Wong Kah Seng

5 Cekap 2012

SMK Batu Lintang

additional mathematics project work 2|2012

CONTENT

Index Title Page

1 COVER PAGE 1

2 CONTENT 2

3 APPRECIATION 3

4 OBJECTIVES 4

5 INTRODUCTION 5

6 PART A 7

7 PART B 10

8 PART C 15

9 PART D 18

10 FURTHER EXPLORATION 21

11 CONCLUSION 22

12 REFLECTION 23

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APPRECIATION

First of all, I would like to thank God for giving me energy, strength and health to carry out this project work.

Next, I would like to thank my school for the opportunity of getting the task of completing this assignment. The school itself had also been a great medium of discussion among my friends and I when I meet any difficulties and problems faced in the project work itself.

Not forgetting my beloved parents who provided everything needed in this project work, such as money, Internet, books, computer and so on. They had contribute their time and spirit on sharing their experience with us. Their support had raised my hopes and motivated me in completing the project. Having more experience, I have gained numerous tips from them in the event of completing the project.

After that, I would like express my heartfelt thanks to my Additional Mathematics teachers, Mdm. Lucy Chan and Mr. Tan for guiding me throughout this project. When I faced some difficulties on doing the tasks, they will try their best to teach me patiently until I have grasped the concept and what I am supposed to achieve to complete the project work with perfection.

Lastly, I would like to thank my classmates who had shared ideas and provided some help on solving problems. Together we solved numerous tricky questions and were able to move on to completing the project. Once again I would love to thank everyone in making this assignment such a success.

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OBJECTIVES

All of our students in Form 5 are required to carry out an Additional Mathematics Project Work during the mid-term holiday. The project was assigned to individuals to be carried out. Upon completion of the Additional Mathematics Project Work, I had gained valuable experiences and was able to:

Solve routine and non-routine problems.Improve my thinking skills.Knowledge and skills are applied in meaningful ways in solving real-life problems.Expressing ones mathematical thinking, reasoning and communication are highly encouraged and expected.Stimulates and enhances effective learning.Acquire effective mathematical communication through oral and writing and to use the language of mathematics to express mathematical ideas correctly and precisely.Enhance acquisition of mathematical knowledge and skills through problem-solving in ways that increase interest and confidence.Prepare ourselves for the demand of our future undertakings and in workplace.Realise that mathematics is an important and powerful tool in solving real-life problems and hence develop positive attitude towards mathematics.Train myself not only to be independent learners but also to collaborate, to cooperate, and to share knowledge in an engaging and healthy environment.Use technology especially the ICT appropriately and effectively.Train ourselves to appreciate the intrinsic values of mathematics and to become more creative and innovative.Realize the importance and the beauty of mathematics.Practise our self to complete a report for the use once I reach higher standards.

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INTRODUCTION

A Brief History and Introduction to Interest

Economically, the interest rate is the cost of capital and is subject to the laws of supply and demand of the money supply. The first attempt to control interest rates through manipulation of the money supply was made by the French Central Bank in 1847.

The first formal studies of interest rates and their impact on society were conducted by Adam Smith, Jeremy Bentham and Mirabeau during the birth of classic economic thought.[citation needed] In the late 19th century leading Swedish economist Knut Wicksell in his 1898 Interest and Prices elaborated a comprehensive theory of economic crises based upon a distinction between natural and nominal interest rates. In the early 20th century, Irving Fisher made a major breakthrough in the economic analysis of interest rates by distinguishing nominal interest from real interest. Several perspectives on the nature and impact of interest rates have arisen since then.

The latter half of the 20th century saw the rise of interest-free Islamic banking and finance, a movement that attempts to apply religious law developed in the medieval period to the modern economy. Some entire countries, including Iran, Sudan, and Pakistan, have taken steps to eradicate interest from their financial systems entirely.[citation needed] Rather than charging interest, the interest-free lender shares the risk by investing as a partner in profit loss sharing scheme, because predetermined loan repayment as interest is prohibited, as well as making money out of money is unacceptable. All financial transactions must be asset-backed and it does not charge any "fee" for the service of lending.

Types of interest

Simple interest

Simple interest is calculated only on the principal amount, or on that portion of the principal amount that remains unpaid.

The amount of simple interest is calculated according to the following formula:

where r is the period interest rate (I/m), B0 the initial balance and m the number of time periods elapsed.

To calculate the period interest rate r, one divides the interest rate I by the number of periods m.

For example, imagine that a credit card holder has an outstanding balance of RM2500 and that the simple interest rate is 12.99% per annum. The interest added at the end of 3 months would be,

Isimp = (. RM 2500) .3 = RM 81.19

and they would have to pay RM2581.19 to pay off the balance at this point.

If instead they make interest-only payments for each of those 3 months at the period rate r, the amount of interest paid would be,

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I = (. RM 2500).3 = (RM 27.0625/month). 3 = RM 81.19

Their balance at the end of 3 months would still be RM2500.

In this case, the time value of money is not factored in. The steady payments have an additional cost that needs to be considered when comparing loans. For example, given a RM100 principal:

Credit card debt where RM1/day is charged: 1/100 = 1%/day = 7%/week = 365%/year. Corporate bond where the first RM3 are due after six months, and the second RM3 are

due at the year's end: (3+3)/100 = 6%/year. Certificate of deposit (GIC) where RM6 is paid at the year's end: 6/100 = 6%/year.

There are two complications involved when comparing different simple interest bearing offers.

1. When rates are the same but the periods are different a direct comparison is inaccurate because of the time value of money. Paying RM3 every six months costs more than RM6 paid at year end so, the 6% bond cannot be 'equated' to the 6% GIC.

2. When interest is due, but not paid, does it remain 'interest payable', like the bond's RM3 payment after six months or, will it be added to the balance due? In the latter case it is no longer simple interest, but compound interest.

A bank account that offers only simple interest, which money can freely be withdrawn from is unlikely, since withdrawing money and immediately depositing it again would be advantageous.

Composition of interest rates

In economics, interest is considered the price of credit; therefore, it is also subject to distortions due to inflation. The nominal interest rate, which refers to the price before adjustment to inflation, is the one visible to the consumer (i.e., the interest tagged in a loan contract, credit card statement, etc.). Nominal interest is composed of the real interest rate plus inflation, among other factors. A simple formula for the nominal interest is:

Where i is the nominal interest, r is the real interest and is inflation.

This formula attempts to measure the value of the interest in units of stable purchasing power. However, if this statement were true, it would imply at least two misconceptions. First, that all interest rates within an area that shares the same inflation (that is, the same country) should be the same. Second, so the lenders know the inflation for the period of time that they are going to lend the money.

PART A

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The School Cooperative in one of the schools in your area made a profit of RM 50 000 in the year 2011. The cooperative plans to keep the money in a fixed deposit account in a bank for one year. The interest collected at the end of this period will be the poor students in the school. As a member of Board of Cooperative you are to find the total interest which can be collected from different banks.

Given below are the interest rates offered by 3 different banks: Bank A, Bank B and Bank C. You are to calculate the interest that can be obtained based on the given rates, if the money is to be kept in the bank for a period of one year for monthly auto renewable, three months auto renewable, six months auto renewable and twelve months auto renewable without withdrawal. Compare and discuss which bank will you choose and explain why.

SOLUTION

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Period Bank A (% p.a.) Bank B (% p.a.) Bank C (% p.a.)1 month 3.10 3.00 3.002 month 3.10 3.00 3.003 month 3.15 3.05 3.054 month 3.15 3.05 3.055 month 3.15 3.10 3.056 month 3.20 3.10 3.107 month 3.20 3.10 3.108 month 3.20 3.10 3.109 month 3.20 3.10 3.1010 month 3.20 3.10 3.1011 month 3.20 3.10 3.1012 month 3.25 3.15 3.20

Calculating compound interest and comparing which bank would yield the highest end product at the end of the given timeline.

Using formula:

P (1+)n

P is the present value. In this context for calculation we set the present value as RM 50,000.i is the interest rate as given in the above table per annum.n  is the number of investment periods, for example if the banks offer 3 monthly renewal per year,

n is calculated by: iA= , n= 4. Since the total amount is renewed 4 times in a year corresponding to 3 months renewal in a year.

Respective interests earned at the end of each period:

Monthly Renewal

Bank A - IA : RM 50,000 (1+ )12 – 50,000 = RM 1572.21

Bank B = Bank C - IB = IC : RM 50,000 (1+ )12 – 50,000 = RM 1520.80

3 Monthly Renewals

Bank A - IA : RM 50,000 (1+ )4 – 50,000 = RM 1593.70

Bank B = Bank C - IB = IC : RM 50,000 (1+ )4 – 50,000 = RM 1542.53

6 Monthly Renewals

Bank A- IA : RM 50,000 (1+ )6 – 50,000 = RM 1621.49

Bank B = Bank C- IB = IC : RM 50,000 (1+ )6 – 50,000 = RM 1570.16

1 Year Renewal

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Bank A - IA : RM 50,000 (1+ ) = RM 1625.00

Bank B - IB : RM 50,000 (1+ ) = RM 1575.00

Bank C- IC : RM 50,000 (1+ ) = RM 1600.00

Tabulated and interpreted data:

Period of RenewalMonthly 3 Monthly 6 Monthly 12 Monthly

Bank A RM51,572.21 RM51,593.70 RM51,621.49 RM51,625.00Bank B RM51,520.80 RM51,542.53 RM51,570.16 RM51,575.00Bank C RM51,520.80 RM51,542.53 RM51,570.16 RM51,600.00

Having comparing the three banks and their respective periods of renewal, we can conclude that Bank A will be chosen as it yielded the most amount of interest at the end of the timeline when compared to Bank B and Bank C which had a lower amount.

ANSWER

Thus, Bank A is chosen.

Monthly 3 Monthly 6 Monthly 12 MonthlyRM51,460.00

RM51,480.00

RM51,500.00

RM51,520.00

RM51,540.00

RM51,560.00

RM51,580.00

RM51,600.00

RM51,620.00

RM51,640.00

Bank ABank BBank C

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PART B

(a) The Cooperative of your school plans to provide photocopy service to the students of your school. A survey was conducted and it is found out that rental for a photo copy machine is RM 480 per month, cost for a rim of paper (500 pieces) is RM 10 and the price of a bottle of toner is RM 80 which can be used to photocopy 10 000 pieces of paper.

(i) What is the cost to photocopy a piece of paper?

SOLUTION

Finding the answer using mathematical methods and question analysis:

Variables in question: (To photocopy 10 000 pieces of paper)

Rental for photocopy machine = RM 480 per monthCost for a rim of paper (500 pieces) = RM 10Cost for a bottle of bottle of toner = RM 80

To find the cost of photocopying a piece of paper-

Cost of a piece of paper (from a rim of paper) = = 2 sens

Cost of toner per piece of paper photocopied = = 0.8 sens

Cost of rental of machine per piece of paper = = 0.048 sens

Total cost to photocopy a piece of paper ≈ 2.8 sens (Rounded figure)

ANSWER

The cost to photocopy a piece of paper is estimated to RM 0.028

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(ii) If your school cooperative can photocopy an average of 10 000 pieces per month and charges a price of 10 cent per piece, calculate the profit which can be obtained by the school cooperative.

SOLUTION

Finding the answer using mathematical methods and question analysis:

For 1 month, 10 000 pieces of paper are being photocopied on average.

* 500 pieces of paper = 1 rim10 0000 pieces of paper = 20 rim

Cost toner as of above, 10 000 pieces of paper = RM 80

Total cost = RM (20 X 10) + RM 80 + RM 480- Rental of photocopying machine

= RM 760

Charging a price of RM 0.10 per piece of paper photocopied,

Total income= RM 0. 10 X 10 000 = RM 1 000

Thus, profit = RM 1 000 – RM 760

= RM 240

ANSWER

The profit obtained by the school cooperative is RM 240.00

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(b) For the year 2013, the cost for photocopying 10 000 pieces of paper increased due to the increase in the price of rental, toner and paper as shown in table below:

(i) Calculate the percentage increase in photocopying a piece of paper based on the year 2012, using two different methods.

SOLUTION

First method using Price Index,

I= Ῑ� = ∑ IW

∑W

Paper price increase = (RM 240 – RM 200) = RM 40

For 20 rims of paper

Toner price increased by RM 20 for 10 000 pieces of paper photocopied.

Cost of paper per piece photocopied = = 2.4 sens

Cost of toner per piece of paper photocopied = = 1 sen

Total cost to photocopy a piece of paper = 3.4 sens

Percentage in increase in cost = X 100%

= X 100%

= 121.43%

ANSWER

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Cost 2012 (RM) Cost 2013 (RM)

Rental 480 500

Toner 80 100

Paper 200 240

The percentage of increase in photocopying a piece of paper based on the year 2012 is 21. 43 %.

Second method using mathematical solutions and question analysis:

Percentage of increase of cost to photocopy a piece of paper in 2013 as of 2012:

In 2012, the cost was RM 0.028

In 2013, the cost is RM 0.034

Therefore the percentage of increase is calculated as: 3.4 sens – 2.8 sens = 0.6 sens

= X 100% = 21.43%

ANSWER

The percentage of increase in photocopying a piece of paper based on the year 2012 is 21. 43 %.

(ii) If the school cooperative still charges the same amount for photocopying a piece of paper, how many pieces of paper should the cooperative photocopy in order to get the same amount of profit?

SOLUTION

Calculating unknown using equations:

It is noted that to maintain the same amount of profit which was RM 240, the school cooperative would then have to photocopy more pieces of paper.

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Let z pieces of extra copies of photocopied paper to balance the increase in cost.

Extra cost = RM 20 + RM 40 + RM 20 = RM 80

Rental Paper Toner

z X RM 0.10 – z (3.4 sens) = RM 80

z = 1212Total pieces of paper needed to keep same amount of profit = 10 000 + 1212

=11 212

ANSWER

11 212 pieces of paper should be photocopied by the school cooperative in order to get the same amount of profit.

(iii) If the cooperative still maintain to photocopy the same amount of paper per month, how much profit can the Cooperative obtain?

SOLUTION

Finding the profit using simple mathematical method:

Original profit = RM 240

Increase in cost = RM 80

New profit = RM 240 – RM 80

= RM 160

ANSWER

Therefore, with the same amount of paper photocopied per month, the Cooperative could have gained a total profit of RM 160.00.

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PART C

The population of the school is increasing. As a result, the school cooperative needs more space for keeping the increasing amount of stock. Therefore the school cooperative plans to expand the store-room.

It is estimated that cost for renovation is RM 150 000. Make a conjecture on which is a better way for the school cooperative to pay, whether to pay the whole lump sum in cash or keep the RM 150 000 in a fixed deposit account at a rate of 6% p.a. in a bank then borrow the RM 150 000 from a bank and pay for the hire purchase for a period of 10 years with an interest rate of 4.8% p.a. and withdraw monthly to pay for the hire purchase every beginning of a month. Make a conclusion and give your reason.

SOLUTION

Choosing whether to place the sum of money in a fixed deposit account or pay the sum periodically for a period of 10 years with an interest rate of 6% per annum through monthly withdrawals.

Using formula:

A = R [a n˥i]

A is the present value. In this context for calculation we set the present value.

R is the unknown value which is the amount of monthly payment needed to be paid.

[a n˥i] is defined as [ ] where i is the interest rate given and n is the period of month. (10 years = 120 months)

We can find out the monthly payment which is need to be paid by the school each month and also the total amount needed to be paid by the school at the end of the period.

RM 150 000 = R []

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R = RM 1576.36

X 120 = RM 189 163.20 Amount to be paid at the end of the period of 10 years.

Interest paid = RM 189 163.20 – RM 150 000 = RM 39 163.20

Therefore, if the school borrowed RM 150 000 with an interest of 4.8% per annum and withdraw monthly to pay for the hire purchase, at the end of the period, the school would have to pay a total of RM 189 163.20.

To calculate the total amount of money owned by the school cooperative at the end of the 10 years when RM 150 000 is placed in a fixed deposit account of interest 6% per annum.

Using arithmetic progression:

P is the present value. In this context for calculation we set the present value as RM 50,000.i is the interest rate as given in the above table per annum.

P1 RM 150 000 X = RM 750 I1

RM 150 000 + RM 750 = RM 150 750.00

Minus: - RM 1576.36 (Monthly withdrawals)

P2 RM 149 173.64 X = RM 745.87 I2

T120 = a + 119n= 750 – 119 (4.5) I2 – I1 ≈ 4.5= 750 – 535.50= 215

T1 T2 T120

750.00 745.87 215.00

S120 = (a+l)

= (750 + 215)

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Method 1 Method 2Amount paid (RM) 150 000 189 163.20

Interest earned 0 58 000Balance 0 18 836.80

= 60(965)

= 58 000.

From the calculations shown, at the end of the 10 year period, the school cooperative will have an interest of RM 58 000. In other words, their amount of RM 150 000 will increase to RM 208 000 in 10 years.

From such amount, we deduct the money which is borrowed with an interest of 4.8% annually which was RM 189 163.20. Bringing our balance of RM 18 836.80.

ANSWER

Instead of paying the whole lump sum in cash of RM 150 000, the school should keep the RM 150 000 in a fixed deposit account at a rate of 6% p.a. in a bank then borrow the RM 150 000 from a bank and pay for the hire purchase for a period of 10 years with an interest rate of 4.8% p.a. and withdraw monthly to pay for the hire purchase every beginning of a month.

Reason: Using the second way, the school would then have a balance of RM 18 836.80.

PART D

The cooperative of the school also has another amount of RM 50 000. The cooperative plans to keep the money in a bank. The bank offered a compound interest rate of 3.5% per annum and a simple interest rate of 5% per annum.

Explain the meaning of “compound interest” and “simple interest”. Suggest a better way of keeping the money in this bank. State a suitable period for keeping the money for each plan. Explain why.

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SOLUTION

Source: Dictionary (source: Oxford Advanced Learner’s Dictionary 6th Edition)

ANSWER

Compound interest

Interest that is paid both on the original amount of money saved and on the interest that has been added to it.

Simple interest

Interest that is paid only on the original amount of money that you invested, and not on any interest that is earned.

SOLUTION

Comparing how should the use of compound and simple interest be and how it would affect the sum of money over a period of fixed time.

Using mathematical solutions and question analysis:

Period of 20 years:

Simple interest

I1 = RM 50 000 X X 1 = RM 2 500

Total interest after 20 years = RM 2 500 X 20 = RM 50 000

Total amount of money in account = RM 50 000 + RM 50 000

= RM 100 000.

Compound interest

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RM 50 000 (1 + )20 – 50 000 = RM 49 489.00

Total interest earned after 20 years = RM 49 489.00

Total amount of money In account = RM 50 000 + RM 49 489.00

Period of > 20 years

Simple Interest

I1 = RM 50 000 + RM 2 500 = RM 52 500

Compound interest

RM 50 000 ( 1 + )21 – 50 000 = RM 52 971.57

Compound Interest Earned (RM) Simple Interest Earned (RM)20 years RM49,489.00 RM50,000.00

> 20 years RM52,971.57 RM52,500.00

Graph comparison:

20 years > 20 yearsRM47,000.00

RM48,000.00

RM49,000.00

RM50,000.00

RM51,000.00

RM52,000.00

RM53,000.00

RM54,000.00

Compound Interest Earned (RM)Simple Interest Earned (RM)

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After 21 years, the compound interest earned is MORE than the simple interest earned.

ANSWER

Having compared both ways of keeping the money in a bank, it is suggested that the school place the money in a simple interest account for a period of less than 20 years under the circumstances that they might need the money in the span of 20 years. But if the school were to opt for the compound interest account, then the suggested period would be more than 20 years as the yield is higher compared to the amount gained through simple interest provided that the money in this account were not to be withdrawn over the span of more than 20 years.

FURTHER EXPLORATION

When Ahmad was born, his parents invested an amount of RM 5 000 in the Amanah Saham Bumiputera (ASB) for him. The interest rate offered was 8.0% p.a. At what age will Ahmad have a saving of RM 50 000, if he keeps the money without withdrawal?

SOLUTION

Finding the age of Ahmad by the time he has a saving of RM 50 000 when he first started out with RM 5 000.Using geometrical progression:

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* It is noted that T1 did not start when Ahmad was just born. His age was still in the context 0, therefore we start from his age 1.

T0 Age 0 RM 5 000 (8% interest per annum)

T1 Age 1 RM 5 400 (8% interest per annum)

Tn = arn-1

Tn = arn-1 > RM 50 000

RM 5 400 (1.08)n-1 > RM 50 000

5 400 (1.08n-1) > 50 000 ∴ 1.08n-1 > 10

log 1.08n-1 > log 9.259(n-1) log 1.08 > log 9.259

n-1 > log 9.259log 1.08

n-1 > 28 n = 29

ANSWER

Ahmad will be 29 years old by the time he has an amount of RM 50 000 in his account.

CONCLUSION

After doing numerous researches, answering the questions, planning tables and some problem solving, we saw that usage of index number is important in our daily business activity. It is not just widely use in the business segment but also in banking skills. We learnt a lot of lesson from this Additional Mathematics Project Work such as banking account skills, loaning technique, counting the cost of a product, predict the future plans of money and so on. Without this, shopkeepers will suffer loses in their respective business activities. We would like to thank the ones who contributed the idea of index number to help us in this project. Moreover, I myself have seen the importance of using formulas and techniques used in everyday life. The subject itself is not forgotten as almost every single aspect and topics I have learnt is being put to use in this project.

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As a conclusion, index numbers and problem solving together with question analysis are one of the many important techniques needed to master in the event of solving everyday problems. Besides that, many moral values can also be mirrored from the timeline of completing the project such as perseverance, cooperation, understanding and many more. To end it, I would like to again thank everyone for their kind contributions and help in any ways given to me in completing the project, with such I myself am glad with the fact that I have and was able to complete the task given. Thank you.

REFLECTION

After spending countless hours, days and night to finish this project and also sacrificing my time for play over the holidays I was able to adept numerous moral values and lessons over the timeline. As my final regards to everyone out there:

I really would love to say that although Additional is a killer subject to most people, we all have to learn to study hard on it. It seemed easier at first and looked so interesting at first impression and of course had its distinct features which stands out from other subjects, we should all learn to embrace it and love it like any other subjects learnt. Sacrificing so much time for the project, spirit and energy for the project, now I have realized something important from it. Additional Mathematics is our true friend, part of our simple lives and is needed anywhere.

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Without, many things wouldn’t be able to be solved, countless of things will go unknown. For the sake of the future and for own good, I have gotten and fully understood the plain fact that above all, Additional is part of us, one of us and we should love it among everything else.

©Copyrights reserved. No part of this publication maybe replicated and issued in public. This project work may only be used as reference to aid others in completing their respective tasks. Wong Kah Seng, Additional Mathematics Project

2|2012.

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