ADDITIONAL MATHEMATICS

PROJECT WORK

2/2013

FORM 5

FAMILY’S MONTHLY

EXPENDITURE

PREPARED BY :

NAJAA SYAIRAH MAHYUDIN

FORM :

5 PUTRA

SCHOOL :

SMK SEKSYEN 19

PREPARED FOR :

SOMU A/L PANTINAIDU

INDEX NUMBE

RPREPARED BY :

NAJAA SYAIRAH MAHYUDIN

FORM :

5 PUTRA

SCHOOL :

SMK SEKSYEN 19

PREPARED FOR :

SOMU A/L PANTINAIDU

CONTENT

Num. Title Page 1. Objectives2. Introduction 3. Part 14. Part 25. Part 36. Part 47. Further Explorations8. Reflection 9. Conclusion

JOKES !

OBJECTIVESWe are students from Sekolah Menengah Kebangsaan Section

19 taking Additional Mathematics that are required for us to carry

out a project work while we are in Form 5. This year the

Curriculum Development Division, Ministry of Education

has prepared four tasks for us but we are assigned to choose and

complete only one task based on our area of interest.

This project can be done in groups or individually, but each of

us are expected to submit by individually a written report. Upon

completion of the Additional Mathematics Project Work, we can

gain valuable experiences and enable us:

i. To  apply and  adapt a variety of problem-solving strategies to solve

problems. 

ii. To improve thinking skills.

iii. To promote effective mathematical communication

iv. To  develop  mathematical  knowledge  through  problem  solving in

a way that increases students interest and confidence. 

v. To use the language of mathematics to  express  mathematical

ideas precisely.

vi. To  provide  learning  environment  that  stimulates  and  enhances

effective learning.

vii. To  develop positive  attitude towards  mathematics.

INTRODUCTION

Assalamualaikum and hello ,

My name is Najaa Syairah . I am 17 years old and from 5 Putra . I am

honour to execute my responsibility in order to conduct this project work.

Thanks for giving me a chance to complete my Additional Mathematics

project work and as a representative for my school , SMK SECTION 19.

First of all, I would like to say Alhamdulillah, for giving me the strength

and health to do this project work and finish it on time. Besides, not forgotten

to my parents and family for providing everything, such as money, to buy

anything that are related to this project work, their advise, which is the

most needed for this project and facilities such as internet, books, computers

and others. They also supported me and encouraged me to complete this

task so that I will not procrastinate in doing it. Next, I would like to thank to

my teacher, Sir Somu for guiding me throughout this project. Even I had

some difficulties in doing this task, but he taught me patiently until I knew

how to managed the problem well. He tried his best to teach me until I

understand what I am supposed to do with the project work. Furthermore, my

friends who always supporting me, eventhough this project is assign

individually but we are cooperate with each other for this project especially in

discussion and sharing ideas to ensure our task will finish successfully. Last

but not least, any party which involved either directly or indirect in completing

this project work.

Thank you everyone.

HISTORY OF INDEX NUMBER

Index numbers are meant to study the change in the effects

of such factors which cannot be measured directly. Thus, index

numbers are used to measure the changes in some quantity

which we cannot observe directly’. For example, changes in

business activity in a country are not capable of direct

measurement but it is possible to study relative changes in

business activity by studying the variations in the values of some

such factors which affect business activity, and which are capable

of direct measurement.

Index numbers are commonly used statistical device

for measuring the combined fluctuations in a group related

variables. If we wish to compare the price level of consumer items

today with that prevalent ten years ago, we are not interested in

comparing the prices of only one item, but in comparing some

sort of average price levels. We may wish to compare the present

agricultural production or industrial production with that at the

time of independence. Here again, we have to consider all items

of production and each item may have undergone a different

fractional increase (or even a decrease). How do we obtain a

composite measure?

Basically, this composite measure is provided by index

numbers which may be defined as a device for combining the

variations that have come in group of related variables over a

period of time, with a view to obtain a figure that represents the

‘net’ result of the change in the constitute variables.

In addition, Index numbers may be classified in terms of the

variables that they are intended to measure. In business, different

groups of variables in the measurement of which index number

techniques are commonly used are :

i. price

ii. quantity

iii. value 

iv. business activity.

Thus, we have index of wholesale prices, index of consumer

prices, index of industrial output, index of value of exports and

index of business activity, etc. Here we shall be mainly interested

in index numbers of prices showing changes with respect to time,

although methods described can be applied to other cases.

In general, the present level of prices is compared with the

level of prices in the past. The present period is called the

current period and some period in the past is called the

base period.

Statistics Today

During  the  20th  century, the  creation  of  precise 

instruments for  agricultural research, public health concerns

(epidemiology, biostatistics, etc.), industrial quality control, and

economic and social purposes (unemploymentrate, econometry,

etc.) necessitated substantial advances in statistical practices.

Today the use of statistics has broadened far beyond its origins.

Individuals and organizations use statistics to understand data

and make informed decisions throughout the natural and social

sciences, medicine, business, and other areas. Statistics is

generally regarded not as a subfield of mathematics

but rather as  a distinct, albeit allied, field. Many  universities

maintain separate mathematics and statistics departments.

Statistics is also taught in departments as diverse as psychology,

education, and public health.

 Index Number

Index numbers are today one of the most widely used

statistical indicators. Generally used to indicate the state of the

economy, index numbers are aptly called “barometers of

economic activity ”. Index numbers are used in comparing

production, sales or changes exports or imports over a certain

period of time. The role-played by index numbers in Indian trade

and industry is impossible to ignore. It is a very well known fact

that the wage contracts of workers in our country are tied to the

cost of living index numbers. By definition, an index number is a

statistical measure designed to show changes in a variable or a

group or related variables with respect to time, geographic

location or other characteristics such as income, profession, etc.

Characteristics of an Index Numbers

These are expressed as a percentage:

Index number is calculated as a ratio of the current value to a

base value and expressed as a percentage. It must be clearly

understood that the index number for the base year is always

100. An index number is commonly referred to as an index.

Index numbers are specialized averages:

  Index number is an average with a difference. An index

number is used for purposes of comparison in cases where the

series being compared could be expressed in different units i.e. a

manufactured products index (a part of the whole sale price

index) is constructed using items like Dairy Products, Sugar,

Edible Oils, Tea and Coffee, etc. These items naturally are

expressed in different units like sugar in Kgs, milk in liters, etc.

The index number is obtained as a result of an average of all

these items, which are expressed in different units. On the other

hand, average is a single figure representing a group expressed in

the same units.

Index numbers measures changes that are not directly

measurable:

  An index number is used for measuring the magnitude of

changes in such phenomenon, which are not capable of direct

measurement. Index numbers essentially capture the changes in

the group of related variables over a period of time. For example,

if the index of industrial production is 215.1 in 1992-93 (baseyear

1980-81) it means that the industrial production in that year was

up by 2.15 times compared to 1980-81. But it does not, however,

mean that the net increase in the index reflects an equivalent

increase in industrial production in all sectors of the industry.

Some sectors might have increased their production more than

2.15 times while other sectors may have increased their

production only marginally.

Uses of index numbers

Establishes trends

Index numbers when analyzed reveal a general trend of the

phenomenon under study. For instance, Index numbers of

unemployment of the country not only reflects the trends in the

phenomenon but are useful in determining factors leading to

unemployment.

Helps in policy making

It is widely known that the dearness allowances paid to the

employees is linked to the cost of living index, generally the

consumer price index. From time to time it is the cost of living

index, which forms the basis of many a wages agreement

between the employees union and the employer. Thus index

numbers guide policy making.

Determines purchasing power of the rupee

Usually index numbers are used to determine the purchasing

power of the rupee. Suppose the consumers price index for urban

non-manual employees increased from 100 in 1984 to 202 in

1992, the real purchasing power of the rupee can be found out as

follows: 100/202=0.495 It indicates that if rupee was worth 100

paise in 1984 its purchasing power is 49.5 paise in 1992.

Deflates time series data

Index numbers play a vital role in adjusting the original data

to reflect reality. For example, nominal income (income at current

prices) can be transformed into real income (reflecting the actual

purchasing power) by using income deflators. Similarly, assume

that industrial production is represented in value terms as a

product of volume of production and price. If the subsequent

year’s industrial production were to be higher by 20% in value,

the increase may not be as a result of increase in the volume of

production as one would have it but because of increase in the

price. The inflation which has caused the increase in the series

can be eliminated by the usage of an appropriate price index and

thus making the series real.

Part 1

a)

i. Price Index

Price index is a an index that traces the relative

changes in the price for a given class

of goods or services in a given region, during a given

interval of time. It is a statistic designed to help to

compare how these prices, taken as a whole, differ

between time periods or geographical locations or other

characteristics such as income, profession, etc.

Eg :

ii. Weightage

Weightage is a stock index in which each stock

influences the index in proportion to its price per share.

The value of the index is generated by adding the

prices of each of the stocks in the index and dividing

them by the total number of stocks. Stocks with a

higher price will be given more weight and, therefore,

will have a greater influence over the performance of

the index.

iii. Composite Index

Composite index number is a number that

measures an average relative changes in a group of

relative variables with respect to a base.

b) There are four ways of weightage representations. They are :

Laspeyre’s Index Number:

In this index number the base year quantities are used

as weights, so it also called base year weighted index.

Paasche’s Index Number:

In this index number, the current (given) year quantities are used

as weights, so it is also called current year weighted index.

Fisher’s Ideal Index Number:

Geometric mean of Laspeyre’s and Paasche’s index numbers is

known as Fisher’s ideal index number. It is called ideal because it

satisfies the time reversal and factor reversal test.

                          

                         

Marshal-Edgeworth Index Number:

In this index number, the average of the base year and current

year quantities are used as weights. This index number

is proposed by two English economists Marshal and Edgeworth.

 

Example:

Compute the weighted aggregative price index numbers for 2011 with 2010 as base year using :

(1) Laspeyre’s Index Number

(2) Paashe’s Index Number

(3) Fisher’s Ideal Index Number

(4) Marshal Edgeworth Index Number

CommodityPrices Quantities

2010 2011 2010 2011

Solution:

Commodity

Prices Quantity

2010 2011 2010 2011

 

Laspeyre’s Index Number

                        

Paashe’s Index Number

                        

Fisher’s Ideal Index Number

Marshal Edgeworth Index Number                       

Part 2

We often hear complaints from the public about inflation. It causes an increase in the household expenditure in a family.

The household expenditure for every family is different.

a) My family’s monthly expenditure for the year 2013.

ItemAverage Monthly Expenditure for the year 2013(to the nearest

RM)

Percentage of monthly

expenses (to the nearest %)

Food 3000 36.1Accommodation(Rental / Loan)

1000 12

Transportation (Petrol/Loan/Bus fare etc)

2000 24.1

Clothing 100 1.2Education 650 8Recreation 450 5.4

Utilities (Water/Electricity/Telephone)

1000 12

Medication 50 0.6Miscellaneous 50 0.6

TOTAL 8300 100

TABLE 1

b) If we want to compare the cost of living from one year to another, we have to calculate the price index that involves some of the items mentioned above.

i. In order to calculate the price index of all the items above, we have to consider the average monthly expenses of any previous year as the base year.

ii.

Item

Average Monthly Expenditure for

the year 2010 as the base year

(RM)

Average monthly expenses for the year 2013 (RM)

Food 2500 3000Accommodation(Rental / Loan)

1000 1000

Transportation (Petrol/Loan/Bus fare etc)

2000 2000

Clothing 80 100Education 525 650Recreation 380 450

Utilities (Water/Electricity/Telephone

)

1000 1000

Medication 50 50Miscellaneous 50 50

TOTAL 7585 8300

TABLE 2

c) i.

ItemPrice indices for the year 2013 based on year

2010

Weightage

Food 120 60Accommodation(Rental / Loan)

100 20

Transportation (Petrol/Loan/Bus fare etc)

100 40

Clothing 125 2Education 124 13Recreation 118 9

Utilities (Water/Electricity/Telephone

)

100 20

Medication 100 1Miscellaneous 100 1

TOTAL 987 166

TABLE 3

ii. Calculate the composite index for the average monthly expenditure in the year 2013 based on year 2010.Answer :

Composite index , I = ∑ IW

∑W

120(60)+100(20)+100(40)+125(2)+124(13)+118(9)+100(20)+100(1)+100(1)¿

60+20+40+2+13+9+20+1+1 18324¿

166

¿ RM 110.4

d) My family’s expenditure based on my findings is the average

monthly expenditure for the year 2013 increase by

RM110.4 . This is because some items in the year 2010

increased in the year 2013.

Part 3

Question 1

Your family is planning to buy a new television set.

a) You have conducted a survey on the price of the

television for two different brands from three different shops.

You would like to make a comparison between two modes of

payment, namely, cash payment and payment by

installment.

Table 4(a) shows the price of televisions by cash payment in

three different shops whereas Table 4(b) shows the prices of

televisions by installment.

TABLE 4(a)

BrandSize of

Television

(inches)

Price (RM) Mean

Price(RM)

Standard

Deviation

(RM)

Shop A

Shop B

Shop C

BrandSize of Televisi

on (inches)

Price (RM) Mean

Price(RM)

Standard

Deviation

(RM)

Shop A

Shop B

Shop C

Samsung24 380 399 350 376.3 532.232 650 650 600 633.3 895.740 900 1900 2999 1933 2733.8

Panasonic

24 300 450 250 333.3 471.432 500 700 1400 866.7 1225.640 980 1300 1300 1193.

31687.3

Samsung24 31.6 33.3 29.2 31.4 44.332 54.2 54.2 50 52.8 74.740 75 158.3 249.9 161.1 227.8

Panasonic

24 25 37.5 20.8 27.8 39.232 41.7 54.3 116.6 70.9 100.240 81.7 108.3 108.3 99.4 140.6

TABLE 4(b)

b) I have decided to buy a Samsung television with 40 inches from shop A because Television with 40 inches has a good quality of the image ,

sound and screen.

Television with 40 inches in shop A is much cheaper than

shop B and C and it is worth it.

Shop A sell with RM 900 which is my family are affordable

to buy the television based on the money expenditure in

Table 1.

My family also can pay the prices of television by

installment and we can use other money for other uses.

c) If I am the panels of the Fair Price Shop Award, the shop that

deserve the award is shop A.

No. I don’t consider the value of mean and the value of

standard deviation of television because based on the table

mean and standard deviation is based on 3 shops. So, shop

A sell with reasonable price. Therefore I assume that,

consumers are able to buy the television at shop A.

In my research, factors that influenced the prices of

goods in the shops is such as the location of the shop, the

population of the customers, the status of the shop, the size

of the shop, and the rent for the shop.

Part 4

Question 2

a) Your family has a fixed monthly income. In order to buy the

television, your family needs to make some adjustment on

the various types of expenditure.

ItemAverage Monthly Expenditure for the year 2013

(to the nearest

Percentage of monthly

expenses (to the nearest %)

RM)Food 3000 36.2

Accommodation(Rental / Loan)

1000 12

Transportation (Petrol/Loan/Bus fare etc)

2000 24.1

Clothing 100 1.2Education 650 7.3Recreation 450 5

Utilities (Water/Electricity/Telephone

)

1075 13

Medication 50 0.6Miscellaneous 50 0.6

TOTAL 8375 100

SOLUTIONSI choose to pay the television by installment.

b) Assuming you have just started working with a monthly

salary of RM2500. You intend to save 10% of your salary

every month.

Plan your monthly expenditure in a Table 1 above and add

other items such as savings and contributions to your

parents.

ItemAverage Monthly Expenditure for the year 2013

(to the nearest RM)

Percentage of monthly

expenses (to the nearest %)

Food 3000 28

Accommodation(Rental / Loan)

1000 9

Transportation (Petrol/Loan/Bus fare etc)

2000 19

Clothing 100 0.9Education 650 6Recreation 450 4

Utilities (Water/Electricity/Telephone

)

1000 9

Medication 50 0.5Miscellaneous 50 0.5

Savings 250 2.3Contributions 2250 20.8

TOTAL 10800 100

REFLECTIONAfter spending countless hours, days and night to finish this project and also sacrificing my time video games and magazine in this mid year holiday, there are several things that I can say...

From the day I born...From the day I was able to holding pencil...From the day I start learning...And ...From the day I heard your name...

I always thought that you will be my greatest obstacle and rival in excelling in my life...But after countless of hours...Countless of days ...Countless of nights ...

After sacrificing my precious time just for you...

Sacrificing my Computer Games...Sacrificing my Video Games...Sacrificing my Facebook ...Sacrificing my Internet...Sacrificing my Anime...Sacrificing my magazine...

I realized something really important in you...I really love you...You are my real friend...You my partner...You are my soulmate...I LOVE U ADDITIONAL MATHEMATICS...

CONCLUSION After doing research, answering questions, complete the table

and all the problem solving, we can conclude that the planning of

money expenditure for the family is important. In connection, if

one family make a planning of their financial expenditures, it will

give them many advantages such as :-

a. Help them stabilize and cope with challenges that are

associated with that stage of life.

b. The changes will protect their family and make it

possible to still reach their financial goals even with the

added expenses of children in their life.

c. Boost household savings and investments.

d. Help families get the right insurance.

e. Financial backup when job changes occur.

f. Estate planning for families.

g. Assist in retirement planning.

Therefore, parents should plan their family’s financial in order to

have stability in life. Also, parents can be a good role model by

demonstrate to their children on how to spend money wisely . As

a result, the family would not face any critical financial problems

and their childrens will have a quality life for their future.