Antoine Ledoux Étienne Meyer Dominic Paul Nathalie Ricard Vincent Roy

volume

3

SUPPLEMENT Systems of equations and inequalities . . . . . . IV Arithmetic and algebra

SECTION 1.0 Half-planes in the Cartesian plane . . . . . . . . . . . 1 • First-degree inequality in two variables • Half-plane

SECTION 5.4 Other financial contexts . . . . . . . . . . . . . . . . . . 61 • Final value, initial value, term and rate

CHRONICLE OF THE PAST Some financial mathematicians . . . . . . . . . . . . 70

IN THE WORKPLACE Portfolio managers . . . . . . . . . . . . . . . . . . . . . . 72

OVERVIEW . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

SUPPLEMENT Equivalent figures and the law of cosines . . . . 12

BANK OF PROBLEMS . . . . . . . . . . . . 82

Geometry

SECTION 2.5 The law of cosines . . . . . . . . . . . . . . . . . . . . . . 13 • Trigonometric relations in triangles: the law of cosines

Probability and random experiments . . . . . . . 84 Probability

REVISION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

Logarithms and financial mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . 20 Arithmetic and algebra

REVISION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 • Exponents • Exponential function • Inverse

• Random experiment • Events: compatible, incompatible and complementary • Probability of an event • Random experiment with several steps

SECTION 6.1 Methods of enumeration . . . . . . . . . . . . . . . . . 92 • Factorial • Random experiment with or without order • Permutation, arrangement and combination

SECTION 5.1

SECTION 6.2

Logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

Subjective probability and odds . . . . . . . . . . . 104

• Logarithmic function • Equivalence between the exponential form and the logarithmic form • Specific logarithms • Common logarithm, natural logarithm and change of base • Solving an exponential or logarithmic equation

SECTION 5.2 Simple interest . . . . . . . . . . . . . . . . . . . . . . . . . 40 • Financial vocabulary • Compounding, discounting, term of a simple interest investment or loan • Simple interest rate

SECTION 5.3 Compound interest . . . . . . . . . . . . . . . . . . . . . 48 • Compounding, discounting, term of a compound interest investment or loan • Compound interest rate • Incomplete interest period

• Theoretical, experimental and subjective probability • Odds for and odds against • Distinguishing between probability and odds

SECTION 6.3 Mathematical expectation . . . . . . . . . . . . . . . 116 • Mathematical expectation • Fairness

CHRONICLE OF THE PAST Pierre-Simon de Laplace . . . . . . . . . . . . . . . . . 126

IN THE WORKPLACE Financial analysts . . . . . . . . . . . . . . . . . . . . . . 128

OVERVIEW . . . . . . . . . . . . . . . . . . . . . . . . . . 130 BANK OF PROBLEMS . . . . . . . . . . . 137 INDEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

Table of contents

III

Systems of equations and inequalities What is the minimum number of vehicles you need to book for a trip in order to minimize costs? How do you determine the values that optimize a company’s sales? What constraints influence a decision and how do you express them mathematically? In “Vision 1,” you will learn to express a situation as an inequality in order to define the constraints applicable to a specific case. In this way, you will obtain a system of inequalities and be able to determine the optimal solution or solutions according to the given context.

Arithmetic and algebra • First-degree inequality in two variables • System of first-degree inequalities in two variables • Polygon of constraints • Optimizing function • Optimizing a situation and making a decision using linear programming

Geometry

Graphs

Probability

1.0

Half-planes in the Cartesian plane

Electricity consumption

problem

The electricity bill represents an important part of an average household’s budget. Therefore, changing consumption habits can lead to substantial savings. Electricity is measured in kilowatt hours (kWh). One kilowatt hour corresponds to 1000 watts consumed per hour.

Below is some information about the Tremblay household’s electricity consumption: Electricity consumption for July

Appliance

Power (W)

Duration of use (h)

3500

120

Lighting

15

280

Refrigerator

300

744

Stove

4500

30

Washer/dryer

2650

15

Television

300

95

Other

350

100.75

Water heater

The rule C 5 0.0824k 1 17 indicates the cost of electricity C (in $) as a function of the number of kilowatt hours k consumed during the month of July.

Do the Tremblays stay within their budget if they allocate less than $90/month for electricity?

The amount of energy produced by a wind turbine depends primarily on the strength of the wind, the surface area swept by the blades, and the air density. For a turbine to produce electricity, it needs an intake of wind of at least 12 to 14 km/h with full production at 50-60 km/h. Wind speeds over 90 km/h damage the equipment, so in those conditions electricity production has to be interrupted.

Section 1.0

1

Equivalent figures and the law of cosines How do you maximize a tent’s living space while using as little canvas as possible to fabricate it? How do you calculate the measurements of a triangular piece of aluminum? How do you determine the ideal shape of a container? How do you measure the length of a bicycle frame? Many activities involve trigonometric relations and the optimization of surfaces and space. In “Vision 2,” you will design objects that save on materials and make optimal use of space.

You will also learn about a law that shows the relationship between the measurements of the angles and sides of a triangle.

Arithmetic and algebra

Geometry • Equivalent figures • Optimizing surfaces and space • Trigonometric relations in triangles: the law of cosines

Graphs

Probability

2.5 problem

The law of cosines

Della Falls

Located on Vancouver Island, Della Falls is the highest waterfall in Canada and can be reached by seaplane or by a relatively difficult hiking trail.

Positioned at a certain distance from the base of the waterfall, Tamara looks toward the top of the falls at an angle of elevation of 59°. Standing between Tamara and the base of the waterfall, Guillaume looks toward the top of the falls at an angle of elevation of 62°.

Della Falls

59°

62°

30.36 m Tamara

Guillaume

What is the height of Della Falls?

Fed by water from Della Lake in Strathcona Provincial Park, Della Falls is eight times higher than Niagara Falls.

Section 2.5

13

Logarithms and financial mathematics How do you determine what the value of an RRSP will be in a few years? How do you calculate the change in the price of goods over time? How do you calculate the term of a loan or an investment? How do you determine the interest rate in situations involving financial mathematics? Many financial situations can be modelled using exponential formulas. In “Vision 5,” you will solve exponential and logarithmic equations. You will also deal with the concepts of simple interest and compound interest in order to calculate the future value and the present value of investments or loans.

Arithmetic and algebra • Powers and logarithms • Solving exponential or logarithmic equations • Calculating, interpreting and analyzing financial situations • Simple and compound interest • Interest period • Discounting (present value) • Compounding (future value)

Geometry

Graphs

Probability

Financial planning

Some financial mathematicians

Portfolio managers

Prior learning

1

The Zika virus

The Zika virus was first identified in humans in the 1950s in Africa and Asia. In 2015, the virus appeared in South America, particularly in Brazil where major outbreaks occurred. By the end of 2015, 1 500 000 Brazilians were infected with the Zika virus. In that country, the virus spread according to a model whereby the number of people infected increased by 50% per month starting January 1, 2015. The following graph represents the number of Brazilians infected with the Zika virus according to the time elapsed since January 1, 2015 (in months).

Spread of the Zika virus in Brazil in 2015

Number of Brazilians infected 1 600 000 1 400 000 1 200 000 1 000 000 800 000 600 000 400 000 200 000 0

1

2

3

4

5

6

7

8

9

10

11

a.

According to what functional model did the virus spread?

b.

Determine the rule of the function that allows you to calculate the number N of Brazilians infected with the Zika virus according to the time m elapsed (in months) since January 1, 2015.

c.

How many Brazilians were already infected on January 1, 2015?

d.

How many Brazilians were infected:

e.

22

1)

by the end of June 2015?

2)

by the end of September 2015?

In 2016, the population of Brazil was approximately 206 million. Would it be true to say that if the virus had continued to spread at the same rate, over 90% of the population would have been infected by the end of 2016? Justify your answer.

Vision 5

12 Time elapsed since January 1, 2015 (months)

The Zika virus was first discovered in a monkey in Uganda in 1947. Zika is the name of a forest situated south of the capital Kampala. As with other tropical infections, the Zika virus is transmitted through mosquito bites. Most people who are infected do not realize that they have the virus. The Zika virus can cause malformations such as microcephaly, whereby a baby is born with a small cranium, which affects cognitive development.

Prior learning

2

Montréal–Trudeau Airport

The Montréal–Pierre Elliott Trudeau International Airport plays an important role in the economy of Greater Montréal and the province of Québec. In 2003, this airport served 9 million travellers.

Since then, this number has increased exponentially every year so that by 2015, for the first time in its history, the airport served 15.5 million travellers.

a.

Determine the rule of the exponential function that allows you to calculate the annual number of travellers (in millions) according to the time elapsed (in years) to 2003.

b. Use a graph to represent this function for the period from 2003 to 2015. c.

Determine the properties of the function in b and what they represent in this context: 1)

domain

2)

range

3)

x-intercept

4)

y-intercept

5)

variation

6)

sign

d.

Is the inverse of this function also a function? Justify your answer.

e.

How many travellers went through the Montréal–Trudeau Airport in: 1)

2010?

2) 2013?

f.

If this trend were to continue, would it be true to say that the number of travellers at the Montréal–Trudeau Airport will exceed: 1)

20 million in 2020?

2)

30 million in 2030?

The Montréal–Trudeau Airport is an essential infrastructure for business, commerce and tourism. There are approximately 200 businesses and organizations located in the airport facilities, providing 27 000 jobs directly and another 28 000 jobs indirectly, for a total of 55 000 jobs.

The economic activities of these businesses generate 5.5 billion dollars annually. The Montréal–Trudeau Airport ranks second in Canada in terms of air service provided, offering direct flights to 140 destinations, 80 of which are international.

Revision

23

EXPONENTS Power The equality a n 5 p means that a n is the nth power of a and that this power is equal to p. a n 5 a 3 a 3 … 3 a if n ∈ N and n  2 n times

Below are a few special cases involving powers: Special case

Example

a 0 5 1 if a  0

90 5 1

a1 5 a

71 5 7

a m 5 1 n

1 if a  0 am

53 5 1 4

n

a 5 a if a  0 and n  0

1 1 5 53 125 4

16 5 16 5 2

Laws of exponents The laws of exponents allow you to perform operations on expressions written in exponential form.

Law

Example

Product of powers with the same base a m 3 a n 5 a m 1 n if a  0

83 3 84 5 83 1 4 5 87 5 2 097 152

Quotient of powers with the same base am 5 a m 2 n if a  0 an

67 5 67 2 3 5 64 5 1296 63

Power of a product (ab)m 5 a mb m if a  0 and b  0

(4 3 3)2 5 42 3 32 5 16 3 9 5 144

Power of a power (a m )n 5 a mn if a  0 Power of a quotient

() a b

24

m

am 5 m if a  0 and b  0 b

Vision 5

(23 )5 5 23 3 5 5 215 5 32 768

()5 24 5

24 16 5 5 0.0256 54 625

EXPONENTIAL FUNCTION • An exponential function is a function defined by a rule in which the independent variable is an exponent.

• Graphically, the curve associated with an exponential function approaches the horizontal asymptote.

• The rule of a basic exponential function can be written as f(x) 5 cx, where c  0 and c  1.

Graphically, the curve associated with this function passes through point (0, 1).

E.g. f (x) 5 4x

Table of values x

f (x )

2

0.0625

1

0.25

0

1

1

4

2

16

Graph

Properties Domain: R

f(x ) 16

Range: ]0, 1[ x-intercept: None

12

y-intercept: 1

8 Asymptote

4 4

2

Sign: Positive over R Variation: Increasing over R

0

2

Extremum: None

x

4

• The rule of a transformed exponential function can be written as f(x) 5 acx, where a  0, c  0 and c  1. Graphically, the curve associated with this function passes through point (0, a).

E.g. f (x) 5 3(4)x

Table of values

Graph f(x )

x

f (x )

2

 0.1875

1

 0.75

10

0

3

20

1

12

2

48

Properties Domain: R

4

2

0

2

x

4

Range: ]2, 0[ x-intercept: None

Asymptote

y-intercept: 3 Sign: Negative over R

30

Variation: Decreasing over R

40

Extremum: None

• To determine the rule of an exponential function f(x) 5 acx, replace a with the initial value and determine the value of c by replacing x and f(x) with the coordinates of a point belonging to the function.

INVERSE • To find an inverse, invert the values of each ordered pair in a relation between two variables.

E.g. f (x) 5 3x and its inverse y f (x ) 3x

• The curves of a relation and its inverse are symmetrical relative to the straight line of the equation y 5 x.

• The inverse may or may not be a function. If the inverse of function f is a function, then it is written as f 1.

4

Axis of symmetry

2 4

2

0 2

x

f 1(x ) 3

2

4

x

Inverse

4

Revision

25

1 Calculate each of the following expressions. b) 82

a) 73 1

c) 6.23

d) 90

e) 07

27 g) 13 h) 56 i) 45 j) 29

f)

4

2

1

3

2 Determine the value of x for each of the following cases.

1 9

a) x 3 5 8

b) x 5 5 1024

c) x 4 5 1296

d) x 2 5

e) 3x 5 27

f) 8x 5 64

g) 9 x 5 3

h) 8x 5 2

12 

i)

x

5

1 1 x 5 27 j) 16 3

 

k)

 161 

x

1 4

1 25

x

5 l)   55

3 Determine whether each of the following statements is true or false. a8 a

4 a) a 2 3 a 7 5 a 9 b) 2 5 a , where a  0

d) (a 2a 9)3 5 a 33 g)

a 6 3 a 3 a 4 3 a 2

5 a 5, where a  0

c) (a 3)3 5 a 6 a5 b

5

a b

e) (a 6)0 5 a 6 f )   5 5 , where b  0 h) ((a 4a 5)2)3 5 a 14 i) a 5 3 b 5 5 (ab)5

4 Rewrite each expression in the form of a base with a positive exponent. a) 3 3 35 3 33

b) 74 3 7 2 3 70

c) (52)4 3 (53)1

d)

90(9)894 27 5 50 3 5 2 e) 3 f ) 9(9) (9) 2 54

g)

95 

  3 8

 

3 8

4

h)    

4

6

i) 3 

36 32



1

5 Rewrite each expression in the form of a power with the smallest possible base. a) 274

b) 4 3 83 3 162

c) 812 3 34 3 93

3433 49

642 8

5122 2

43n 1 2 2

93a 3

274a 81

d) 362 3 65 3 2161 e) 3 f ) 5 3 9 g) 252a 1 5 3 1254a 2 3 h) n 1 4 i) 2a 3 a

6 Simplify each algebraic expression and use positive exponents to express your answer. a) b 4 3 b 3 b 3 b) c 0 3 (c 4)2 a5 a

a0 a

d) (a 3b 4)5 e) 3 3 2 , where a  0 g)

26

e5 f3

 

2

, where e  0 and f  0 h)

Vision 5

n4 3 n1 3 n0 , n6 3 n4 3 n2

c) 32n 1 3 3 3n 1 5 f )

3

2

m  3

where n  0 i)

5

dc  , where d  0 m4 1 4

8

, where m  0

7 For each of the graphs below: 1)

represent the inverse of the function graphically

2)

indicate if the inverse is a function y

a)

10 8

6

4

2

y

b)

10

10

8

8

6

6

4

4

2

2

0 2

2

4

6

10 x

8

10 8

6

4

2

0 2

4

4

6

6

8

8

10

10

2

4

6

10 x

8

8 For each of the two exponential functions below: 1) calculate

f(5) and f(22)

2)

represent function f graphically

3)

represent the inverse of function f graphically

4)

determine if the inverse is a function and explain your answer 1 4

x

f(x) 5   a) f(x) 5 4x b)

9 For each of the functions below: 1)

represent the function graphically

2)

determine the properties of the function: • domain

• range

• x-intercept

• y-intercept

• sign

• variation

a) f(x) 5 2(3)x

b) g(x) 5 4

2 3

x

h(x) 5 3.4(0.6)x  c)

• extremum d) i(x) 5 3(2.5)x

10 Determine the rule of the exponential function associated with each of the following tables of values. a) b) c) x f (x ) x g(x)

x

h(x )

1

7 12

1

0.4

1

4

0

3.5

0

2

0

3

1

21

1

10

1

2.25

2

126

2

50

2

1.6875

Revision

27

11 Determine the rule of the exponential function for each of the following cases. a) b) c) g(x) f (x) 0 (0, 3.2)

x

h(x ) ( 1, 10)

(2, 45.9)

0

(0, 4)

x

(1, 19.2)

(0, 5.1) 0

x

12 ALZHEIMER’S DISEASE Alzheimer’s disease is a neurodegenerative disease that worsens over time and can eventually cause death. Age is the greatest risk factor for this disease. In Canada, the risk of developing Alzheimer’s disease at age 65 is 2%. After this age, the risk increases by 15% per year until the age of 90. a) Determine the rule of the function that allows you to calculate the risk R (in %) of developing Alzheimer’s disease, according to the time elapsed t (in years) from the moment a person turns 65. b) Determine the risk of developing Alzheimer’s disease at the age of: 1) 75

2) 85

c) Use a graph to represent the function determined in a) for people aged 65 to 90. d) Would it be true to say that the risk of developing the disease doubles approximately every 5 years from age 65 to 90? Justify your answer. e) In Canada, approximately 305 000 people were 71 years old in 2016. How many of these people may have developed Alzheimer’s disease?

13 VIDEO GAME INDUSTRY Québec’s video game industry began to develop in the 1980s.

From 2002 to 2012, the number of jobs in this sector increased exponentially. In 2002, 1200 people were employed in this industry; 10 years later, by 2012, 9000 people worked in this field in Québec. a) Determine the rule of the function that allows you to calculate the number of jobs N in the video game industry according to the time elapsed t (in years) since 2002. b) Use a graph to represent the function determined in a) for the years from 2002 to 2012. c) Would it be true to say that the number of jobs tripled from 2005 to 2010? d) If the number of jobs in the video game industry had continued to increase at the same rate, how many jobs would there have been in Québec in 2016?

28

Vision 5

In 2016, Québec had 230 companies in the video game industry. Most of these companies are located throughout Greater Montréal. The city is world-renowned in the field of multimedia. Montréal is in fact the fifth hub in the videogame industry after Tokyo, London, San Francisco and Austin. Since 2012, the number of jobs in this sector in Québec has stabilized at around 10 000.

14 A city had a population of 80 000 in 2006. Since that time, the population has increased by an average of 2.4% per year. What was the population of this city in 2016?

15 Here is some information about three motorcycle models: Model A

Model B

Model C

• Price: $16,000

• Price: $31,000

• Price: $24,000

• Average annual rate of depreciation: 12%

• Average annual rate of depreciation: 16%

• Average annual rate of depreciation: 14%

Which model will have the highest resale value 15 years after its purchase?

16 PLASMA CONCENTRATION When medication is administered intravenously, its concentration in the blood, called plasma concentration, is initially optimal.

It then decreases according to an exponential model. The graph on the right represents the plasma concentration (in mg/L) of a drug according to the time elapsed (in h) after an intravenous injection.

Plasma concentration (mg/L) 10 9 (0, 8) 8

Plasma concentration of medication

7 (7, 5.6529)

6 5 4 3 2

When a drug’s concentration decreases by half over a given period of time, this period is called a halflife. Would it be true to say that the half-life of this drug is 12 h? If not, determine its half-life.

1 0

1

2

3

4

5

6

7

8

9 10 Time elapsed (h)

17 ELASTICITY FACTOR The different balls used in sports are able to bounce due to their elasticity. The following information concerns three types of balls whose bouncing capacity follows an exponential model.

Ball B

Ball A

The ball is dropped from a height of 25 m. It 3 bounces back up to 4 of the height of the previous bounce.

Ball C

Number of bounces

Height of the bounce (m)

0

28

1

19.6

15

2

13.72

10

3

9.604

5

Which ball bounces the highest on its 6th bounce?

Height of the bounce 25 (m) (0, 22) 20

0

(2, 12.7072)

1

2

3

4

5 Number of bounces

Revision

29

5.3

Compound interest

This section is related to LES 9 and 10.

problem

Three proposals

Several factors influence the maturity value of an investment, including how often the interest is calculated. The following is a conversation between an investor and his financial advisor: Here are three proposals.

Keep in mind that the interest earned over each period is, at the end of the period, added to the capital for the next calculation of interest.

I would like to invest $5,000 over 2 years.

Proposal A

Proposal B

We invest the amount at an annual interest rate of 6% and the interest is calculated at the end of each 1-year period.

We invest the amount at a half-yearly interest rate of 2% and the interest is calculated at the end of each 6-month period.

I will take Proposal C .

In my opinion, the more often the interest is calculated over a given period, the higher will the maturity value of the investment be.

What do you think about this claim?

48

Vision 5

Proposal C We invest the amount at a quarterly interest rate of 0.5% and the interest is calculated at the end of each 3-month period.

Activity

1

Interest that generates interest

$10,000 is invested at an annual compound interest rate of 4%. The following procedure allows you to calculate the accumulated capital at the end of each of the first four years.

a.

1

Capital accumulated in 1 year: 10 000 3 1.04 5 10 400 Therefore, $10,400

2

Capital accumulated in 2 years: 10 000 3 1.04 3 1.04 5 10 816 Therefore, $10,816

3

Capital accumulated in 3 years: 10 000 3 1.04 3 1.04 3 1.04 5 11 248.64 Therefore, $11,248.64

4

Capital accumulated in 4 years: 10 000 3 1.04 3 1.04 3 1.04 3 1.04  11 698.59 Therefore, $11,698.59

The calculation written in blue corresponds to what amount in: 1) Step 2

?

2) Step 3

?

3) Step 4

?

b.

What does each amount correspond to in a?

c.

Given that the expression 10 000(1.04)4 allows you to calculate the capital accumulated in 4 years, determine the expression for calculating the capital accumulated in: 1)

1 year or after 1 compounding period

2)

2 years or after 2 compounding periods

3)

3 years or after 3 compounding periods

d. What link can you make between the number of compounding periods and the exponent of each expression determined in c?

e.

Determine the accumulated capital of this investment in: 1)

8 years

2)

12 years

3)

25 years

Section 5.3

49

Activity

2

Back to square one

To start up his business 25 years ago, Philippe took out a loan at a half-yearly compound interest rate of 1.45%. The following procedure can be used to determine the amount borrowed.

2

513 502.52 5 C0(1 1 1.45%)50

3

C0 5

4

C0 5 513 502.52(1.0145)50

5

C0  250 000

a.

Cn 5 C0(1 1 i )n

1

Therefore, $250,000

Given the context, in Equation

b.

c.

513 502.52 1.014550

2

, what corresponds to:

1)

the amount on the left member?

2)

the exponent 50?

Indicate as specifically as possible how the following are obtained: 1) Equation 3

based on Equation

2

2) Equation 4

based on Equation

3

How much did Philippe borrow 25 years ago?

To finance the purchase of new equipment, Philippe takes out a new loan at a quarterly compound interest rate of 0.9%.

d.

Use Equation 4 as a model to determine the initial sum if he expects to pay back in 10 years an amount of: 1)

$71,551.16

2)

$97,309.57

3)

$264,739.28

According to the Institut de la statistique du Québec, approximately 17% of owners of small and medium-sized enterprises (SMEs) in Québec were aged 40 or younger in 2014. These young entrepreneurs often resort to financing to help their businesses grow. Among the different types of financing, a personal loan is most often used before resorting to a commercial loan or a financial lease. Furthermore, the government can grant subsidies to SMEs that meet certain criteria.

50

Vision 5

Some graphing calculators have an application for calculating different financial values.

Screen 1 These screens allow you to select the Finance‌ option to calculate different values.

Screen 2

Screen 3 This screen shows the accumulated capital of a $2,500 investment over 5 years at an annual compound interest rate of 3.5%.

Screen 4 This screen shows the term of a $3,000 investment that earns an accumulated capital of $3,312.24 at an annual compound interest rate of 2%.

Screen 5

This screen shows the annual compound interest rate of a $5,600 investment that earns $7,085.79 after 6 years.

a.

b.

c.

Referring to Screen 3, determine: 1)

the accumulated capital

2)

the three numbers, in order, that you need to use with the function tvm_FV to calculate the accumulated capital of a $6,900 investment over 9 years at an annual compound interest rate of 1.75%, and the value of this accumulated capital.

Referring to Screen 4, determine: 1)

the term of the investment

2)

the four numbers, in order, that you need to use with the function tvm_N to calculate the term of a $5,000 investment that earns an accumulated capital of $6,948.83 at an annual compound interest rate of 4.2%, as well as this term.

Referring to Screen 5, determine: 1)

the interest rate

2)

the four numbers, in order, that you need to use with the function tvm_I% to calculate the annual compound interest rate of a $25,000 investment that earns $38,106.97 after 15 years, as well as this interest rate.

Section 5.3

51

5.3

COMPOUND INTEREST When interest that is earned over a certain period is added to the capital before the next period is calculated for its interest, it is called compound interest. In other words, interest is earned on interest.

COMPOUNDING INTEREST Compounding interest can be calculated as follows.

After a term of one period:

C1 5 C0(1 1 i ) 5 C0(1 1 i )1

After a term of two periods: C2 5 C0(1 1 i )(1 1 i ) 5 C0(1 1 i )2 After a term of three periods: C3 5 C0(1 1 i )(1 1 i )(1 1 i ) 5 C0(1 1 i )3 After a term of n periods:

...

...

Cn 5 C0(1 1 i )(1 1 i ) … (1 1 i ) 5 C0(1 1 i )n

n times

The following formula is obtained.

– Cn is the accumulated capital – C0 is the initial capital Cn 5 C0(1 1 i )n , where: – i is the compound interest rate – n is the term (that is, the number of periods) Note : If necessary, the term n is transformed so as to obtain the same unit of time as the interest rate i.

E.g. We borrow an initial capital of $700 at a monthly compound interest rate of 1.5%.

We want to determine what the accumulated capital will be in 2 years.

Here, i 5 1.5% and C0 5 $700 n 5 2 3 12 5 24 months Cn 5 C0(1 1 i )n C24 5 700(1 1 1.5%)24 C24 5 700(1.015)24 C24  1000.65 Therefore, $1,000.65 In 2 years, the accumulated capital will be $1,000.65.

52

Vision 5

DISCOUNTING WITH COMPOUND INTEREST Discounting with compound interest is obtained from the compound interest formula.

Cn 5 C0(1 1 i )n

Cn 5 C0 (1 1 i )n

Cn(1 1 i )n 5 C0

The following formula is therefore obtained.

– C0 is the initial capital – Cn is the accumulated capital C0 5 Cn(1 1 i )n , where:

– i is the compound interest rate – n is the term (that is, the number of periods) Note : If necessary, the term n is transformed so as to obtain the same unit of time as the interest rate i.

E.g. We incur a debt and, 3 years later, pay back $5,436.28. Given that the annual compound interest rate was 4%, we want to determine the value of the initial capital borrowed.

Here, n 5 3 years, i 5 4% and C3 5 $5,436.28 C0 5 Cn(1 1 i )n C0 5 5436.28(1 1 4%)3 C0 5 5436.28(1.04)3 C0  4832.83 Therefore, $4,832.83 The initial capital was $4,832.83.

TERM OF A COMPOUND INTEREST INVESTMENT OR LOAN It is possible to calculate the term of a compound interest investment or loan by using logarithms to isolate the variable n in the compound interest formula.

E.g. We invest $1,200 at an annual compound interest rate of 1.75%.

We want to determine in how many years the accumulated capital will be $1,264.11.

Here, C0 5 $1,200, i 5 1.75% and Cn 5 $1,264.11 Cn 5 C0(1 1 i )n 1264.11 5 1200(1 1 1.75%)n 1264.11 5 1200(1.0175)n

1264.11 5 1200

1.0175n

1.0534  1.0175n n  log1.0175 1.0534 log1.0534 log1.0175

n  n 3

Therefore, 3 years The accumulated capital will be $1,264.11 in 3 years.

Section 5.3

53

COMPOUND INTEREST RATE The compound interest rate of an investment or loan can be calculated by isolating the variable i in the formula for compound interest.

E.g. We borrowed $108,000 and, after 5 years, the accumulated capital is $135,880.51. We want to determine the annual compound interest rate of this loan.

Here, n 5 5 years, C0 5 $108,000 and C5 5 $135,880.51 Cn 5 C0(1 1 i )n 135 880.51 5 108 000(1 1 i )5 135 880.51 5 108 000 1 135 880.51 5

5 108 000





(1 1 i )5 11i

135108880.51 000 

i 5

1 5

21

i  0.047 i  4.7% Therefore, 4.7% The annual compound interest rate is 4.7%.

INCOMPLETE INTEREST PERIOD If the term of a compound interest investment or loan includes several interest periods that are already completed as well as one incomplete interest period, the accumulated capital can be calculated using the following procedure.

Procedure 1. Calculate the accumulated capital with compound interest for the completed interest periods using the formula Cn 5 C0(1 1 i )n.

E.g. We invest an initial capital of $11,500 at an annual compound interest rate of 2%. We want to calculate the accumulated capital in 5 years and 9 months.

Here, n 5 5 years, i 5 2% and C0 5 $11,500 Cn 5 C0(1 1 i )n C5 5 11 500(1 1 2%)5 C5 5 11 500(1.02)5 C5  12 696.93 Therefore, $12,696.93 After 5 years, the accumulated capital will be $12,696.93.

2. Using the result obtained in the previous step, calculate the accumulated capital with simple interest for the incomplete period using the formula Cn 5 C0(1 1 n 3 i ).

Simple interest applies over a period of 9 months.

9 n5 5 0.75 year 12

Cn 5 C0(1 1 n 3 i ) C0.75 5 12 696.93(1 1 0.75 3 2%) C0.75 5 12 696.93(1.015) C0.75  12 887.38 Therefore, $12,887.38 After 5 years and 9 months, the accumulated capital will be $12,887.38.

54

Vision 5

5.3

1 Calculate the accumulated capital for each of the following situations. a) We borrow $4,500 over 2 years at a

b) We borrow $12,000 over 4 years at a

weekly interest rate of 0.15%.

c) We invest $7,000 for 10 years at a

d) We invest an initial capital of $2,000 for

monthly compound interest rate of 0.2%.

e) We invest $21,000 for 11 years at a

7 years at an annual compound interest rate of 3.75%.

f) We invest an initial capital of $6,500 for

quarterly compound interest rate of 1.3%.

semi‑annual compound interest rate of 3%.

9 years at a monthly compound interest rate of 0.15%.

2 Calculate the initial capital for each of the following situations. a) At a semi-annual compound interest rate

of 3.5%, the repayment of a debt will be $10,559.92 after 3.5 years.

c) At a quarterly compound interest rate

of 2.75%, the accumulated capital of a loan will be $14,115.01 in 4 years and 3 months.

e) At a weekly compound interest rate of

0.08%, the repayment of a debt will be $10,341.34 after 5 years.

b) In 4 years, the repayment of a loan at an

annual compound interest rate of 5% will be $6,563.73.

d) In 8 years, the accumulated capital of an investment at an annual compound interest rate of 2% will be $7,029.96.

f) In 20 years, the repayment of a loan at a

monthly compound interest rate of 0.4% will be $625,608.04.

3 Determine the term of the investment or loan for each of the following situations. a) An initial capital of $3,000 earns an

accumulated capital of $3,312.24 at an annual compound interest rate of 2%.

c) The repayment of a $500 debt at a weekly

compound interest rate of 0.15% is $727.29.

e) An initial capital of $7,600 earns an

accumulated capital of $10,610.23 at a quarterly compound interest rate of 1.4%.

b) A $1,400 investment yields $2,691.50

at a monthly compound interest rate of 3.5%.

d) The repayment of a $13,500 loan at a

daily compound interest rate of 0.024% is $18,665.01.

f) The repayment of a $13,000 debt at a

semi-annual compound interest rate of 2.4% is $18,999.53.

Section 5.3

55

4 Calculate the compound interest rate for each of the following situations. a) The repayment of an initial capital of

$25,300 at a semi-annual compound interest rate is $35,035.03 after 6 years.

c) The repayment of a $7,100 loan at a

quarterly compound interest rate is $8,729.90 after 4 years.

e) An initial capital of $7,800 at a weekly

compound interest rate earns an accumulated capital of $13,045.74 after 5.5 years.

b) The repayment of an initial capital of

$17,546 at an annual compound interest rate is $25,950.21 after 7 years.

d) A $4,700 investment at an annual

compound interest rate yields $8,699.37 after 8 years.

f) An initial capital of $3,500 at a monthly

compound interest rate earns an accumulated capital of $4,064.90 after 2.5 years.

5 Calculate the accumulated capital for each of the following situations. a) We invest a capital of $10,100 for 4 years

and 6 months at an annual compound interest rate of 2%.

c) We invest $12,000 for 10 years and

3 months at an annual compound interest rate of 4.25%.

e) We invest a capital of $11,800 for 7 years

and 3 months at an annual compound interest rate of 2.4%.

b) We borrow $17,800 for 6 years and

9 months at an annual compound interest rate of 3.55%.

d) We invest an initial capital of $40,000

for 17 years and 5 months at an annual compound interest rate of 5.75%.

f) We borrow $200,000 for 25 years and

4 months at an annual compound interest rate of 4.15%.

6 In anticipation of the renovations she wants to do in a few years, Emmanuelle estimates that she will need $15,000. a) How much should she invest now in order to raise the money needed for the renovations in 2 years, if the annual compound interest rate is: 1) 2.5%?

2) 3.5%?

3) 5%?

b) How much should she invest now at annual compound interest rate of 4.55% in order to raise the money needed for the renovations in:

56

1)

3 years?

2)

7 years?

3)

10 years?

Vision 5

7 Léonie borrows $24,500 to buy a car. Given that the monthly compound interest rate is 0.9%, what is the term of the loan if Léonie’s debt grows to $33,825.80?

8 EQUITY FUNDS An equity fund is a type of financial investment whose assets consist of shares from various companies. An investor places $6,700 in an equity fund. a) Calculate the average annual compound interest rate of this fund if in 8 years the value of the investment is: 1)

$9,528.07

2)

$8,055.62

3)

$10,050.81

b) If the average quarterly compound interest rate of this fund is 0.9%, determine what the value of the investment will be in: 1)

3 years

2)

5 years

3)

10 years

An equity fund is intended for investors with a relatively high-risk profile who are interested in long-term growth.

9 A worker receives a $500 bonus, which she invests at an annual compound interest rate of 3%. What will the value of this investment be in 4 years?

10 The following table indicates the value of a compound interest investment over time.

Investment value Time (years) Value ($)

0

2

4

6

15 000

16 380.38

17 887.78

19 533.90

Given that the value V of this investment (in $) varies according to the rule V 5 C0(1 1 i )t, where C0 is the initial value and t is the time (in years), determine the annual compound interest rate i of this investment.

Section 5.3

57

11 GUARANTEED INVESTMENT CERTIFICATE (GIC) A GIC is an investment where the amount invested and the interest earned are guaranteed.

Sophie invests $12,000 in a GIC at a monthly compound interest rate of 0.58%. a) What will be the value of this investment in: 1)

9 months?

2)

5 years?

3)

10 years?

b) Will the difference in this investment’s value from the 4th to 6th year be the same as that from the 8th to 10th year? Explain your answer.

A GIC is usually offered over different terms and is intended for investors with a low-risk profile.

12 David and Anaïs enjoy camping and the great outdoors. They decide to borrow $12,000 to buy a camper. They are offered an annual compound interest rate of 4.5%. How much will their debt be if they repay it in 2 years and 9 months?

13 Julien inherits $35,000. He invests this sum for 9 years at a quarterly compound interest rate of 1.25%. After 9 years, he intends to reinvest the accumulated capital for 9 more years at a weekly compound interest rate of 0.2%. How much will the accumulated interest be after these 18 years?

14 Olivier invests $2,000 in a financial institution to enable his 5-year-old son to buy a car when he comes of age. What must the quarterly compound interest rate of the investment be so that the sum triples by his son’s 18th birthday?

15 Myriam takes out a loan at a semi-annual compound interest rate of 2.66%. If she repays the loan in 14 years by paying back $252,000, what sum did she borrow?

16 Amina wants to contribute a maximum of $3,000 to an RRSP. These offers are from two banks:

Bank A recommends that she invest $3,000 in order to obtain $5,800 in 5 years.

Bank B offers a monthly compound interest rate of 0.35% for a 5-year term.

Which offer should Amina take?

Explain your answer.

The Québec Pension Plan (QPP) is a compulsory public insurance plan for most workers. Its purpose is to provide basic financial protection upon retirement. People who want to have additional capital upon retirement can invest their savings in a Registered Retirement Savings Plan (RRSP).

58

Vision 5

17 Florence is 18 years old and dreams of travelling around the world for her 30th birthday.

In order to raise $20,000 for the trip, she makes an initial investment for 8 years at an annual compound interest rate of 4.65%, with which she expects to reach 35% of her goal. She will then invest her accumulated capital in a second investment for the remaining years. Determine: a) the initial capital that Florence invested b) the monthly compound interest rate of the second investment that will allow her to reach her goal This is an illustration of some of the most famous monuments in the world.

18 Félix buys a boat. To do this, he borrows $7,700 at a semi-annual compound interest rate of 3%. When the time comes to repay his loan, Félix will have to pay back $10,658.60.

How much time will it take Félix to repay his loan?

19 Alex borrows $23,000 from his mother at an annual compound interest rate of 5.5%. He intends to repay his loan in 5 years and 6 months. How much will he have to pay back his mother at that time?

20 In order to pay his income tax, Christophe borrows $8,000 for a period of 6 years. He is offered the following two options.

Option A

Option B

A monthly compound interest rate of 0.9%

An annual compound interest rate of 11.350 96%

Which option should Christophe choose? Justify your answer.

21 When her daughter is born, Valérie invests $4,800 in the Québec Savings Bonds at a daily compound interest rate of 0.04%. Given that a year has 365 days, how old will her daughter be by the time this amount quadruples?

Section 5.3

59

Some financial mathematicians Louis Bachelier (1870–1946) Louis Jean-Baptiste Alphonse Bachelier was a French mathematician. A precursor of the modern theory of probabilities, he is credited with being the first to apply financial mathematics to financial markets. In 1900, in his PhD thesis The Theory of Speculation, he applied a mathematical model based on biology known as Brownian motion to finance. His work went unrecognized for many years, but its true value finally came to be appreciated towards the end of his life.

Value

Brownian motion is a mathematical description of a big particle’s trajectory as it collides with smaller particles in a liquid. Louis Bachelier was the first to use it to model the dynamics of the stock market.

Brownian motion

Value of share ($) 55

24 20 16

50

12

45

8 4 0

Share price of Company A

40 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0

Time

50

100

Benoît Mandelbrot (1924–2010)

H

B

20

G

10

A 0

70

F

D

30

E

C 2

4

Vision 5

6

8

10 Time elapsed since purchase (years)

Value of share ($) 70 60 50 40 30 20 10 0

Share price of Company C

Value of share ($) 140 120 100 80 60 40 20

20 01 20 02 20 03 20 04 20 05 20 06 20 07 20 08 20 09 20 10

Share price of Company B

Time elapsed since purchase (days)

Fractals are geometric objects that can be broken down into fragments, where each fragment has the same form as the whole. For example, instead of following a share price over 5 years, we can follow it over one year, a few months, or even a few days. Whether we consider a stock market price over a long or a short period of time, fractals show that the general appearance is similar.

Benoît Mandelbrot was a French-American mathematician. In 1958, after obtaining a post-doctorate at Princeton, he settled in the United States to work at the IBM Research Center. He invented a new class of mathematical objects called fractals.

Mandelbrot became interested in modeling the evolution of prices on the stock market by analyzing the price of cotton. He believed that the amplitude of market fluctuations can remain independent from one day to the next while still correlating over very long periods of time.

Value of share ($) 40

150

Years

0

Share price of Company D

2

4

6

8

10 12 14 16 Time elapsed since purchase (years)

Fischer Black (1938–1995) Fischer Black was an American economist. In 1964, he received a PhD in applied mathematics from Harvard University. He became a finance professor at the University of Chicago in 1971, and then at the Massachusetts Institute of Technology (MIT) in 1975.

Myron Scholes (b. 1941) Myron Samuel Scholes is a Canadian economist. In 1964, he received a Master of Business Administration from the University of Chicago, followed by a PhD in 1969. He is known for his work on determining the value of derivatives, particularly options.

The Black-Scholes model Drawing on the work of Robert Merton, Fischer Black and Myron Scholes published an article in 1973 that would revolutionize financial mathematics and the functioning of financial markets. Their article “The Pricing of Options and Corporate Liabilities” is known as the Black-Scholes model. For discovering this model, which uses Brownian motion among other things, Scholes and Merton received the Nobel Prize in Economic Sciences in 1997. Since the prize cannot be awarded posthumously, Black did not receive the prize, as he had died two years prior.

Context for using the Black-Scholes model: “A buyer proposes a stock option to a seller for the right to buy or sell stock within a certain period of time. The seller demands a premium from the buyer for this privilege.” The BlackScholes model allows the seller to calculate the necessary premium so as not to lose money.

1. Use Brownian motion to answer the following questions.

a) In the graph for the share price of Company A , what interval of time corresponds to the generally decreasing segment from 0.18 to 0.38 of Brownian motion? b) Describe the general appearance of the share prices of Company A from the 200th day to the 230th day.

2. Use Mandelbrot’s theory to answer the following questions.

a) The share prices of Company B show a fractal cycle of 10 years indicated by the line segments joining points A-D-E-H. For this company, name two cycles that have the same form over shorter periods of time. b) What should an investor do with shares of Company C : 1) in 2008? 2) in 2010? c) What will the value of the shares of Company D be 17 to 22 years after their purchase?

3. A buyer proposes to a seller a stock option of

$15 per share, to use for one year, for 1000 shares whose present value is $18 each. The seller accepts the offer and demands a premium of $2,000. Use the Black‑Scholes model to analyze the positions of the buyer and the seller if, over the course of the year, the value of the shares: a) decreases by 20% b) increases by 10%

Chronicle of the past

71

Portfolio managers The profession Portfolio managers are responsible for managing the portfolio of a client, investor or institution. These managers must make qualitative and quantitative analyses of the companies and securities in which the client wants to invest in order to choose the right combination that will maximize investor return for a given level of risk.

Portfolio managers work in different financial services sectors, such as banks, insurance companies, investment management firms and investment fund companies.

The skill set Portfolio managers must be able to analyze a situation and have great communication skills. They must have an excellent knowledge of markets and investment products.

The education Portfolio managers can begin their careers as investment analysts after obtaining a university degree or a Master in Business Administration (MBA). They can then obtain their Chartered Financial Analyst (CFA) designation after taking three courses from the CFA Institute. Another option is to obtain their Canadian Investment Manager designation from the Canadian Securities Institute (CSI).

72

Vision 5

Related professions The field of finance offers many other professions besides portfolio management. For example, a person can work as a stockbroker, trader, money market dealer, economist, chartist or financial planner.

The Institut québécois de planification financière (IQPF) is the only organization in Québec authorized to grant a financial planning diploma, which gives the graduate the right to bear this title. Every year, the IQPF awards the Prix de journalisme en littératie financière to a journalist in Québec who has contributed to promoting financial planning.

Portfolio management is a constantly moving and evolving field.

Some managers acquire specializations, such as the management of shares or bonds. Others work with both types of securities.

A portfolio manager advises a client to invest a certain amount of money at an annual compound interest rate in a portfolio of company shares. The following table of values shows the evolution of this portfolio over a given period.

Evolution of a portfolio of company shares Time elapsed (years) Accumulated capital ($)

2

3

4

4984.13

5078.83

5175.33

According to the manager’s estimates, the accumulated capital Cn (in $) of this portfolio should evolve on average according to the rule Cn 5 C0(1.019)n, where n represents the time elapsed (in years) and C0, the initial capital (in $).

The portfolio manager also advises the client to invest $5,400 in a portfolio of savings bonds at an annual compound interest rate of 0.8% for the 1st year, 0.9% for the next 2 years, and 1% for the last 3 years.

1. What is the initial capital of

the portfolio of company shares?

2. What will be the

accumulated capital of the portfolio of company shares in: a) 5 years? b) 8 years and 3 months? c) 10 years and 6 months?

3. What will be the

accumulated capital of the portfolio of savings bonds in 6 years?

In the workplace

73

1 Which of the following expressions are false?

4 5 log3 81

A E

1 3

5 log1331 11

B 25 5 log5 125

C

7 5 log2 49

G

F

1 3

2 5 log8

81 5 log9 2

2 Which of the following expressions are equivalent to

log10 15 log10 3

D

2 5 log8 64

H 5 5 log1 32 2

?

A

log 15

B

log 3

C

log3 15

D

log 5

E

log10 3 log10 15

F

log 15 log 3

G

 2.465

H

 0.4057

3 Rewrite the exponential expressions below in logarithmic form. a) 154 5 50 625

1

b) 123 5 1728

c) 77765 5 6

d) 95 5

1 59 049

4 Rewrite the logarithmic expressions below in exponential form. a) log16 4096 5 3

b) log20 400 5 2

c) log4

1 16 384

5 7

d) log161 051 11 5

1 5

5 Without using a calculator, determine the value of each of the following expressions. a) log5 25 2 log4 64 2 log 100 1 log7 7 1

1 3

1 36

c) 4 log5 52 1 log2 64 1 4 log6

1 8

b) 3 log2 16 1 6 log16 4 2 5 log 1 1 2 log9 93

2 log1 2

d) logp p 1 logr 1 2 logs s 3 1 2 logt t 5

6 Solve the following exponential equations. a) 5(7)x 5 38

b) 4(0.9)x 5 25

3(10)2x 5 82.5 c) 2(6)3x 5 40 d)

7 Solve the following logarithmic equations. a) 5 log8 4x 5 10

b) 7 log9 10x 5 21

c) 6 log3 x 2 5 24

d) 4 log0.5 (16x 2 176) 5 20

4

8 Calculate the accumulated capital for each of the following situations. a) We invest $6,300 for 4.25 years at an

annual simple interest rate of 9%.

c) We invest $2,800 for 9 years at an annual

compound interest rate of 7.85%.

e) We invest $5,250 for 7.5 years at a

74

monthly compound interest rate of 1.05%.

Vision 5

b) We borrow $4,750 for 6.75 years at a

monthly simple interest rate of 0.7%.

d) We borrow $9,700 for 8 years at a semi

annual compound interest rate of 4.25%.

f ) We borrow $420 for 10 weeks at a daily

compound interest rate of 0.02%.

9 Calculate the initial capital for each of the following situations. a) In 9 years, the accumulated capital of an

b) In 2 years, the repayment of a debt will be

investment will be $5,067.60. The semiannual simple interest rate is 3.6%.

$3,434.11 at a weekly simple interest rate of 0.37%.

c) In 6 years, the accumulated capital of

an investment will be $16,030.55. The quarterly compound interest rate is 2.05%.

e) At a monthly compound interest rate of

1.9%, the accumulated capital will be $31,726.17 in 4 years and 5 months.

d) In 7 years, the repayment of a debt will be

$4,476.28 at a weekly compound interest rate of 0.11%.

f ) At an annual compound interest rate of

8.06%, the loan will cost $62,681.48 after 9 years.

10 Calculate the term of the investment or loan for each of the following situations. a) An initial capital of $2,400 generates an

accumulated capital of $3,750 at a semiannual simple interest rate of 3.75%.

c) An initial capital of $3,700 earns an

accumulated capital of $7,186.56 at an annual compound interest rate of 11.7%.

e) An initial capital of $33,800 generates an

accumulated capital of $63,768.97 at a monthly compound interest rate of 0.85%.

b) The repayment of a $5,975 debt at a

quarterly simple interest rate of 2.1% is $8,735.45.

d) The repayment of a $6,975 loan will cost

$8,250.57 at a daily compound interest rate of 0.06%.

f ) The repayment of a $8,800 debt at a

quarterly compound interest rate of 2.45% is $15,355.28.

11 Calculate the interest rate for each of the following situations. a) A $4,550 investment at an annual simple

b) A $5,900 investment yields $9,715.60

interest rate yields $8,685.95 after 9 years.

c) The repayment of a $21,300 loan at a

weekly simple interest rate is $27,779.46 after 4.5 years.

e) An initial capital of $13,350 at a weekly

compound interest rate generates an accumulated capital of $24,903.73 after 7.5 years.

after 6 years. The compound interest rate is quarterly.

d) We borrow an initial capital of $3,850. In

90 days, its repayment will cost $3,948.24.

The compound interest rate is daily.

f ) The repayment of an initial capital of

$22,700 at a semi-annual compound interest rate is $91,535.90 after 11.5Â years.

Overview

75

12 Calculate the accumulated capital for each of the following situations. a) We invest $16,500 for 6 years and

9 months at an annual compound interest rate of 7%.

c) We invest $37,700 for 5 half-years and

3 months at a semi-annual compound interest rate of 4.6%.

b) We borrow $4,300 for 4 years and

6 months at an annual compound interest rate of 8.3%.

d) We borrow $5,600 for 3 years and

1 month at a quarterly compound interest rate of 3.8%.

13 Calculate the final value for each of the following situations. a) An asset’s value of $7,297 increases by an

average of 2.8% per year over 6 years.

c) The value of a share bought at $18.47

increases by an average of 6.4% per year over 5 years.

b) An asset’s value of $2,899 decreases by an

average of 3.1% per year over 7 years.

d) The value of a share bought at $7.33

decreases by an average of 9.2% per year over 8 years.

14 Calculate the initial value for each of the following situations. a) An asset’s value has increased by an

average of 11.3% per year over 3 years to reach $893.45.

c) A share’s value has increased by an

average of 3.5% per year over 9 years to reach $22.28.

b) An asset’s value has decreased by an

average of 7.7% per year over 10 years to reach $1,833.68.

d) A share’s value has decreased by an

average of 8.97% per year over 4 years to reach $19.03.

15 We invest $4,800 at an annual compound interest rate of 7.1% for 6 years. Over the same term, we invest $5,450 at a monthly simple interest rate of 1.4%. What is the total value of the accumulated capital?

16 During a school trip to China, a student posts a photograph online. On the first day, 38 people see the photo. The number increases by an average of 22% per day. a) How many people will see the photo on the 14th day?

The Great Wall of China is the largest monument ever built by humans. The wall is 5 to 7 m wide and 5 to 17 m high, depending on the section. The construction of the Great Wall of China stretched over almost 2 millennia.

b) On what day will 12 140 people see the photo?

17 Jérémy works as a lifeguard at the municipal pool. His hourly rate is $21.72. His employer gives him a raise of 2.85% per year over the next several years. What will be Jérémy’s hourly rate in 6 years?

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Vision 5

18 CALCIUM CHLORIDE Calcium chloride is used to de-ice the roads during Québec’s winters. Under certain conditions, this salt can reduce the thickness of a layer of ice by 12% per hour after being applied.

During an ice storm, 7.5 cm of ice accumulates on the roads before the The Ministère des Transports du Québec (MTQ) has de-icer spreads the salt. almost 8000 vehicles and equipment. The MTQ has a a) How thick will the ice be 4 h after the de-icer spreads the salt?

highly diversified vehicle fleet with a variety of cars that include pickup trucks, trucks, electric vehicles, and a variety of heavy machinery.

b) After how many hours will the thickness of the ice be 3 cm?

19 Manon borrowed money in order to buy some land. In 7 years, Manon’s debt will be $86,753.50. If the monthly simple interest rate is 0.95%, how much money did Manon borrow?

20 SUPER BALLS Super Balls are made out of a synthetic rubber invented by chemist Norman Stingley in 1964. They bounce much higher than other balls. At each bounce, a Super Ball loses 8% of the maximum height reached during the previous bounce. In an experiment, one of these balls is dropped from a height of 450 cm. a) How high will the ball be on its 5th bounce? b) Which bounce will have a height of 165.45 cm?

21 A jeweller estimates that the value of necklaces increases by an average of 8% per year, while the value of watches increases by an average of 6% per year. If a necklace and a watch, which cost $245 and $318 respectively, are sold after 7 years, how much profit will he make?

22 The hull of Marie-Claude’s sailboat is in need of repair. She borrows $2,700 from her friend at an annual simple interest rate of 9.3%, which she will repay in 3 years and 9 months. How much money will Marie-Claude need in order to pay back her friend?

The Transat Québec Saint-Malo is a sailing transoceanic race where crewed sailboats sail non-stop across the Atlantic, from west to east.

Every four years since 1984, several boats sail out on the St. Lawrence River, between Québec City and Lévis, and head for France.

23 In a municipality in 1995, the average property value was $173,250. In 2015, it was $409,905. A couple’s property was worth $192,700 in 1995. If this asset followed this municipality’s average increase in property value, how much was it worth in 2015?

Overview

77

24 Benoît wants to invest $4,000 at an annual interest rate of 9% over 4 years. In his opinion, a compound interest rate will yield at least $200 more than simple interest. Is Benoît right?

Justify your answer.

25 Two friends plan to travel to Alaska ten years after they graduate from high school. They estimate that the trip will cost $4,200. If one friend has $2,500, how long after graduating from high school should he invest this sum at an annual compound interest rate of 9.03% in order to raise the money needed for the trip?

26 Paul collects sculptures. After selling a sculpture he invests the proceeds from the sale at an annual simple interest rate of 5.25% over 4 years. Given that the accumulated value of this investment is $2,879.80, what was the sculpture’s selling price?

The Thinker by Auguste Rodin is one of the most famous bronze sculptures. It is of a man who appears to be deep in thought, perhaps contemplating an important dilemma. Created around 1880, the original plaster casting measures 71.5 cm in height. Over twenty castings of the sculpture are displayed in museums throughout the world. Most of the castings were made in the sculptor’s lifetime.

27 Donald inherited $12,000, which he would like to invest at a semi-annual simple interest rate of 5%. How many years will it take for Donald’s initial capital to double?

28 Stéphanie sells her hockey card collection for $5,625. She invests the amount at a quarterly simple interest rate of 3.1% over 3 years. She then reinvests the capital at a monthly compound interest rate of 0.75% to obtain an accumulated capital of $11,046.87. What is the term of Stéphanie’s 2nd investment?

A 1979 hockey card of Wayne Gretzky, the famous Canadian hockey player, was sold at a record price in August 2016. An anonymous buyer purchased it for US$465,000. It was the first card ever made of the player.

Nicknamed “The Great One,” Gretzky wore number 99 for his entire career. He is the only player whose number has been retired from the National Hockey League (NHL).

29 François finds two banknotes of the same value on the street. Instead of spending them, he invests them at an compound interest rate of 1.05% per month over 4 years and 3 months. François knows that the accumulated capital will be $340.71. What is the value of each banknote that François found?

30 PERFORMING ARTS In Québec from 2011 to 2013, the average price of a live show increased considerably. In 2 years, the price rose by $9 from $32 to $41. If the trend continues, what will be the average price of a show in 2025?

In 2014, according to the Institut de la statistique du Québec, 6 800 000 spectators attended 17 100 paying shows throughout the province. All together, the ticket sales increased by 4% to reach $238M.

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Vision 5

31 In order to renovate his home, Antoine borrowed $17,800 and had to pay back $25,543. If it took him 5 years to repay his loan, what was the semi-annual simple interest rate of Antoine’s loan?

Home renovation is a very important area of economic activity in Québec. Most house owners renovate certain rooms approximately every seven years. The most common home renovations are updating the kitchen and bathroom and changing the windows and doors.

32 Andrée-Anne has a passion for flying. Having obtained her pilot licence, she borrows $156,000 to buy an ultralight aircraft. She takes out the loan at a semi-annual compound interest rate of 3.3%. To repay it, Andrée-Anne will have to spend $308,483.08. How many years will it take Andrée-Anne to repay her loan?

33 OIL PRICES The crude oil price per barrel is listed on NYMEX (New York Mercantile Exchange). Prices can vary several times in one day. From June 2014 to January 2016, the crude oil price per barrel dropped by an average of 6.74% per month to reach a floor price of $30. What was the crude oil price per barrel 19 months earlier, that is, in June 2014?

The significant oil price drop in 2015 and 2016 can be partly explained by the global economic downturn as well as increased production.

However, according to the National Energy Board (NEB), the crude oil price per barrel is expected to rise to US$80 by 2020 and reach US$105 by 2040.

34 MUTUAL FUNDS Mutual funds are portfolios of Canadian and foreign stocks, bonds, mortgage-backed securities and money market securities. Hubert sold his skydiving equipment for $4,750. He invested the money in a mutual fund at a financial institution.

Its present value is $8,193.58. Its annual return is 8.1%. How long ago did Hubert invest his money?

35 In order to make their capital grow, a couple invests $6,400 with an investment advisor.

After 6 years, the accumulated capital and compounded interest amount to $9,712.89.

What is the compound interest rate applied: a) annually? b) quarterly? c) monthly?

Overview

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36 CAMPING INDUSTRY The camping industry is booming in Québec. The province has almost 900 campgrounds with over 115 000 campsites.

In 2012, the total spending per stay of campers in Québec reached $530 million. A person borrows $9,600 to buy a camper trailer. At maturity, the loan results in a debt of $14,184.16. What was the loan’s term, in years, at: a) a monthly simple interest rate of 0.6%? b) an annual compound interest rate of 5.85%? c) a weekly compound interest rate of 0.15%?

Over 2 million Quebecers go camping.

Approximately 40% of campers use a tent, 50% use a recreational vehicle and 10% stay at ready-to-camp sites. Between the four of them, the Laurentians, Eastern Townships, Québec City and Montérégie regions boast almost half of Québec’s campsites.

37 Alyson needs to borrow some money to buy a two-seater ROV. Her bank offers her a compound interest rate of 8.65% per year for 6.5 years. At the end of the term, Alyson will have to pay back $16,218.05. How much did Alyson borrow?

38 CANADIAN OIL PRODUCTION Canada is the 6th largest producer of crude oil in the world. However, its share of the market is only 4%. In 2015, Canada produced 3.87 million barrels per day (MBD). Despite increasingly strict environmental and climate policies, experts believe that Canada’s oil production will continue to grow, reaching 7 MBD by 2040. If this According to the National Energy Board (NEB), the oil produced from prediction is correct, what is the the tar sands represents 90% of Canadian oil production. There are two average annual increase of the main regions of oil production in Canada: one in Western Canada, which number of barrels of oil? includes Alberta, Saskatchewan, British Columbia and Manitoba, and the other in Eastern Canada, in Nova Scotia and Newfoundland and Labrador.

39 Martin wants to build a garage. He needs to borrow $21,000 to finance the materials and tools required. A bank offers him a loan at a compound interest rate of 0.68% per month over an 8-year term. What would be an annual simple interest rate equivalent to the one offered?

40 MICROWAVE OVENS The first microwave ovens hit the market in 1975. By 2015, over 80% of households had one and their average retail price was $150. The price of a microwave oven has been decreasing by an average of 5.2% per year since 1975. What was the average retail price for a microwave oven in 1975?

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Vision 5

41 ELECTRIC CARS In 2016, there were 8000 electric cars in Québec. In order to reduce greenhouse gas emissions, the Government of Québec encourages Quebecers to buy electric vehicles by In 2016, in order to encourage people helping them financially. The goal is to to choose less-polluting vehicles, have 100 000 electric cars in the the Government of Québec offered province by 2020. If this objective is a rebate of up to $8,000 on the purchase of an electric or hybrid car reached, what would have been the and an additional $600 on installing a average annual increase of electric cars charging station. since 2016?

42 The price of a company’s shares has fluctuated in recent years. In 2005, the share price was $42.12, and it increased by an average of 8.42% per year until 2011. The company then underwent some difficulties due to a recession. From 2011 to 2015, the value of the shares dropped by 5.1% per year on average. What was the value of a share in 2015?

43 A university awarded Tanya a $3,500 scholarship for her high grades. She would like to invest this sum over 8 years. She is offered the following three investments.

Investment A

Investment B

Investment C

An annual compound interest rate of 6%

A semi-annual compound interest rate of 2.9%

A monthly compound interest rate of 0.48%

Which investment is the most advantageous for Tanya? Justify your answer.

44 In 1990, a family in Québec needed $12,000 per year to provide for their basic needs.

Since then, the cost of living has increased by an average of 3.2% per year. In what year did a family in Québec need $25,000 to provide for their basic needs?

45 COLLECTIBLE CARS In the first 20 years after a car is purchased, its value decreases by an average of 14.5% per year. The value then remains stable for the next 10 years. Then, if the car has been carefully maintained and has no mechanical problems, its value begins to increase by an average of 6.8% per year. How many years after a car is purchased for $48,000 will it reach its initial value again?

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bankof problems

1 ANTIBIOTICS Antibiotics are used in medicine to fight bacterial infections. However, taking antibiotics also affects the bacterial population of our digestive system.

A patient is prescribed antibiotics for 10 days. The following rules are used to calculate the bacterial population P (in percent) of the patient’s digestive system according to the time t (in days) elapsed since the treatment began.

Evolution of the bacterial population during the treatment

Evolution of the bacterial population after the treatment until it reaches a normal level

P 5 100(0.85)t

P 5 Pend of treatment (1.15)(t 2 10)

According to a doctor, once the treatment ends, the time needed for the digestive system’s bacterial population to return to a normal level is greater than the duration of the treatment. Prove whether this claim is true or false.

Antibiotics are molecules that destroy or limit bacterial growth. They are used on human beings as well as on animals in veterinary medicine.

2 Buying a house is one of life’s significant milestones. The following formula allows you to calculate the number of payments needed to repay a mortgage loan.

– M is the amount of the mortgage (in $) M 5 P 3

12



1 11i

i

n

 , where: – P is the amount of each payment (in $) – n is the number of monthly payments – i is the monthly interest rate

To buy their first house, a couple takes out a mortgage loan of $192,500 at an annual interest rate of 5.25%. Acting as their mortgage broker, explain to these people the possible savings they would have if they opt for monthly payments of $1,100 instead of $1,000.

3 An investor places $1,700 in Investment A at a monthly simple interest rate of 0.25%, whose maturity value will be $1,904. Over the same term, she also places an amount in Investment B at a quarterly compound interest rate of 1.75%, whose maturity value will be $3,233.83. What was the initial capital of Investment B ?

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Vision 5

4 In the following two simultaneous situations, a city’s population varies as a function of the time elapsed since the beginning of 2012.

City A

City B

Every year since 2012, the population has increased by 30% relative to the previous year.

The population P evolves according to the rule P 5 P0e , where P0 represents the population at the beginning of 2012, and t represents the time elapsed (in years).

3t 10

Relative to the beginning of 2012, which city’s population will triple first and how much time before the other city will it do so?

According to the Institut de la statistique du Québec, from 2011 to 2015 the average annual population growth rate in Québec was 7.9%. Over the same period, the regions of Laval and the Laurentians also underwent a population growth of 11.5% and 11.2% respectively, while the Gaspésie–Îles-de-la-Madeleine region experienced a population decline of 7.2%.

5 The following statement is spoken by a fund manager to an investor.

“Over the course of the next 5 years, you will deposit $2,000 every January 1st in a mutual fund. The annual compound interest rate will be 3%, and the interest will be calculated at the end of each year.”

The fund manager claims that the maturity value of the investment will be greater than $10,500. Prove whether this claim is true or false.

6 In 2000, the price per litre of gas at the pump was $0.78. From 2000 to 2008, this price increased by an average of 8.34% per year. In 2009 and 2010, this price decreased to reach $1.05. What was the average percentage decrease per year of the average price per litre of gas at the pump for 2009 and 2010?

7 The value of an investment made at a quarterly compound interest rate of 1.2% is $14,200 at maturity after a term of 12 years. Show that that accumulated capital of the investment will be less than 1.6 times its initial value, if made at an annual simple interest rate of 4.3%.

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83

Probability and random experiments Is it possible to predict or even control random events? How do you determine the probability that it will rain tomorrow? How do you know whether or not a game of chance is fair?

How does probability influence the expected gain of certain companies? In “Vision 6,” you will calculate probabilities pertaining to meteorology, financial analysis and other areas. You will use mathematical expectation to determine whether a game of chance is fair, and you will make changes to an unfair game in order to make it fair.

Arithmetic and algebra

Geometry

Graphs

Probability • • • •

Methods of enumeration Subjective probability Odds for and odds against Distinguishing between probability and odds • Mathematical expectation and fairness

The human immunodeficiency virus

Pierre-Simon de Laplace

Financial analysts

student book volume

The 3rd volume of the Secondary Cycle Two, Year Three Visions series, Cultural, Social and Technical option, was designed in the spirit of the Québec Education Program and addresses an update to the program.

Student Book, volume 3

• Vision 1 Supplement, including inequalities

• Vision 2 Supplement, including the law of cosines • Vision 5, a chapter on financial mathematics, including logarithms, simple interest and compound interest in different financial contexts

• Vision 6, a chapter on probabilities, including subjective probabilities, odds for, odds against and mathematical expectation

Dominique Boivin • Richard Cadieux • Claude Boivin • Antoine Ledoux Étienne Meyer • Dominic Paul • Nathalie Ricard • Vincent Roy

Cultural, Social and Technical

• Many

reproducible sheets and their answer keys (Support, Consolidation, Enrichment, Snapshot)

• LES including the logbooks and evaluation grids • Tests and their answer keys

Secondary Cycle Two, Year Three

• Teaching notes for each chapter, including detailed lesson plans

MATHEMATICS

Teaching Guide, volume 3

volume Cultural, Social and Technical

• Volume 1, Student Book • Volume 2, Student Book • Volume 3, Student Book • Volumes 1 and 2, Teaching Guide • Volume 3, Teaching Guide

3

The components of the Visions series:

MATHEMATICS Secondary Cycle Two, Year Three Dominique Boivin Richard Cadieux Claude Boivin Antoine Ledoux Étienne Meyer Dominic Paul Nathalie Ricard Vincent Roy