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Statistics 5 Mark Each

INDEX NUMBER · Gujarat Board (GSEB) STD 12 Commerce (GSEB - English Medium). 23 questions.

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23 questions · timed · auto-graded

Question 15 Marks
The data about sale of three grain flour wheat, bajri and chana at a flour mill from the year $2011$ to $2015$ are as follows. Compute the general index number using simple average with $(i)$ fixed base method $($taking base year $2011)$ and $(ii)$ Chain base method.
Answer
$(i)$ Fixed base method :
Fixed base index number $I=\frac{\text { Value of variable in current year (period) }}{\text { Value of variable in base year (period) }} \times 100$
$(ii)$ General index number by chain base method :
Chain base index number $I=\frac{\text { Value of variable in current year (period) }}{\text { Value of variatle in preceding year (period) }} \times 100$
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Question 25 Marks
The details of expenses on fuel for a group of workers of a region are as follows :
Prepare the index number of the group of fuel expenditure from these data. If the expenditurefor food, clothing, house rent and miscellaneous groups in the year $2015$ are $3, 2.5, 4.5$ and $3.25$ times respectively that of the year $2012,$ and if the expenditures on these groups are $42 \%, 15 \%, 10 \%$ and $12 \%$ respectively of the total expenditure then prepare the cost ofliving index number for the workers.
Answer
First of all, we shall prepare the group index number for fuel-expenditure from its details. We will take the base year $2012$ and obtain the index number by total expenditure method.Note : The method of family budget can also be used here for calculation.
$ \text { Index number for fuel expenditure } =\frac{\Sigma p_{1} q_{0}}{\Sigma p_{0} q_{0}} \times 100$
$ =\frac{1185}{1044} \times 100$
$ =113.5057$
$ \simeq 113.51 $
The expenditure for the groups of food, clothing, house rent and miscellaneous are $3, 2.5, 4.5$ and $3.25$ times respectively than the base year.
Hence, the index numbers of these four groups are $(3 \times 100) = 300; (2.5 \times 100) = 250; (4.5 \times 100) = 450$ and $(3.25 \times 100) = 325$ respectively.
The index number of fuel category is obtained as $113.51.$
We will take the percentage expenditure for all the five groups as the weights for their corresponding index numbers to find the cost of living index number.
The expenditures for the groups of food, clothing, house rent and miscellaneous are given here as $42 \%, 15 \%, 10 \%$ and $12 \%$ respectively.
These will be taken as their respective weights W. Total expenditure is $100 \%.$
Hence, the weight for fuel expenditure index number will be $100 - (42 + 15 + 10 + 12) = 21 \%.$
The calculation of cost of living index number is as follows :
$ \text { Cost of living index number } =\frac{\Sigma I W}{\Sigma W}$
$ =\frac{27133.71}{100}$
$ =271.3371$
$ \simeq 271.34 $
Thus, it can be said that there is a rise of $(271.34 - 100) = 171.34 \%$ in the cost of living in the current year as compared to the base year.
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Question 35 Marks
Compute Laspeyre’s, Paasehe’s and Fisher’s index numbers for the year $2016$ from the data given below by taking $2015$ as the base year.
Answer
The base year is $2015$ and the current year is $2016 .$
Hence, we will take price $p_{0}$ and quantity $q_{0}$ for the year $2015 ,$ price $p_{1}$ and quantity $q_{1}$ for the year $2016 .$
The price of item $A$ is per $20 \ kg$ here whereas the unit for quantity is kg.
The price ofitem $B$ is per quinta] but the unit for the quantity is kg.
The price of item $C$ is per kg whereas the unit for quantity is gram.
The price for item Fis per dozen whereas the unit for quantity is piece.
The calculation of the price per item of these four items will be as follows :
The price of item $A$ in the year $2015$ is $₹ 300$ per $20 \mathrm{~kg}$.
Hence, its price $=\frac{300}{20}=₹ 15$ per $\mathrm{kg}$.
Similarly, the price of item $A$ in $2016$ is $\frac{440}{20}=₹ 22$ per $\mathrm{kg}$.
It is convenient to express the price of item $B$ per $\mathrm{kg}$ than quintal.
Hence, the price for the year $2015=\frac{500}{100}=₹ 5$ per $\mathrm{kg}$ and the price for the year $2016=\frac{700}{100}=₹ 7$ per $\mathrm{kg}$.
The price of item $(I)$ is per kg.
Hence, it is convenient to express its quantity in kg.
Thus, the quantity in the year $2015=\frac{1200}{1000}=1.2 \mathrm{~kg}$ and the quantity for the year $2016=\frac{2000}{1000}=2 \mathrm{~kg}$.The price of item $F$ is per dozen which is convenient to express in per piece.
Hence, the price of the year $2015=\frac{30}{12}=₹ 2.5$ per piece and the price for the year $2016=\frac{36}{12}=₹ 3$ per piece.
Now, the index number will be calculated as follows :

${\text{Laspeyre's index number}} ~I_{L} =\frac{\sum p_{1} q_{0}}{\Sigma p_{0} q_{0}} \times 100$
$ =\frac{1167}{1006.75} \times 100$
$ =115.9175$
$ \simeq 115.92 $
Thus, we can say that there is a rise of $(115.92 - 100) = 15.92 \%$ in the prices in the year $2016$ as compared to the year $2015.$
${\text{Paasche's index number}} ~I_{P} =\frac{\sum p_{1} q_{1}}{\sum p_{0} q_{1}} \times 100$
$ =\frac{1916}{1656.25} \times 100$
$ =115.6830$
$ \simeq 115.68 $
Thus, it can be said that there is $(115.68 - 100) = 15.68 \%$ rise in the prices in the year $2016$ as compared to the year $2015.$
$ {\text{Fisher's index number}} ~I_{F} =\sqrt{I_{L} \times I_{P}}$
$ =\sqrt{115.92 \times 115.68}$
$ =115.7999$
$ \simeq 115.8 $
Thus, it can be said that there is $(115.8 - 100) = 15.8 \%$ rise in the prices in the year $2016$ as compared to the year $2015.$
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Question 45 Marks
Find Laspeyre’s, Paasche’s and Fisher’s index numbers for the year $2016$ with base year $2015$ from the data about price and consumption of food items given below.
Answer
We will take price $p_{0}$ and quantity $q_{0}$ for the base year, price $p_{1}$ and quantity $q_{1}$ for the current year.
$ \text { Laspeyre's index number } I_{L} =\frac{\sum p_{1} q_{0}}{\sum p_{0} q_{0}} \times 100$
$ =\frac{740}{669} \times 100$
$ =110.6128$
$ \approx 110.61 $
Thus, there is a rise of $(1 1 0.61 - 100) = 10.61 \%$ in prices ofthe year $2016$ as compared to the base year $2015.$
$ {\text{Paasche's index number}}~ I_{P} =\frac{\sum p_{1} q_{1}}{\Sigma p_{0} q_{1}} \times 100$
$ =\frac{629}{571} \times 100$
$ =110.1576$
$ \simeq 110.16 $
Thus, there is arise of $(110.16- 100) = 10.16 \%$ in prices ofthe year $2016$ as compared to the base year $2015.$
${\text{Fisher's index number}} ~I_{F} =\sqrt{I_{L} \times I_{P}}$
$ =\sqrt{110.61 \times 110.16}$
$ =110.3847$
$ \simeq 110.38 $
Thus, there is a rise of $(110.38 - 100) = 10.38 \%$ in prices of the year $2016$ as compared to the base year $2015.$
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Question 55 Marks
The data about per unit price and weight of four different items in the year $2014$ and $2015$ are as follows. Compute the index number of the year $2015.$
Item Weight Year $2014$ Price per unit $(Rs.)$ Year $2015$ Price per unit $(Rs.)$
$A$ $40$ $32$ $40$
$B$ $25$ $80$ $100$
$C$ $20$ $24$ $30$
$D$ $15$ $4$ $6$
Answer
Table of calculation is as under :
Image
Index number of the year $2015$
$ \mathrm{I}=\frac{\Sigma I W}{\Sigma W}$
$=\frac{12875}{100}$
$=128.75$
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Question 65 Marks
The data about the industrial production quantity and weights for the year $2015$ are given below. Compute the index number of industrial production and interpret it.
Industry
Mine		 Unit
Lakh tons		 Year $2013$ production
$10$		 Year $2015$ production
$15$		 Weight
Rs.
Textile Crore meters $20$ $25$ $6$
Engineering Lakh tons $30$ $25$ $30$
Chemicals Hundred tons $40$ $50$ $3$
Food Lakh tons $50$ $60$ $4$
Answer
Image
Index number of industrial production $I =\frac{\Sigma I W}{\Sigma W}$
$=\frac{4705}{47}$
$=100.10$
Interpretation: There is $( 100.10-100=0.10 \%)$ increase in the industrial production of the year $2015$ as compared to the year $2013.$
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Question 75 Marks
The following data are given about the index numbers and weights for the items of living of industrial workers in a city in the year $2014.$ Find the cost of living index number for industrial workers. If the average monthly salary paid to these workers in the year $2012$ was $₹ 6,000,$ what should be the monthly salary in the current year $2014$ to maintain the same standard of living $?$Image
Answer
Image
Cost of living index number for industrial workers
$I=\frac{\Sigma I W}{\Sigma W}$
$=\frac{23144}{100}$
$=231.44$
$\rightarrow $ Base year $= 2012$
There is $(231.44 -100 =) 131.44\%$ increase in the cost of living expenditure in $2014$ as compared to the base year $2012.$
$\rightarrow $ The average monthly salary of the workers in the year $2012$ is $₹ 6,000.$
$\therefore $ The average monthly salary in the current year $2014$ to maintain same standard of living
$=\frac{\text { Current year's ( i.e. 2014) cost of living index number }}{\text { Base year's (i.e. 2012) cost of living index number }} \times \text { Average salary in the year 2012}$
$=\frac{231.44}{100} \times 6000$
$= ₹ 13886.40$
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Question 85 Marks
The data about index numbers an weights for groups of items for the living of industrial workers in Ahmedabad city in the year $2014$ and $2015$ are as follows. Find the cost of living index number of industrial workers. If the wages of workers are increased by $5\%$ in the year $2015$ then is this rise sufficient to compensate the rise in the year $2015 ?$
Group Food Fuel and Electricity Housing Clothing Miscellaneous expense
Weight $31$ $14$ $22$ $10$ $23$
Index Number of $2014$ $270$ $168$ $205$ $174$ $303$
Index Number of $2015$ $281$ $178$ $210$ $177$ $337$
Answer
Image
Index number for the year $2014=\frac{\Sigma I_1 \mathrm{~W}}{\Sigma \mathrm{W}}=\frac{23941}{100}=239.41$
Index number for the year $2015=\frac{\Sigma 2_1 \mathrm{~W}}{\Sigma \mathrm{W}}=\frac{25344}{100}=253.44$
$\rightarrow$ There is an increase of $(253.44-239.41 ) 14.03 \%$ in the cost of living expenditure of $2015$ as compared to that of the year $2014 .$
$\rightarrow$ The wages of workers are increased by $5 \%$ in the year $2015 .$
The actual percentage increase in the cost of living expenditure of workers in the year $2015$ as compared to the year $2014$
$=\frac{\text { Percentage increase in cost of living expenditure in the year } 2015}{\text { Cost of living index number for the year 2014 }} \times 100$
$=\frac{14.03}{239.41} \times 100$
$=0.0586 \times 100$
$=5.86 \%$
But the increase in wage is $5 \%$.
Hence, the increase in the wages of workers in the year $2015$ is not sufficient and is short by $(5.86\ 5=) 0.86 \%$.
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Question 95 Marks
Compute the index number for the year $2015$ by total expenditure method and family budget method and state whether both the index numbers are same.
Item Unit Year $2013$
Consumption $($Quantity$)$		 Year $2013$
Price $(Rs.)$		 Year $2015$
Price $(Rs.)$
Wheat Quintal $100 \ kg$ $1,800$ $2,400$
Rice $20 \ kg$ $40 \ kg$ $700$ $800$
Sugar kg $40 \ kg$ $30$ $36$
Oil kg $60 \ kg$ $108$ $120$
Pulses $20 \ kg$ $40 \ kg$ $2,000$ $2,400$
Ghee kg $36 \ kg$ $400$ $480$
Answer
Here, base year is $2013$ and current year is $2015 .$
$\therefore p _0=$ Price in $2013, q _0=$ Quantity in $2013, p _1=$ Price in $2015$ and $q _1=$ Quantity in $2015$ The units of price and quantity for the items wheat, rice and pulses are not equal. We compute the index number for the year $2015$ by total expenditure method and family budget method after
making the units of these items equal.
Image
Total expenditure method :
Index number for the year $2015$
$=\frac{\Sigma p_1 q_0}{\Sigma p_0 q_0} \times 100$
$=\frac{34720}{29280} \times 100$
$=118.58$
Family budget method:
Index number for the year $2015$
$=\frac{\Sigma I W}{\Sigma W}$
$=\frac{3471992.8}{29280}$
$=118.58$
Thus, the index numbers obtained by total expenditure method and family budget method are equal.
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Question 105 Marks
Compute the Laspeyre’s, Paasche’s and Fisher’s index number for the year $2015$ from the given below :
Item Quantity Price $(Rs.)$
Year $2014$ Year $2015$ Year $2014$ Year $2015$
$A$ $25 \ kg$ $32 \ kg$ $42$ $45$
$B$ $15$ Liter $20$ Liter $28$ $30$
$C$ $10$ Pieces $20$ Pieces $30$ $36$
$D$ $8$ Meter $15$ Meter $20$ $25$
$E$ $30$ Litre $36$ Litre $60$ $65$
Answer
Here, $P_0=$ Price of the year $2014;$
$P_1=$ Price of the year $2015;$
$q_0=$ Quantity of the year $2014$
$q_1=$ Quantity of the year $2015.$
Table of calculation is as under.
Item Year $2014$ Year $2015$ $p_1 q_0$ $p_0 q_0$ $p_1 q_1$ $p_0 q_1$
$p_0$ $q_0$ $p_1$ $q_1$
$A$ $42$ $25$ $45$ $32$ $1,125$ $1,050$ $1,440$ $1,344$
$B$ $28$ $15$ $30$ $20$ $450$ $420$ $600$ $560$
$C$ $30$ $10$ $36$ $20$ $360$ $300$ $720$ $600$
$D$ $20$ $8$ $25$ $15$ $200$ $160$ $375$ $300$
$E$ $60$ $30$ $65$ $36$ $1,950$ $1,800$ $2,340$ $2,160$
Total $-$ $-$ $-$ $-$ $\sum p_1 q_0
= 4,085$		 $\sum p_0 q_0
= 3,730$		 $\sum p_1 q_1
= 5,475$		 $\sum p_0 q_1
= 4,964$
Laspeyre's, Index Numbers
$I_L=\frac{\sum p_1 q_0}{\sum p_0 q_0} \times 100$
$=\frac{4085}{3730} \times 100$
$=109.5174$
$=109.52$
$\therefore I_L=109.52$
Paasche's Index Numbers
$I_p=\frac{\sum p_1 q_1}{\sum p_0 q_1} \times 100$
$=\frac{5445}{4964} \times 100$
$=110.2941$
$=110.29$
$\therefore I_p=110.29$
Fisher's index numbers
$I _{ F }=\sqrt{I_L \times I_P}$
$=\sqrt{109.52 \times 110.29}$
$=\sqrt{12078.9608}$
$=109.9043$
$=109.90$
$\therefore I _{ F }=109.90$
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Question 115 Marks
Compute the Fisher’s index number from the data given below about six different items.
Item
Unit		 $A
20 \ kg$		 $B$
Quintal		 $C$
kg		 $D$
Litre		 $E$
Meter		 $F$
Dozen
Year $2013$ Quantity $5 \ kg$ $10 \ kg$ $1200\ gm$ $30$ litre $12$ meter $20$ piece
Year $2013$ Price (Rs.) $600$ $1600$ $60$ $52$ $8$ $30$
Year $2013$ Quantity $12 \ kg$ $12 \ kg$ $2000\ gm$ $36$ litre $20$ meter $16$ piece
Year $2013$ Price (Rs.) $880$ $2400$ $75$ $32$ $12$ $36$
Answer
Here, $P_0=$ Price of the year $2013$;
$P_1=$ Price of the year $2015;$
$q_0=$ Quantity of the year $2013$
$q_1=$ Quantity of the year $2015$
First of all, we shall make uniform units for the price and quantity of each item.
In the year $2013$, price of $20\ kg $ of an item $A =Rs.600$
Price per $kg =\frac{600}{20}=Rs.30$
Price per quintal of an item $B=Rs. 1,600$
Price per $kg =\frac{1600}{100}=Rs.16$
Quantity of an item $C =1,200\ gm$
Quantity $=\frac{1200}{1000}=1.2\ kg$
Price per dozen of an item $F=Rs.30$
Price per piece $=\frac{30}{12}=Rs.2.5$
Same as in the year $2015,$
Per kg price of item $-A=\frac{880}{20}=Rs.44$
Per kg price of item $-B=\frac{2400}{100}=Rs.24$
Quantity of item$-C =2,000$ gram
Quantity $=\frac{2000}{1000}=2\ kg$
Per piece price of item$-F=\frac{36}{12}=Rs.3$
Table of calculation is as under:
Item Unit Year $2013$ Year $2015$ $p_1q_0$ $p_0q_0$ $p_1q_1$ $p_0q_1$
$p_0$ $q_0$ $p_1$ $q_1$
$A$ kg $30$ $5$ $44$ $12$ $220$ $150$ $528$ $360$
$B$ kg $16$ $10$ $24$ $12$ $240$ $160$ $288$ $192$
$C$ kg $60$ $1.2$ $75$ $2$ $90$ $72$ $150$ $120$
$D$ Litre $52$ $30$ $32$ $36$ $960$ $1560$ $1152$ $1872$
$E$ Meter $8$ $12$ $12$ $20$ $144$ $96$ $240$ $160$
$F$ piece $2.5$ $20$ $3$ $16$ $60$ $50$ $48$ $40$
Total $-$ $-$ $-$ $-$ $-$ $\Sigma p_1q_0
= 1714$		 $\Sigma p_0q_0
= 2088$		 $\Sigma p_1q_1
= 2406$		 $\Sigma p_0q_1
= 2744$
Fisher's index numbers
$I _F=\sqrt{\frac{\sum p_1 q_0}{\sum p_0 q_0} \times \frac{\sum p_1 q_1}{\sum p_0 q_1}} \times 100$
$=\sqrt{\frac{1714}{2088} \times \frac{2406}{2744}} \times 100$
$=\sqrt{0.719766} \times 100$
$=0.848390 \times 100$
$=84.84$
$I _F=84.84$
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Question 125 Marks
The quantity consumed and total expenditure of four different items are as given below. Find Paasche’s and Fisher’s index number for the year $2015$ with respect to the year $2013.$
Item Base year $2013$ Current year $2015$
  Total expenditure
(Rs.)		 Consumption (Quantity) Total expenditure
(Rs.)		 Consumption (Quantity)
$A$ $360$ $60 \ kg$ $375$ $25 \ kg$
$B$ $160$ $10$ litre $416$ $30$ litre
$C$ $480$ $15 \ kg$ $613.2$ $6 \ kg$
$D$ $336$ $3 \ kg$ $400$ $2.5 \ kg$
Answer
Here, quantity (consumption) of items and their total expenditure are given. We obtain the price per unit of item using the formula, Price per unit $=\frac{\text { Total expenditure }}{\text { Quantity }}$
We take $p_0=$ Price in $2013,$
$p_1 =$ Price in $2015,$
$q =$ Quantity in $2013$ and
$q_1=$ Quantity in $2015.$
 The table for calculation is prepared as follows :
Image
Paasche's index number:
$ I_p=\frac{\Sigma p_1 q_1}{\Sigma p_0 q_1}$
$=\frac{1804.2}{942}$
$=1.53$
Fisher's index number:
$\mathrm{I}_{\mathrm{F}}=\sqrt{\frac{\Sigma p_1 q_0}{\Sigma p_0 q_0} \times \frac{\Sigma p_1 q_1}{\Sigma p_0 q_1}} \times 100$
$=\sqrt{\frac{3121}{1336} \times \frac{1804.2}{942}} \times 100$
$=\sqrt{\frac{5630908.2}{1258512}} \times 100$
$=\sqrt{4.4742} \times 100$
$=2.1152 \times 100$
$=211.52$
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Question 135 Marks
The index number of food and clothing among the different groups of cost of living are $150$ and $224.7$ respectively for the year $2015.$ The price of fuel has increased by $220\%.$ The expense on rent has increased from $Rs. 4,000$ to $Rs. 6,000$ and miscellaneous expenses increased by $1.75$ times. If the expenditure for the first four groups are $40\%, 18\%, 12\%$ and $20\%$ respectively, find the cost of living index number for the year $2015$ and interpret it.
Answer
$\rightarrow$ Index number of fuel $=100+$ increase of $220 \%$ for the year $2015=320$
$\rightarrow$ Index number of rent for the year $2015=\frac{6000}{4000} \times 100=150$
$\rightarrow$ Index number of miscellaneous expenses for the year $=100+(100 \times 1.75)=100+175=275$
$\rightarrow$ Percentage expenditure on miscellaneous expenses $=100-(40+18+12+20)=100-90=10$
Table of calculation is as under :
Image
Cost of living index number for the year $2015\ \mathrm{I}=\frac{\Sigma I W}{\Sigma W}$
$=\frac{19634.6}{100}$
$=196.35$
Interpretation: There is $(196.35 - 100=) 96.35 \%$ increase in the cost of living expenditure of the year $2015$ as compared to the base year.
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Question 145 Marks
Find the Laspeyrer’s and Paasche’s index number using the following data for the year $2015$ by taking the year $2014$ as the base year. Also find the Fisher’s index number and interpret it.
Item Base year $2014$ Current year $2015$
Per unit price $(Rs.)$ Total expenditure $(Rs.)$ Per unit price $(Rs.)$ Total expenditure $(Rs.)$
Wheat $16$ $224$ $18$ $270$
Rice $35$ $140$ $40$ $200$
Tuver Dal $100$ $200$ $120$ $360$
Oil $108$ $432$ $120$ $600$
Answer
Here, price of items and its total expenditure are given. We obtain the consumption (quantity) of items using the formula,
$=\frac{\text { Total expenditure }}{\text { Price per unit }}$
We take $p_0$ = Price in $2014$,
$p_1$ = Price in $2015$,
$q_0$ = Quantity in $2014$ and
$q_1$ = Quantity in $2015$.
The table for calculation is prepared as follows :
Image
Laspeyre's index number:
$I _{ L }=\frac{\Sigma p_1 q_0}{\Sigma p_0 q_0} \times 100$
$=\frac{1132}{996} \times 100$
$=113.65$
Paasche's index number:
$I _p=\frac{\Sigma p_1 q_1}{\Sigma p_0 q_1} \times 100$
$=\frac{1430}{1255} \times 100$
$=113.94$
Fisher's index number:
$I _{ F }=\sqrt{ I _{ L } \times I _{ P }}$
$=\sqrt{113.65 \times 113.94}$
$=\sqrt{12949.281}$
$=113.79$
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Question 155 Marks
The details of expenditure on clothing for the worker class of a region are as follows. Find the index number for clothing by the total expenditure and family budget method.
Answer
Here, $p_0$ = price in $2010$,
$q_0$ = quantity in year $2010$ and
$p_1$ = price in year $2014$.
The table for calculation is prepared as follows:

Index number of clothing by total expenditure method:
$I = \frac{\Sigma p_{1} q_{0}}{\Sigma p_{0} q_{0}} \times 100$
$= \frac{3917}{3021.5} \times 100$
$= 1.2964 \times 100 = 129.64$
Index number of clothing by family budget method:
$I = \frac{\Sigma \mathrm{IW}}{\Sigma \mathrm{W}}$
$= \frac{391692.05}{3021.5}$
$= 129.64$
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Question 165 Marks
Compute the general index number for the production using the following data:
Answer

General index number of production:
$I = \frac{\Sigma \mathrm{IW}}{\Sigma \mathrm{W}}$
$=\frac{21320}{100}$
$= 213.20$
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Question 175 Marks
Compute the cost of living index number by the method of total expenditure from the following data:
Answer
Here, $p_0=$ Price in $2014,$
$q_0=$ Quantity in $2014$ and
$p_1=$ Price in $2015.$
After making the units of price and quantity same, we will compute index number.Explanation:
Item $A:$ The unit of price is quintal and the unit of quantity is $kg.$
$\therefore \ln 2014$. the price per $kg =\frac{1200}{100}=₹ 12$
In $2015,$ the price per $kg =\frac{1700}{100}=₹ 17$
Item $B :$ The unit of price is $20 \ kg$ and the unit of quantity is $kg .$
$\therefore$ In $2014.$ the price per $kg =\frac{340}{20}=₹ 17$
In $2015,$ the price per $kg =\frac{380}{20}=₹ 19$
Item $C:$ The unit of price is $10$ litre and the unit of quantity is litre.
$\therefore \ln 2014$, the price per litre $=\frac{30}{10}=₹ 3$
In $2015,$ the price per litre $=\frac{40}{10}=₹ 4$
Item $D :$ The unit of price is dozen and the unit of quantity is a piece.
$\therefore \ln 2014$, the price per piece $=\frac{15}{12}=₹ 1.25$
In $2015,$ the price per piece $=\frac{24}{12}=₹ 2$
The table for calculation is prepared as follows:​​​​​​​

Cost of living index number by total expenditure method:
$I = \frac{\Sigma p_{1} q_{0}}{\Sigma p_{0} q_{0}}× 100 = \frac{1504}{1135} \times 100 = 1.3251 \times 100 = 132.51$
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Question 185 Marks
Find the index number for the year $2014$ by the method of family budget from the following data about prices and consumption of food items and interpret it:
Answer
Here, $p_0 =$ Price in $2010,$
$q_0 =$ Quantity in $2010$ and
$p_1 =$ Price in $2014.$
The table for calculation is prepared as follows:

Index number of $2014$ by family budget method:
$I = \frac{\Sigma \mathrm{IW}}{\Sigma \mathrm{W}}$
$=\frac{464006}{3610}$
$= 128.53$
Interpretation: Index number of $2014$ is $128.53.$
So there is $(128.53 – 100 =) 28.53\%$ increase in the prices of food items for the year $2014.$
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Question 195 Marks
The following data are obtained from the family budget inquiry of middle class people. State the change in the cost of living in the year $2015$ with respect to the year $2013$ by finding the index number. If the average monthly disposable income of a family in the year $2013$ is ₹ $15000$, then obtain the estimate of the necessary average monthly disposable income in the year $2015$.
Answer
Here, $p_0$ = Expenditure in $2013$ and
$P_1$ = Expenditure in $2015$.
The table for calculation is prepared as follows:

Cost of living index number:
$I = \frac{\Sigma \mathrm{IW}}{\Sigma \mathrm{W}}=\frac{13564}{100} = 135.64$
The cost of living in the year $2015$ has been increased $(135.64 – 100 =) 35.64 \%$
The necessary average monthly disposable income in the year $2015$
$= \frac{(\text { The average monthly income in 2013) } \times(\text { The cost of living index number in 2015) }}{100}$
$= \frac{15000 \times 135.64}{100}$
$= \frac{2034600}{100}$
$= ₹ 20346$
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Question 205 Marks
Find Laspeyre's, Paasche's and Fisher's index numbers for the year 2024 with base year 2023 from the data about price and consumption of food items given below :
ItemUnitYear 2024 Year 2023
Price (₹)QuantityPrice (₹)Quantity
Rice20 Kg.8001.5 Kg.7801 Kg.
MilkLitre4410 Litre4012 Litre
BreadKg.501.5 Kg.452 Kg.
BananaDozen361.5 Dozen302 Dozen
Answer
ItemUnit$p_1$$q_1$$p_0$$q_0$$p_1 q_0$$p_0 q_0$$p_1 q_1$$p_0 q_1$
Ricekg401.539140396058.5
MilkLitre44104012528480440400
Breadkg501.545 100907567.5
BananaDozen361.530272605445
Total 740669629571

1. $I_L=\frac{\sum p_1 q_0}{\sum p_0 q_0} \times 100$
$=\frac{740}{669} \times 100$
$I_L \approx 110.61$
2. $I_P=\frac{\sum p_1 q_1}{\sum p_0 q_1} \times 100$
$=\frac{629}{571} \times 100$
$I_P \approx 110.16$
3. $I_F=\sqrt{I_L \times I_P}$
$I_F=\sqrt{110.61 \times 110.16}$
$I_F=\sqrt{12184.80}$
$I_F \approx 110.38$
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Question 215 Marks
Find the Paasche's and Fisher's index numbers for the year 2023 with the base year 2022 using the data given below :
ItemsABCDE
Year 2022Price (₹)100100150180250
Total Expenditure40050060010801000
Year 2023Price (₹)120120160200300
Total Expenditure72060080010001200
Answer
Item$p_0$$q_0$$p_1$$q_1$$p_0 q_0$$p_1 q_1$$p_1 q_0$$p_0 q_1$
A10041206400720480600
B10051205500600600500
C15041605600800640750
D18062005108010001200900
E250430041000120012001000
$\operatorname{Sum}\left(\sum\right)$3580432041203750
$I_P=\frac{\sum p_1 q_1}{\sum p_0 q_1} \times 100$
$=\frac{4320}{3750} \times 100=115.2$
$I_L=\frac{\sum p_1 q_0}{\sum p_0 q_0} \times 100=\frac{4120}{3580} \times 100 \approx 115.08$
$I_F=\sqrt{I_L \times I_P}=\sqrt{115.08 \times 115.2} \approx 115.14$
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Question 225 Marks
Answer
Item$q_0$$q_1$$p_0$$p_1$$p_1 q_0$$p_0 q_0$$p_1 q_1$$p_0 q_1$
A253242451125105014401344
B15202830450420600560
C10203036360300720600
D8152025200160375300
E303660651950180023402160
4085373054754964
$I_L=\frac{\sum p_1 q_0}{\sum p_0 q_0} \times 100$
$I_L=\frac{4085}{3730} \times 100 \approx 109.52$
$I_P=\frac{\sum p_1 q_1}{\sum p_0 q_1} \times 100$
$I_P=\frac{5475}{4964} \times 100 \approx 110.29$
$I_F=\sqrt{I_L \times I_P}$
$I_F=\sqrt{109.52 \times 110.29}=\sqrt{12078.96} \approx 109.90$
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Question 235 Marks
Find Laspeyre's, Paasche's and Fisher's index number from the data given below about consumption and total expenditure of five different items :
ItemsBase Year Current Year
ConsumptionTotal ExpenditureConsumptionTotal Expenditure
A100 kg5000120 kg8400
B240 kg1200280 kg1400
C60 litres66040 litres400
D40 kg72030 kg600
E10 kg8010 kg100
Answer
Item$q_0$$p_0$$q_1$$p_1$$p_1 q_0$$p_0 q_0$​$p_1 q_1$$p_0 q_1$​
A10050120707000500084006000
B240528051200120014001400
C60114010600660400440
D40183020800720600540
E10810101008010080
Total 97007660109008460
$I_L=\frac{\sum p_1 q_0}{\sum p_0 q_0} \times 100$
$=\frac{9700}{7660} \times 100 \approx 126.63$
$I_P=\frac{\sum p_1 q_1}{\sum p_0 q_1} \times 100$
$=\frac{10900}{8460} \times 100 \approx 128.84$
$I_F=\sqrt{I_L \times I_P}$
$=\sqrt{126.63 \times 128.84}$
$=\sqrt{16315.01} \approx 127.73$
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