OP Malhotra Class 11 Maths Solutions Chapter 30 Index Numbers Ex 30(a)

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S Chand Class 11 ICSE Maths Solutions Chapter 30 Index Numbers Ex 30(a)

Question 1.

Commodities	A	B	C	D	E
1993 prices (in ₹)	50	40	10	5	2
1995 prices (in ₹)	80	60	20	10	6

Solution:
We construct the table of values is given as under :

Commodities	Price in 1993 P0	Price in 1995 P1
A	50	80
B	40	60
C	10	20
D	5	10
E	2	6
	ΣP0 = 107	ΣP1 = 176

Using simple aggregate method,
P₀₁ = \(\frac{\Sigma \mathrm{P}_1}{\Sigma \mathrm{P}_0} \times 100\) = \(\frac{176}{107}\) × 100 = 164.486

Question 2.

Commodities	A	B	C	D	E	F
1990 prices (in ₹)	40	60	20	50	80	100
1998 prices (in ₹)	50	60	30	70	90	110

Solution:
We construct the table is as under:

Commodities	Prices P0	Prices P1
A	40	50
B	60	60
C	20	30
D	50	70
E	80	90
F	100	110
	ΣP0 = 350	ΣP1 = 410

∴ required price index number = P₀₁
= \(\frac{\Sigma \mathrm{P}_1}{\Sigma \mathrm{P}_0} \times 100\) = \(\frac{410}{350} \times 100\) = 117.143

Question 3.

Commodities	A	B	C	D
Price in 1997	90	40	90	30
Price in 1998	95	60	110	35

Solution:
We construct the table is as under:

Using simple aggregate method
Price index = P₀₁ = \(\frac{\Sigma \mathrm{P}_1}{\Sigma \mathrm{P}_0} \times 100\) = \(\frac{300}{250} \times 100\) = 120

Question 4.
Using 2005 as base year, the index numbers for the price of a commodity in 2006 and 2007 are 118 and 125. Calculate the index numbers for 2005 and 2007 if 2006 is taken as the base year.
Solution:
Let prices in year 2005, 2006 and 2007 are P₀, P₁ and PP₂ respectively.
Given \(\frac{\mathrm{P}_1}{\mathrm{P}_0} \times 100\) = 118 …(1)
and \(\frac{\mathrm{P}_2}{\mathrm{P}_0} \times 100\) = 125 …(2)
Thus index number for 2005 with 2006 as base year = \(\frac{P_0}{P_1} \times 100\) = \(\frac{100}{118} \times 100\) = 84.745
Index number for 2007 with 2006 as base year

Question 5.
Compute a price index for the following by using price relative method.

Commodities	A	B	C	D	E
price in 1991  (in ₹)	20	40	60	80	100
price in 1992 (in ₹)	70	45	70	90	105

Solution:
We construct the table as under :

Question 6.

Commodities	cement	timber	steel	bricks
price in 1969  (in ₹)	5	9.5	35	12
price in 1970 (in ₹)	8	14.3	42	24

Solution:
We construct the table as under:

using price ralative method, price index = \(\frac{\Sigma\left(\frac{\mathrm{P}_1}{\mathrm{P}_0} \times 100\right)}{n}\) = \(\frac{630.53}{4}\) = 157.6325

Question 7.
The index number for the following data for the year 2008 , taking 2004 as base year was found to be 116. The simple aggregate method was used for calculation. Find the numerical value of x and y if the sum of the prices in the year 2008 is ₹ 203.

Commodity	Price in (₹) in the year 2004	Price (in ₹) in the year 2008
A	20	25
B	10	30
C	30	15
D	25	45
E	X	35
F	50	y

Solution:
We construct the following table given as under:

Since sum of prices in the year 2011 = ΣP₁ = 150 + y
⇒ 203 = 150 + y
⇒ y = 53
Thus Index number for year 2011 = \(\frac{\Sigma \mathrm{P}_1}{\Sigma \mathrm{P}_0} \times 100\)
⇒ \(116=\left(\frac{150+y}{135+x}\right) \times 100\)
⇒ 116 = \(\frac{203 \times 100}{135+x}\)
⇒ 135 + x = \(\frac{203 \times 100}{116}=175\)
⇒ x = 175 – 135 = 40

Question 8.
Construct index numbers by the simple average of relative method for 1990 and 1991 with 1989 as the base year.

Commodity Price (in ₹)per unit	A	B	C	D	E
1989	100	40	30	10	20
1990	120	45	35	12	22
1991	150	60	45	15	23

Solution:
We construct the table of values is under:

Question 9.
Construct the index number for 1991 taking 1990 as the base year from the following data by simple average of price relative method.

Commodities	A	B	C	D	E
price in 1990 (in ₹)	100	80	160	220	40
price in 1991 (in ₹)	140	120	180	240	40

Solution:
We construct the table as under :

Then by simple average of price relative method,
price index = \(\Sigma \frac{\mathrm{P}_1}{\mathrm{P}_0} \times 100\) = \(\frac{611.591}{5}\) = 122.3182

Question 10.
Construct index number from the following data for 1991 and 1992 taking 1990 as base by using the method of simple average of price relatives :

Group	Price in 1990 (in ₹)	Price in 1991 (in ₹)	Price in 1992 (in ₹)
A	20.00	24.00	22.00
B	1.25	1.50	1.00
C	5.00	8.00	8.00
D	2.00	2.25	2.12

Solution:
We construct a table of values is given as under:

By method of simple average of price relatives, we have
price Index for 1991 = \(\frac{\Sigma \frac{\mathrm{P}_1}{\mathrm{P}_0} \times 100}{n}\) = \(\frac{512.5}{4}\) = 128.125
and price index for 1992 = \(\frac{\Sigma \frac{\mathrm{P}_2}{\mathrm{P}_0} \times 100}{n}\) = \(\frac{456}{4}\) = 114

Question 11.
The following data relate to the price of rice in different years.

Year	1988	1989	1990	1991	1992	1993	1994	1995	1996	1997
Price (in₹)	6	7	7	8	10	14	12	13	14	15

Find out price relatives
(i) taking 1988 as base;
(ii) 1992 as base ;
(iii) taking average of 1988,1989 and 1990 as base.
Solution:
We construct table of values is given as under:

(iii) base year value P₀ = \(\frac{6+7+7}{3}\) = \(\frac{20}{3}\)

Question 12.
Compute a price index for the following by (i) simple aggregate and (ii) average of price relative method.

Commodity	A	B	C	D	E	F
price in 1994 (₹)	20	30	10	25	40	50
price in 1999 (₹)	25	30	15	35	45	55

Solution:
We construct the table of values is given as under :

(i) By simple aggregate method, price index = \(\frac{\Sigma \mathrm{P}_1}{\Sigma \mathrm{P}_0} \times 100\) = \(\frac{205}{175} \times 100\) = 117.143

(ii) By average of price method, we have, price index = \(\frac{\Sigma\left(\frac{\mathrm{P}_1}{\mathrm{P}_0} \times 100\right)}{n}\) = \(\frac{737.5}{6}\) = 122.92

Question 13.
Construct an index for 1998 taking 1997 as base by Average of Relatives.

Commodity	A	B	C	D	E
Price in 1997	5	4	8	11	2
Price in 1998	7	6	9	12	2

Solution:
We construct the table of values is given as under :

Then by average of relative method, Price index = \(\frac{\Sigma\left(\frac{\mathrm{P}_1}{\mathrm{P}_0} \times 100\right)}{n}\) = \(\frac{611.591}{5}\) = 122.32

Question 14.
Construct the consumer price index number for 1990 taking 1989 as the base year and using simple average of price relative method for the following data :

Commodities	Price in 1989	Price in 1990
Butter	20	21
Cheese	16	12
Milk	3	3
eggs	2.80	2.80

Solution:
We construct the table of values is given as under :

Using simple average of price relatives method
required index number = P₀₁ = \(\frac{1}{N} \Sigma\left(\frac{P_1}{P_0} \times 100\right)\) × 380 = 95