Index Numbers - Class 11 Commerce Economics - Extra Questions

The index number by the method of aggregates for the year $$2010$$, taking $$2000$$ as the base year, was found to be $$116$$. If sum of the prices in the year $$2000$$ is Rs. $$300$$, find the values of $$x$$ and $$y$$ in the data given below:
Commodity$$A$$$$B$$$$C$$$$D$$$$E$$$$F$$
Price in the year $$2000$$(rs.)$$50$$$$x$$$$30$$$$70$$$$116$$$$20$$
Price in the year $$2010$$(rs.)
$$60$$$$24$$$$y$$$$80$$$$120$$$$28$$

First we tabulate the given data in the following form:
Commodities     Price in $$2000$$ (in Rs.)                   
$${P}_{0}$$                 		Price in $$2010$$ (in Rs.)
$${P}_{1}$$
$$A$$$$50$$$$60$$
$$B$$$$x$$$$24$$
$$C$$$$30$$$$y$$
$$D$$$$70$$$$80$$
$$E$$$$116$$$$120$$
$$F$$$$20$$$$28$$
$$\sum { { P }_{ 0 } } =286+x$$$$\sum { { P }_{ 1 } } =312+y$$

Given: $$\sum { { P }_{ 0 } } =300$$ $$\Rightarrow$$ $$286+x=300\Rightarrow$$ $$x=14$$
Index number$$=116$$
$$\therefore$$ Using $${P}_{01}=\cfrac{\sum { { P }_{ 1 } } }{\sum { { P }_{ 0 } } }\times 100$$
$$116\times \cfrac{312+y}{300}\times 100$$
$$y=36$$
$$\therefore$$ $$x=14$$ and $$y=36$$


The price relatives and weight of a set commodities are given below:
CommodityABCD
Price relative125120127119
Weight$$x$$$$2x$$$$y$$$$y+3$$
If the sum of the weights is $$40$$ and weighted average of price relatives index number is $$122$$, find the numerical values of $$x$$ and $$y$$. 

 Commodity Weight (w) Price relative (p) wp
 A $$x$$ $$125$$ $$125x$$
 B $$2x$$ $$120$$ $$240x$$
 x $$y$$ $$127$$ $$127y$$
 D $$y+3$$ $$119$$ $$119y+357$$
 Sum $$3x+2y+3$$  $$356x+246y+357$$
Sunm of weights $$=40$$
$$\Rightarrow 3x+2y+3=40$$
$$\Rightarrow 3x+2y=37$$
Index  number $$=122$$
$$\Rightarrow \dfrac{ 365x+246y+357 }{ 40 } =122\\ \Rightarrow 365x+246y=4523......(ii)$$
Solving $$(i)$$ and $$(ii)$$
$$\Rightarrow x=7,y=8$$



Write any two uses of cost of living index number.

Uses of cost of living index number:
(i) It is used in wage negotiations, dearness allowance, bonus etc., to the workers.
(ii) The cost of living index number measures the change in the retail prices of a specified quantity of goods and services.
(iii) It is also useful to the government in framing policies relating to wages.
(iv) It is used as measures of change in the purchasing power of money and real income.


Calculate the cost of living index number of the following data-
ItemQuantity
(in kg)		Cost (in Rs.)
per kg in base year		Cost (in Rs.)
per kg in current year
Sugar$$5$$$$17$$$$16$$
Tea$$1$$$$120$$$$134$$
Pulse$$5$$$$34$$$$40$$
Ghee$$2$$$$180$$$$190$$
Wheat$$30$$$$12$$$$15$$
Rice$$8$$$$20$$$$22$$

Items    Qty kg    Price in Rs./kg
in base year      		Price in Rs./kg
in current year     		Total cost of items
 (in Rs.) in base year   		Total cost of items
 (in Rs.) in current year
Sugar$$5$$$$17$$$$16$$$$85$$$$80$$
Tea$$1$$$$120$$$$134$$$$120$$$$134$$
Pulses$$5$$$$34$$$$40$$$$170$$$$200$$
Ghee$$2$$$$180$$$$190$$$$360$$$$380$$
Wheat$$30$$$$12$$$$15$$$$360$$$$450$$
Rice$$8$$$$20$$$$22$$$$160$$$$176$$
Total$$1,255$$$$1,420$$

Cost of living index $$= \dfrac {\text {Total expenditure in current year}}{\text {Total expenditure in base year}}\times 100$$

$$= \dfrac {1420}{1255}\times 100$$

$$= \dfrac {142000}{1255} = 113.15$$

Thus the cost living index $$= 113.15$$.


Match the following.
IIGroup-AGroup-B
(1)USA(a)was established in 1990
(2)HDI(b) is in the medium HD group
(3)India(c)was established in 1979
(4)PQLI(d)is in the high HD group

1-d, 2-a, 3-b, 4-c.


Explain any three characteristics of index number.

Features of Index numbers are:

  1. Average: They predict or represent the changes that take place in terms of averages.
  2. Quantitative: They offer the accurate measurement of quantitative change.
  3. Measures of relative changes: they measure relative changes over time.


Write the names of construction of index number.

Types of Methods of Construction:

  1. Simple Method: Aggregative and Relative,
  2. Weighted Method  Aggregative and Relative.


Calculate age specific death (A-SDR) rates for the following data:
Age group (in years)Population
$$('000)$$		Number of Deaths
Below $$10$$$$25$$$$50$$
$$10-30$$$$30$$$$90$$
$$30-45$$$$40$$$$160$$
$$45-70$$$$20$$$$100$$

 Age No. of persons$$/1000
\left( { { N }^{ P }x }/{ 100 } \right) $$
 No. of deaths
$$\left( { { N }^{ D }x } \right)$$
 $$SDR=\left( \cfrac { { { N }^{ D }x } }{ { { N }^{ P }x } } \times 1000 \right) $$
 $$<10$$ $$25$$ $$50$$$$=2$$
 $$10-30$$ $$30$$ $$90$$ $$3$$
 $$30-45$$ $$40$$ $$160$$ $$4$$
 $$45-70$$ $$20$$ $$100$$ $$5$$


Calculate the cost of living index from the following data-
ItemsAnnual Quantity Consumed
Base yearGiven year
Rice2.5 Kg per monthRs. 12 per kgRs. 25 per kg
Pulses3 Kg per monthRs. 20 per kgRs. 30 per kg
Oil2 liter per monthRs. 25 per literRs. 35 per liter
Clothing6 meters per monthRs. 5 per meterRs. 7 per meter
Housing12 monthRs. 40 per monthRs. 60 per month
Miscellaneous12 monthRs. 15 per monthRs. 20 per month


2140835_2022178_ans_1844a8c5e931430688f7922e161b8266.png


Calculate Laspeyres index number, Paasches Index number, and Fishers ideal index number.
20152020
Price QuantityPriceQuantity
1001000120800
15001802000100
150500100800
750300700200
12001201000150


2143332_2023474_ans_8d70769e30214deab8b75f080608a6de.jpeg


"Index number is an economical barometers." Explain.

  1. An index number is a number that expresses the relative change in magnitude of a variable or number of variables during a specified period. The variable may be the price of a certain commodity, the quantitative production of certain goods, or the cost of living. 
  2. It is a statistical device to measure the level of certain phenomena in comparison with a certain period known as the base period, which may be a week month, year, or group of years. 
  3. The index numbers are known as economic barometers or economic indicators since they help in understanding the changes in economic conditions of the society.


From the following data construct a price index no. by using - Laspeyre's method and Paasche's
method:
CommodityP0Q0P1Q1
A2846
B51065
C414510
D219215

Solution is as follows:
2146911_2026094_ans_4b2e0b2e619a4bee83d6c296052b74be.PNG


Compute (i) Laspeyre's (ii) Paashe's Index number from the following table:
 ItemsP0 Q0 P1 Q1 
A 10 4 12 6 
B 15 6 20 4 
C 2 5 5 3 
D 4 4 4 4 

Solution is as follows:
2148912_2027005_ans_1c7cd12306e4483c962b39148cca8344.PNG


With the help of the following data calculate the index number for 2010 taking 2009 as the base year with the help of the Simple aggregative method.
CommodityPrice in 2009Price in 2010
A100140
B115155
C145215
D170260
E70130


2139817_2022163_ans_0ed278d4c95944a4af5b891c07a3a85b.png


Calculate Laspeyer's & Paasche's Index number from the following data:
ItemsPrice in 2011Quantity in 2011Price in 2019Quantity in 2019
A10102025
B3534010
C3052015
D1020820
E402405

Solution is as follows:
2146909_2026092_ans_29d780e027bb432dbd4a62708be1efb0.PNG


Construct index of Industrial production of 2012 with the help of the below given data:
IndustriesOutput
(units)
20022012Weights
IT7530025
Auto mobile22026020
Mining15017015
Rubber904510
Consumer producers25050035
Iron and steel 30045030


2157148_2031166_ans_2c7d1df8b67e4138af0cc0e07d4b59f1.png


Calculate the Laspeyre's index number from the following information:

Commodities

Price in 2018 (P0)

Quantity in 2018 (Q0)

Price in 2020 (P1)

A

3

1,000

5

B

2

1,500

4

C

2

1,200

3

D

5

1,800

6



Calculate index number by Laspeyre's method:

Goods

2020

2021`
P0

Q0 

 P1

 Q1

A

2

10

1

6

B

5

5

2

4

C

3

10

1

2

D

4

2

1

1

E

2

1

3

5



Construct the price index by Laspeyre's and Paasche's method from the following information:

Commodities

Price in 2000 ( Rs.)

Quantity in 2000  

Price in 2020 (Rs.)

Quantity in 2020

A

10

30

12

50

B

8

15

10

25

C

6

20

6

30

D

4

10

6

20

Solution is given by:
2158204_2032317_ans_1adb6ee34d524b2ab7f58d77c4627949.PNG


From the following data, calculate price index numbers by: (i) Laspeyer's method and (ii) Paasche's method

Items

Price (P0)

Quantity (Q0)

Price (P1)

Quantity (Q1)

A

20

8

40

6

B

50

10

60

5

C

40

15

50

15

D

20

20

20

25



Calculate index number of the following table, using Paasche's method:

Commodities

Base Year

Current Year

P0

Q0

P1

Q1

A

2

10

4

5

B

5

12

6

10

C

4

20

5

15

D

2

15

3

10



Calculate index number of the following table, using Laspeyre's method:

Commodities

Base Year

Current Year
P0

Q0

 P1

 Q1

A

2

10

4

5

B

5

12

6

10

C

4

20

5

15

D

2

15

3

10



Calculate index number by Paasche's method:

Goods

2020

2021`

P0

Q0 

P1

Q1

A

2

10

1

6

B

5

5

2

4

C

3

10

1

2

D

4

2

1

1

E

2

1

3

5



The price of six different commodities for year $$2009$$ and year $$2011$$ are as follows:
Commodities$$A$$$$B$$$$C$$$$D$$$$E$$$$F$$
Price in $$2009 (Rs)$$$$35$$$$80$$$$25$$$$30$$$$80$$$$x$$
Price in $$2011 (Rs)$$$$50$$$$y$$$$45$$$$70$$$$120$$$$105$$
The Index number for the year $$2011$$ taking $$2009$$ as the base year for the above data was calculated to be $$124$$. Find the values of $$x$$ and $$y$$ if the total price in $$2009$$ is $$Rs. 360$$

CommoditiesPrices in $$2009$$Prices in $$2011$$
$$A$$$$35$$$$50$$
$$B$$$$80$$$$y$$
$$C$$$$25$$$$45$$
$$D$$$$30$$$$70$$
$$E$$$$80$$$$120$$
$$F$$$$x$$$$105$$
$$\sum P_{0} = 250 + x$$$$\sum P_{1} = 390 + y$$

Given, $$\sum P_{0} = 360$$
$$x + 250 = 360$$
$$x = 110$$

$$\dfrac {\sum P_{1}}{\sum P_{0}} \times 100 = P_{01}$$

$$\dfrac {390 + y}{360} \times 100 = 125$$

$$390 + y = \dfrac {125\times 360}{100}$$

$$390 + y = 450$$

$$y = 60$$

Hence, the value of $$x$$ is $$Rs. 110$$ and $$y$$ is $$Rs. 60$$.


Calculate the index number for the year $$2014$$, with $$2010$$ as the base year by the weighted aggregate method from the following data :
Commodity    Price in rupees $$(2010)$$Price in rupees $$(2014)$$  Weight
$$A$$$$2$$$$4$$$$8$$
$$B$$$$5$$$$6$$$$10$$
$$C$$$$4$$$$5$$$$14$$
$$D$$$$2$$$$2$$$$19$$

Commodity               Price in Year
$$2010$$ ($$p_0$$)     		Price in Year
$$2014$$ ($$p_1$$)             		Weight
$$w$$                		$$p_0w$$                           $$p_1w$$
$$A$$$$02$$$$04$$$$08$$$$16$$$$32$$
$$B$$$$05$$$$06$$$$10$$$$50$$$$60$$
$$C$$$$04$$$$05$$$$14$$$$56$$$$70$$
$$D$$$$02$$$$02$$$$19$$$$38$$$$38$$
Total$$\sum p_0w = 160$$$$\sum p_1w = 200$$


Using weighted aggregate method, the required price index $$=\dfrac{\sum p_1w}{\sum p_0w}\times 100=\dfrac{200}{160}\times 100 = 125$$.


Compute (i) Laspeyre's (ii) Paashe's Index number from the following table:
ItemsPrice of base year (Rs.)Quantity of base yearPrice of current yearQuantity of current year
A104126
B156204
C2553
D4444

Solution is as follows:
2150271_2027701_ans_0df077c198134afba23f603d38b561a0.PNG


Calculate Laspeyre's Index number and Paasche's Index number.

 

2019

2020

Commodity

 Price (P0)

Quantity (Q0) 

 Price (P1)

 Price (Q1)

A

100

200

110

150

B

25

400

30

200

C

1,100

10

900

15

D

50

500

40

60

E

10

1,000

12

900

Solution is given by:
2157600_2031928_ans_b2e4c0fc6ebe4b278207a30b2e301f07.PNG


Calculate cost of living index, for the following data, using aggregate expenditure and family budget Method.
CommodityPrice in 2011Price in 2016Quantity in 2011
A101515
B81220
C202410
D32405
E15206
F12182
G8101

Solution is given as:
2150266_2027696_ans_46d9d165f77c4f8aa2d9f61a46ff2b89.jpeg


From the following data, calculate price index numbers for 2016 with 2011 as base by:
  1. Laspeyre's Method,
  2. Paasche's Method,
  3. Fisher's Method.
    CommodityPrice 2011Quantity 2011Price 2016Quantity 2016
    A208406
    B5010605
    C40155015
    D20202025

Solution is give as follows:
2150267_2027697_ans_99a44fa583204a749c8cf3113472a52a.PNG


Construct quantity index number of 2019 from the following data by:
(a) Laspeyre's method, and  (b) Paasche's ethod

 

2018

2019

Commodity

 Price

Quantity 

 Price

 Quantity

A

20

60

24

100

B

16

30

20

50

C

12

40

12

60

D

8

20

12

40

Solution is given by:
2157601_2031929_ans_d110ddb9b4044fe190c3aa5efb5a269f.PNG


Class 11 Commerce Economics Extra Questions