lecture 1 (part a) factorization - holy cross high

REMEMBER!

•Consult text-books for additional examples.

•Attempt as many as possible other similar

examples on your own.

•Compare your methods with those that were

discussed in the Video.

•Repeat this procedure until you are confident.

•Do not forget:

Practice makes perfect!

End of Video on Financial Mathematics

Financial

Mathematics

Grade 10 CAPS Mathematics

Video Series

Outcomes for this Video

In this DVD you will:

• Revise factorization.

LESSON 1.

• Revise simplification of algebraic fractions.

LESSON 2.

• Discuss when trinomials can be factorized.

LESSON 3.

3

In this video the focus will be on:

Simple and Compound Growth

(Lesson 1)

Applications of simple and compound growth

(Lesson 2)

Foreign exchange rates

(Lesson 3)

Simple and

Compound Growth


Video Series

Lesson 1

Outcomes for Lesson 1

In this lesson we will focus on:

• Important terminology

• The meaning of simple interest

• A formula for calculating simple interest

• Simple interest calculations

• The meaning of compound interest

• A formula for calculating compound interest

• Compound interest calculations

occurs wheAppreciat n a valueion incGrowth rea . ses

Important terminology

Two types of growth will be c Simple aonsider nd Comped: d oun

From an amount invEarned:

Interest Owed

ested

From an amount borrowed:I

Percentage Rate for an investment

Percentage Rate charged for an am

Interest rate ount borrowed

r

is the amount borrowed or invested, called the princ l. ipaP

is the accrued or final amount. A P I

is the number of investment (growth) peri s. odn

12 Example: 12% 0.1

1002

100

ri r i

Simple interest is a fixed percentage of the amount invested

or borrowed and is calculated on the original amount.

What is Simple Interest (Growth) ?

When simple interest is applied to an investment (loan), the

value of the investment (loan) increases by an agreed fixed

percentage at specific regular time intervals.

Simple interest entails to the

principal amount

adding a constant amount

at regular intervals.

The constant amount to the added to the principal at regular

intervals is equal to .100

rP i P

Assume that a principal amount is invested (borrowed)

for periods at a interest rate of .

P

n i

Formula to Calculate Simple Growth

1

2 1

3 2

4 3

Accrued amount at different time intervals:

1 1

2 1 (1 ) 1 2

3 1 2 (1 2 ) 1 3

4 1 3 (1 3 ) 1 4

At end of investment (gro

n A P Pi P i

n A A Pi P i Pi P i i P i



wth) period 1 1nA P ni P in

1A P i n

Simple Growth Formula :

I P i n

Interest earned :

Assume that an amount 100 rand is invested (borrowed)

for a period of 10 years at a simple rate of 0.07 (r 7%).

P

n i

Simple Interest: Accrued amount vs Growth period

Use 1 to find the accrued amounts

at differents time intervals .

n nA P i n A

n

Use Casio Calculator in table mode to complete

the ; input-output table 100 1below:

0 1 2 3 4 5 6 7 8 9 10

0.07n

n

yn

A

xA

n

0 1 2 3 4 5 6 7 8 9 10

100 107 114 121 128 135 142 149 156 163 170

7 7 7 7 7 7 7 7 7 7

n

n

n

A

A

Assume that an amount 100 rand is invested (borrowed)

for a period of 10 years at a simple rate of 7% ( 0.07).

P

n r i

Graphical View of Simple Interest Example

0 1 2 3 4 5 6 7 8 9 10

100 107 114 121 128 135 142 149 156 163 170

7 7 7 7 7 7 7 7 7 7

n

n

n

A

A

100

7; 0

n

n

A P P i n

y mx c

A f n

c P

Am P i m

n

An amount of R1 050 is invested at a simple interest rate of

4.75% p.a. for a period of 9 months. Calculate:

1 The simple interest (growth) earned on the investment;

2 Final value of the investment (accrued value).

Calculate Simple Interest Earned

1 050

4.750.0475

100 100

9years 0.75 years

12

P

ri

n

Know that : 1 Amount of growth (interest earned):

1 050 0.0475 0.75 R37.41

I P i n

2 Final value of investment:

R1 050 R37.41

R1 087.41

A P I

Alternatively: Use formula 1 .

1050 1 0.75 0.0475 R1 087.41

A P ni

A

A lady invested R4 000 at 8.75% p.a. in a fixed deposit account.

If simple interest is paid at the end of 4 years, calculate:

1 How much money she will receive;

2 How much interest she will earn.

Calculate Accrued Amount and Interest

4 000

0.0875

4

P

i

n

Know that :

1 1

4 000 1 0.0875 4

R5 400

A P i n

A

2 R5 400 R4 000 R1 400I A P

4 000 0.0875 4 R1 400I P i n or

How much money should you invest in a fixed deposit

account paying 9% p.a. simple interest if you would like

to receive R8 000 at the end of 5 years?

Calculating the Principal Amount

8 000

0.09

5

?

A

i

n

P

Summary : 1

1

A P i n

AP

i n

8 000

1 0.09 5

R5 517.24

P

Tutorial 1: Simple Interest

1 A student borrows R15 000 for 3 years at

simple interest to pay for his studies.

If he has to pay back R20 625 at the end

of the three years, what rate of simple

interest per annum is he paying?

2 If I invest R15 000 in a fixed deposit account

at 10% p.a. simple interest,

how long will it take before

I will have R45 000?

PAUSE VIDEO

• Do Tutorial 1

• Then View Solutions

Tutorial 1: Problem 1: Suggested Solution

1 A student borrows R15 000 for 3 years at

simple interest to pay for his studies.

If he has to pay back R20 625 at the end

of the three years, what rate of simple

interest per annum is he paying?

1

1

1

A P i n

Ai n

P

AA PPi

n n P

15 000

20 625

3

?

P

A

n

i

Summary :

1

20 625 1

15 000 0.1253

A

Pin

i

20 625 15 000 or 0.125 12.5%

3 15 000

A Pi

n P

12.5%r


2 If I invest R15 000 in a fixed deposit

account at 10% p.a. simple interest, how

long will it take before I will have R45 000?

15 000

45 000

0.1

?

P

A

i

n

Summary :

1

1

1

A P i n

Ai n

P

A A Pi n

P P

A Pn

P i

45 000 15 000

15 000 0.1

20 years

A Pn

P i

n

n

Simple interest for each period is always calculated on

the original amount borrowed or invested before it is

added to the accrued amount at the start of that period.

What is Compound Interest (Growth) ?

When calculating compound interest, the

or earned in

.

interest

charged each period applies to the

accrued amount at the start of that period

This means that the principal increases (grows).

Interest earned or charged for the next period

is calculated on the increased principal amount.

Assume that an amount is invested (borrowed)

for periods at a rate of .

P

n i

Formula to Calculate Compound Growth

1

2

2 1 1

2 2 32

3 2 2

3 3 43

4 3 3

Accrued amount at differents time intervals:

1 1

2 1 1 1 (1 ) 1

3 1 1 (1 ) 1 1

4 1 1 (1 ) 1 1

At end

n A P Pi P i

n A A Ai P i P i i P i i P i

n A A A i P i P i i P i i P i

n A A A i P i P i i P i i P i

of investment (growth) period 1n

nA P i

1n

A P i

Compound Growth Formula :

Assume that an amount 100 rand is invested (borrowed) for

a period of 10 years at a compound rate of 0.15( 15%) p.a.

P

n i r

Compound Interest: Accrued amount vs Growth period

Use 1 to find accrued amounts

at differents time intervals .

n

n nA P i A

n

Use Casio Calculator in table mode to complete

the ; input-output table 100 1below:

0 1 2 3 4 5 6 7 8 9 10

0.15n

n

xyn A

n

A

:1

0 1 2 3 4 5 6 7 8 9 10

100 115 132.25 152.09 174.90 201.14 231.31 266.00 305.90 351.79 404.56

1.15 1.15 1.15 1.15 1.15 1.15 1.15 1.15 1.15 1.15A Ann

nAn

Graphical View of Compound Interest Example

1

1

100

1 1.15 1

1.15

n

n

nn

x

n

A P i

y a b

A f n

a P

b i

Ab

A

Exponential Growth

function

Assume that an amount 100 rand is invested (borrowed) for

a period of 10 years at a compound rate of 0.15( 15%) p.a.

P

n i r

0 1 2 3 4 5 6 7 8 9 10

100 115 132.25 152.09 174.90 201.14 231.31 266.00 305.90 351.79 404.56

nAn

Comparison between Compound and Simple Interest

Assume 100 rand, 10 years and 0.15( 15%)p.a.

Compare compound and simple interest at the same rate.

P n i r

Using a table model:

0 1 2 4 6 8 10

Compound 100 115 132.25 174.90 231.31 305.90 404.56

Simple 100 115 130 160 190 220 250

n

n

n

A

A

Same if 1

Compound growth

bigger than Simple

growth if 1

n

n

A lady deposits R5 000 in a fixed account for 7 years

at an interest rate of 9.25% compounded annually.

What amount of money will be available after 7 years?

Calculate Accrued Compound Growth Amount

5 000

0.0925

7

?

P

i

n

A

Summary :

7

7

1

5 000 1 0.0925

5 000 1.0925

R9 287.96

nA P i

A

Use your Casio Calculator

in the computation of .A

What is the annual compound interest rate being

charged by Africa Bank on a loan of R25 000 if

R45 000 is to is to be repaid in 8 years time in full

settlement of the loan?

Calculating the Compound Interest Rate

25 000

45 000

8

?

P

A

n

i

Summary :

1

1

1

1

n

n

n

n

A P i

Ai

P

Ai

P

Ai

P

845 000

125 000

0.07623983

7.624%

i

i

r

Use your Casio Calculator

in the computation of .i

Tutorial 2: Compound Interest

1 An amount of R3 000 is invested at an interest rate of

12% compounded annually.

How long will it take, to the nearest month, to reach a

value of R8 500?

2 How much money should you invest in a fixed deposit

account paying 9% p.a. compound

interest if you would

like to receive R8 000

at the end of 5 years?

PAUSE VIDEO

• Do Tutorial 2



1

1

n

n

A P i

Ai

P

3 000

8 500

0.12

?

P

A

i

n

Summary :

Use the TABLE MODE of your CASIO calculator to determine

the approximate values of which will satisfy 1.12 2.83.nn

85001.12 2.83

3000

n

1 An amount of R3 000 is invested at an interest rate of 12% compounded annually.

How long will it take, to the nearest month, to reach a value of R8 500?

6 7 8 9 10

1.12 1.97 2.21 2.46 2.77 3.11n

n

9 9.1 9.2 9.3 9.4

1.12 2.77 2.80 2.84 2.87 2.90n

n

9.12 9.14 9.16 9.18 9.20

1.12 2.811 2.817 2.824 2.830 2.837n

n

9.185 9.186 9.187 9.188 9.189

1.12 2.8318 2.8321 2.8247 2.8328 2.8331n

n

9.189 years

9 years 2 months

n

n


8 000

5

0.09

?

A

n

i

P

Summary :

1

1

n

n

A P i

AP

i

5

8 000

1.09

R5 199.45

P

P

2 How much money should you invest in a fixed deposit

account paying 9% p.a. compound interest if you would

like to receive R8 000 at the end of 5 years?

Applications of simple

and compound growth


Video Series

Lesson 2


In this lesson we will apply simple and

compound growth to:

• Hire purchase

• Inflation

• Population growth

What is hire purchase?

A hire purchase agreement is a financial arrangement between a buyer and

a seller about how the buyer will pay for the desired product.

Interest on a hire purchase (loan) is charged at a

on the amount owing. simple interest rate

Most loan agreements require that a deposit be paid before the customer can

take ownership of the product. Principal amount of a loan is therefore the

cash price minus the deposit. The accrued amount of the loan is then divided

into monthly payments over the period of the loan.

Customer basicly 'hire' the item from the shop. If customer stop paying

at any stage, the item can be repossessed by the shop. You are only the

owner of the item once the last payment is made. Due to risk factors the

seller sometimes add an insurance premium to the monthly installment.

John decided to purchase a bicycle, advertised for R2 700

in the newspaper. The hire purchase agreement requires a

deposit of 10% and that the loan on which interest is charged

at 8.5% p.a. be settled by making 24 equal monthly payments.

Calculate what John's monthly payments will be.

Hire purchase with deposit

Deposit 10% of R2 700 R270

Cash price Deposit R2 430

0.085

242 years

12

P

i

n

Summary :

1

2430 1 0.085 2

R2 843.10

A P i n

A

Monthly payments

2430 1 0.085 2118.4625 R118.46

24

R2 843.10 R118.46 24 R0.06

Six cents will be added to one of the 24 payments.

Mary buys a TV on a hire purchase agreement. Cash price

1is R8 750 and interest of 16 % p.a. will be charged over

4

the 42 months repayment period. An insurance premium of

R13.75 is added to every monthly payment.

Calculate what Mary's monthly payments will be.

Hire purchase plus insurance

Cash price R8 750

0.1625

42 years

12

P

i

n

Summary : 1

42 8 750 1 0.1625

12

A P i n

A

Monthly payments

428 750 1 0.1625

1213.75 R340.57

42

Tutorial 3: Hire Purchase

Your father wants to buy a fridge on a hire purchase agreement.

The cash price of the fridge is R7 500. He is required to pay a

1deposit of 17 % and pay the balance of the loan off over a

2

period of 39 months at an interest rate of 13.5% p.a.

1 What is the principal amount?

2 What is the accumulated loan amount?

3 What are the monthly repayments?

4 What is the total amount he has paid for the fridge?

PAUSE VIDEO

• Do Tutorial 3


Tutorial 3: Hire Purchase: Suggested Solution

Cash price: R7 500

Deposit: 17.5%

Interest rate: 0.135

39Over 39 months: years

12

i

n

1 Principal amount:

Cash price Deposit

7 500 0.175 7 500

R6 187.50

P

2 Accumulated loan amount:

39 6 187.50 1 0.135

12

R8 902.27

A

3 Monthly repayments:

396187.50 1 0.135

12

39 39

R228.26

A

4 Total paid: Deposit

8 902.27 0.175 7 500

R10 214.77

A

R1312.50

Inflation

There are many factors that influence the change

in the price of an item, one of them is inflation.

Since the rate of inflation increases from year

to year, it is calculated using the formula for

. compound interest

A low annual inflation rate normally is

an indication of very healthy economy.

Inflation is the average increase in the price of

goods each year and is given as a percentage.

The average inflation rate for the next 10 year period,

based on statistical evidence, is estimated at 8.75%.

A loaf of bread currently costs R7.56. How much will it

cost in 10 years time, based on the estimated inflation rate?

Calculate future price based on inflation

7.56

0.0875

10 years

?

P

i

n

A

Summary :

10

1

7.56 1 0.0875

17.49

nA P i

A

Estimation is that a loaf of bread will cost

R17.49 in 10 years time.

Tutorial 4: Inflation

The current average price of a small car is R95 000.

The average rate of inflation, based on statistical data,

over the past 30 years was 10.65%. How much did a

similar car cost 30 years ago?

PAUSE VIDEO

• Do Tutorial 4


Tutorial 4: Past Price based on Average Inflation

95 000

0.1065

30 years

?

A

i

n

P

Summary :

11

nn

AA P i P

i

Price 30 years ago was R4 562.27.

Can you believe this? Ask your grandparents.

The current average price of a small car is R95 000.

The average rate of inflation, based on statistical data,

over the past 30 years was 10.65%. How much did a

similar car cost 30 years ago?

30 30

95 000 95000

1.10651 0.1065P

Population Growth

Family trees grow exponentially as every person

born has the ability to start another family.

For this reason we can also calculate population

growth using the formula.compound interest

Population growth depends on a variety of

factors. Growth rate is normally much lower

in developed countries.

The current population in Port Elizabeth is 1 786 630.

The average population growth rate in PE, based on

statistical evidence, is estimated at 3.16% p.a. over the

next 15 years. What can the city planners expect the

population of PE to be in 15 years time?

Calculate future population based on growth rate

1 786 630

0.0316

15 years

?

P

i

n

A

Summary :

15

1

1 786 630 1 0.0316

2 849 080

nA P i

A

City planners must make provision for housing and

basic service delivery for an additional 1 062 450

people - which is a huge responsibility.

Tutorial 5: Population Inflation

A small town in the Eastern Cape Province is experiencing

a huge increase in births. According to available statistics

the average growth rate for the next 5 years is estimated at

12.65% p.a. How many babies will be born to the current

2 450 residents in each of the next two years?

PAUSE VIDEO

• Do Tutorial 5


A small town in the Eastern Cape Province is experiencing

a huge increase in births. According to available statistics

the average growth rate for the next 5 years is estimated at

12.65% p.a. How many babies will be born to the current

2 450 residents in each of the next two years?

Tutorial 5: New borns in next two years

2 450

0.1265

P

i

Summary :

1n

A P i

22

2 450 1.1265 3 109A

11

2 450 1.1265 2 759.9A

New borns in year one can be 2 759 2 450 309.

New bornes in first two years: 3109 2450 659

Thus 659 309 350 may be born in year two.

Exchange rates


Video Series

Lesson 3


In this lesson we will focus on:

• The meaning of currency and exchange rates

• Different types of currencies around the world

• Exchange rate tables

• Conversion between different currencies

Meaning of currency and exchange rates

Exchange rates refer to the cost of buying currencies

from different countries around the world.

Different countries have their own currencies.

Currency is the type of money that a country

uses to buy and sell goods and services.

Exchange rates are important because countries

trade with each other.

SA imports computers from America and pay them in dollar $ Examples:

SA exports apples to Europe and they pay us in rand R

Exchange rates are important for individuals too.

They travel to foreign countries and will need the

currency of that country to pay for local items.

Currencies of some well-know countries/regions

The currency of South Africa

is the Rand R

The currency of America

is the Dollar $

The currency of the

United Kingdom is the Pound £

The currency of Japan is the

Japanese Yen

The currency of the

European Union is the Euro

Some examples of currencies around the world

€

¥

For more examples consult your local newspapers.

Exchange RateCurrency Meaning?

in Rand

British Pound 13.1702

Euro 10.6090

US Dollar 8.3555

Japanese Yen 0.1076

Table A

First three currencies are stronger than the rand

while the Yen is a weaker currency.

€1 will cost you R10.6090

Exchange rate: Comparing the values of currencies

Consult your daily newspapers for latest exchange rates

Consider table A below:

€

£

¥

$ $1 will cost you R8.3555

¥1 will cost you R0.1076

£1 will cost you R13.1702

Consult your daily newspapers for latest exchange rates

Consider table B below:

Exchange RateCurrency Meaning?

units per R1

British Pound 0.07593

Euro 0.09426

US Dollar 0.11968

Japanese Yen 9.29368

Table B

R1 will buy you €0.09426

Exchange rate: Value of One Rand

€

£

¥

$ R1 will buy you $0.11968

R1 will buy you R9.29368

R1 will buy you £0.07593

Use these extracts to convert:

1 100 rand to pounds 3 100 rand to euro

2 100 rand to yen 4 100 rand to dollar

Currency Conversions

£1 = R13.1702 R1 = €0.09426

¥1 = R0.1076 R1 = $0.11968

Consider the following exchange rate

extracts from a daily newspaper:

13.1702

0.1076

0.09426

0.11968×

(1) R100

(2) R100

(3) R100

(4) R100

£7.5929

¥929.368

€9.426

$11.968

James, a keen golfer, wants to buy a Tiger Woods putter.

He discovered on the Internet that he can purchase this

putter in Hong Kong (Japan) for 21 375 Yen and for

287 dollar in New York (America).

Example of exchange rates

£1 = R13.1702 R1 = €0.09426

¥1 = R0.1076 R1 = $0.11968

If he considers the exchange rates above which offer is the best?

0.1076 R2 299.95 ¥21 375

0.11968 R2 398.06 $287

Hong Kong offer is the best as it is about R100 cheaper.

Tutorial 6: Exchange Rates

Study the following exchange rate table:

Country Currency Exchange Rate

United Kingdom Pounds (£) R14.13

United States Dollars ($) R7.04

Aswin and Cosmo are waiters in a South African

restaurant which attracts many tourists from

abroad.

Aswin receives a £6 tip from a tourist and Cosmo

similarly receives $12.

Who received the better tip? PAUSE VIDEO

• Do Tutorial 6


Tutorial 6: Exchange Rates: Suggested Solution

Aswin and Cosmo are waiters in a South African restaurant

attracting many tourists from abroad. Aswin receives a £6 tip

form a tourist and Cosmo similarly received $12.

Who received the better tip?

Aswin received 6 14.13 R84.78

Cosmo received 12 7.04 R84.48

Aswin got 30 cents more than Cosmo.

Country Currency Exchange Rate

United Kingdom Pounds (£) R14.13

United States Dollars ($) R7.04

lecture 1 (part a) factorization - holy cross high

Documents