Additional Mathematics Module Form 4 SMK Agama Arau, Perlis
Chapter Chapt er 11- Index Number
CHAPTER 11- INDEX NUMBER
11.1 INDEX NUMBER
1. An index number is a statistical measure to show the changes of any variable such as prices, production and many others with respect to time. 2. The formula of to calculate index number is
Q1
I =
Q0
×
100
where Q0 = Quantity at base time
Q1 = Quantity at specific time 11.1.1 Price Index
1. Price index is an example of index number which is widely used. 2. It tells us how the price changes over a fixed period of time. 3. Price index of a certain item, I =
Q1 Q0
×
100 where:
(a) Q0 = Price of the item at the base time or year (b) Q1 = Price of the item at specific time Example 1:
In a year, a copy of newspaper was sold at RM1.20. However, in a year, 2008, a consumer has to pay RM1.50 for the same copy of newspaper. Calculate the price index of the newspaper for the year 2008 if the year 2007 is taken as the base year. Solution:
Q0 (Price of the item at the base base year that is 2007) = RM1.20 Q1 (Price of the item at specific time that is year 2008) = RM1.50
I 08 , 07
=
=
Q08 Q07
×
100
RM 1 .50
RM 1 .20 = 125
×
100
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Additional Mathematics Module Form 4 SMK Agama Arau, Perlis
Chapter 11- Index Number
Example 2:
The price index of a product in the year 1998 is 120 based on the year 1994 and 130 based on the year 1991. If the price of the product in 1994 is RM 650, calculate the price in the year 1991. Solution:
From the information above, given that I 98, 94
I 98 ,94
=
Q98
×
Q94
Q 98
120 , I 98 , 91
=
130 and Q 94
=
RM 650
100
Substitute I 98 , 94
120
=
=
120 and Q 94
=
RM 650 into the formula above,
Q 98
× 100 650 = RM 750 =
I 98 ,91
=
Q 98 Q 91
×
100
Substitute I 98 , 91
130
=
Q 91
=
RM 750 Q 91
=
×
130 and Q 98
=
RM 750 into the formula above,
100
RM 600
The price of the product in the year 199 1 is RM600.
I 91, 94
=
Q 91 Q94
×
100
RM 600
× 100 RM 650 = 92 . 30769231 =
I 94 , 91
=
Q94 Q 91
×
100
RM 650
× 100 RM 600 = 108 .3333333 =
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Additional Mathematics Module Form 4 SMK Agama Arau, Perlis
Chapter 11- Index Number
I 94 , 91
=
I 94 , 94
×
I 91, 94
100
100
=
92 .30769231 = 108 .3333333 I 98 ,91
=
= =
I 98 ,91
×
100
I 98 , 94 I 94 , 91 ×
100
120 × 108 .3333333 100 130
=
I 98 , 94
×
I 91, 94
100
120
=
92 .30769231 = 130
×
100
The other formula to find price index is
I 94 , 91
=
I 94 , 94 I 91, 94
×
100
Besides I 94 ,91 can be calculated by
I 94 , 91
=
Q94 Q 91
×
100
While
I 98 , 91 I 98 ,91
=
I 98 , 94 I 94 , 91 ×
100 I 98 , 94 = × 100 I 91, 94
Besides I 98 ,91 can be calculated by
I 98 ,91
=
Q 98 Q 91
×
100
These three formulae can be used to calculate price index in certain situations.
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Additional Mathematics Module Form 4 SMK Agama Arau, Perlis
Chapter 11- Index Number EXERCISE 11.1
1. A factory produces 10000 cans of drink in the year 2000 and 12000 cans in the year 2001. Calculate the index number for the production of cans of drink i n the year 2001 based on the year 2000. 2. The total salary of Ammar Fauzan in the year 1999 is RM 14400 and in the year 2 000 is RM20000. Calculate the index number for his salary in the year 2000 based on the year 1999. 3. The index numbers for an item in the years 2001 and 2002, based on the year 1997, are 108 and 120 respectively. Calculate: (a) the index number in the year 1997 based on the year 2002. (b) the index number in the year 200 2 based on the year 2001. 11.2 COMPOSTE INDEX NUMBER
1. The formula for composite index is
Info…
∑ Iw ∑w
−
I =
Weightage, w represents the relative importance of different items
where I is price index and w is weightage.
2. For example to find the composite index number is the monthly of a family. (a) In determining the composite index number of the monthly expenditure, we need to consider the price indices and weightages f all the items. (b) Items that we spend more will definitely play m ore significant role and hence, is assigned with a larger weightage. Example:
Things
Price Index
Weightage
Shirt
120
m
Trousers
105
8
Bag
140
4
Shoes
150
6
Table 1 The table shows the price indices and the weightage of four things in 1997 based on the year 1992. Given that the composite index of 1992 is 127. Calculate: (a) The value of m (b) The price of a pair of shoes in 1997 if its price in 1992 is RM60. Solution:
(a) We know that the formula for composite i ndex is −
I =
∑ Iw ∑w Page | 148
Additional Mathematics Module Form 4 SMK Agama Arau, Perlis
Chapter 11- Index Number
Use the information given. −
I = −
I =
120 ( m ) + 105 (8) + 140 ( 4) + 150 (6) m+8+4+6 120 m + 840
+
560
+
900
m + 18 −
Given that I 97 , 92
120 m + 840
+
=
127 ,
560
900
+
= 127 m + 18 120 m + 2300 = 127 m + 2286
7m = 14 m=2 (b)From the information in the table, given that I 97 ,92 ( Shoes )
I 97 , 92 ( Shoes )
=
Q 97 Q 92
×
Q 97
=
Q97
RM 60 = RM 90
×
150 , I 98, 91
=
130 and Q 92
=
RM 60 ,
100
Substitute I 97 , 92 ( Shoes )
150
=
=
150 , I 98, 91
=
130 and Q 92
=
RM 60 into the formula above,
100
The price of shoes in the year 1992 is RM90. EXERCISE 11.2
1. Table 2 shows the index numbers for three items P, Q and R in the years 2 005 and 2006 based on the year 2002. The composite index in the years 2005 and 2006 based on the year 2002 are 115.3 and 123.4 respectively. Item
Year 2005
Year 2006
Weightage
P
115
124
3
Q
120
x
5
R
104
126
y
Table 2 Find the value of x and of y. 2. Table 3 shows the index numbers for 3 items for 3 items in the year 2005 based on the year 2000. Item
P
Q
R
Index Number
115
98
124
Weightage
8
x
12- x
Table 3 Given that the composite index is 117.8, calculate the value of x.
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Additional Mathematics Module Form 4 SMK Agama Arau, Perlis
Chapter 11- Index Number CHAPTER REVIEW EXERCISE
1. Table 4 shows the prices and weightages of four types of items A,B ,C and D. Price(RM) Price(RM) Item Price Index Weightage Year 2003 Year 2005 A
7.00
8.40
w
100
B
13.50
x
130
80
C
y
13.00
115
70
D
11.00
12.10
110
z
Table 4 (a) Calculate the value of w , x and y . (b) The composite index of these items for the year 2 005 based on the year 2003 is 120. Calculate the value of z. (c) The total cost of all these items is expected to increase by 20% from the year 2005 to the year 2007. Find the expected composite index for the year 2007 based on the year 2 003.
2. Table 5 shows the index numbers for 5 items in a town in the year 2005 based on the year 2003 and their weightages. Item
Index Number
Weightage
Food
128
23
Drink
115
5
Clothing
119
5
Rental
124
4
Electricity
110
3
Table 5 (a) Calculate the composite index in year 2005 based on the y ear 2003. (b) If the rental in the year 2005 is RM400, what is the rental in the year 2003? (c) If the total expenditure of a fami ly in the year 2003 is RM320, find the total expenditure in the year 2005.
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