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.
Recommended