GSEB Class 12 Statistics Solutions Chapter 1 Index Number Exercise 1.4

GSEB Solutions Class 12 Statistics Part 1 Chapter 1 Index Number Ex 1.4

 

Question 1. 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 Rs. 15000, then obtain the estimate of the necessary average monthly disposable income in the year 2015.

Group	Food	Fuel-Electricity	Rent	Clothing	Miscellaneous
Weight	45	15	10	20	10
Expenditure in 2013 (Rs.)	3000	1450	1500	600	1600
Expenditure in 2015 (Rs.)	3900	1850	2400	900	1920

Answer: For this problem, \(p_0\) means the expenditure in 2013 and \(p_1\) means the expenditure in 2015.
The calculation table is prepared as follows:

Group	Weight (W)	Year 2013 Expenditure (Rs.) \(p_0\)	Year 2015 Expenditure (Rs.) \(p_1\)	Expenditure relative \(I = \frac{p_1}{p_0} \times 100\)	IW
Food	45	3000	3900	\(\frac{3900}{3000} \times 100 = 130\)	5850
Fuel-Electricity	15	1450	1850	\(\frac{1850}{1450} \times 100 = 127.6\)	1914
Rent	10	1500	2400	\(\frac{2400}{1500} \times 100 = 160\)	1600
Clothing	20	600	900	\(\frac{900}{600} \times 100 = 150\)	3000
Miscellaneous	10	1600	1920	\(\frac{1920}{1600} \times 100 = 120\)	1200
Total	\(\Sigma W = 100\)	-	-	-	\(\Sigma IW = 13564\)

The cost of living index number is found using the formula:
\(I = \frac{\Sigma IW}{\Sigma W}\)
\(I = \frac{13564}{100}\)
\(I = 135.64\)
The cost of living increased by \((135.64 - 100) = 35.64\%\) in 2015 compared to 2013.
To find the required average monthly disposable income in 2015:
Necessary income = \(\frac{\text{(Average monthly income in 2013)} \times \text{(Cost of living index number in 2015)}}{100}\)
Necessary income = \(\frac{15000 \times 135.64}{100}\)
Necessary income = \(\frac{2034600}{100}\)
Necessary income = Rs. 20346
In simple words: We calculated how much prices went up for a middle-class family by finding an index number. Then, we used this number to see how much income a family needs in 2015 to have the same buying power as Rs. 15000 in 2013. The new income needed is Rs. 20346.

🎯 Exam Tip: Remember to clearly define \(p_0\) and \(p_1\) and show all calculation steps for \(I\) and the final income estimate to earn full marks.

 

Question 2. 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:

Item	Year 2010		Year 2014
	Quantity	Price (Rs.)	Price (Rs.)
Wheat	60	15	18
Rice	40	32	40
Bajri	15	12	14
Tuver Dal	25	50	70

Answer: In this problem, \(p_0\) represents the price in 2010, \(q_0\) is the quantity in 2010, and \(p_1\) is the price in 2014.
The calculation table is prepared as follows:

Item	\(q_0\)	\(p_0\)	\(p_1\)	\(W = p_0 q_0\)	Price relative \(I = \frac{p_1}{p_0} \times 100\)	IW
Wheat	60	15	18	900	\(\frac{18}{15} \times 100 = 120\)	108000
Rice	40	32	40	1280	\(\frac{40}{32} \times 100 = 125\)	160000
Bajri	15	12	14	180	\(\frac{14}{12} \times 100 = 116.7\)	21006
Tuver Dal	25	50	70	1250	\(\frac{70}{50} \times 100 = 140\)	175000
Total	-	-	-	\(\Sigma W = 3610\)	-	\(\Sigma IW = 464006\)

The index number for 2014 by the family budget method is:
\(I = \frac{\Sigma IW}{\Sigma W}\)
\(I = \frac{464006}{3610}\)
\(I = 128.53\)
Interpretation: The index number for 2014 is 128.53. This means that the prices of food items increased by \((128.53 - 100) = 28.53\%\) in 2014.
In simple words: We calculated how much food prices changed in 2014 compared to 2010. The index number is 128.53, which means food prices are about 28.53% higher than they were in 2010.

🎯 Exam Tip: When interpreting an index number, always remember to subtract 100 from the index to get the percentage change, and clearly state whether it's an increase or decrease.

 

Question 3. Compute the index number by the method of total expenditure from the following data:

Item	A	B	C	D	E
Unit	Quintal	20 kg	10 litre	dozen	meter
Quantity of year 2014	50 kg	18 kg	12 litre	20 pieces	14 meter
Price of year 2014 (Rs.)	1200	340	30	15	12
Price of year 2015 (Rs.)	1700	380	40	24	16

Answer: Here, \(p_0\) is the price in 2014, \(q_0\) is the quantity in 2014, and \(p_1\) is the price in 2015.
Before calculating the index number, we need to ensure that the units of price and quantity are consistent for each item.
Explanation of unit conversion:
Item A: The unit of price is quintal (100 kg) and the unit of quantity is kg.
In 2014, the price per kg = \(\frac{1200}{100}\) = Rs. 12
In 2015, the price per kg = \(\frac{1700}{100}\) = Rs. 17
Item B: The unit of price is 20 kg and the unit of quantity is kg.
In 2014, the price per kg = \(\frac{340}{20}\) = Rs. 17
In 2015, the price per kg = \(\frac{380}{20}\) = Rs. 19
Item C: The unit of price is 10 litre and the unit of quantity is litre.
In 2014, the price per litre = \(\frac{30}{10}\) = Rs. 3
In 2015, the price per litre = \(\frac{40}{10}\) = Rs. 4
Item D: The unit of price is dozen (12 pieces) and the unit of quantity is a piece.
In 2014, the price per piece = \(\frac{15}{12}\) = Rs. 1.25
In 2015, the price per piece = \(\frac{24}{12}\) = Rs. 2
The table for calculation after unit normalization is as follows:

Item	Unit	Year 2014		Year 2015	\(p_1 q_0\)	\(p_0 q_0\)
		\(q_0\)	\(p_0\)	\(p_1\)
A	kg	50	12	17	850	600
B	kg	18	17	19	342	306
C	Litre	12	3	4	48	36
D	Pieces	20	1.25	2	40	25
E	Meter	14	12	16	224	168
Total	-	-	-	-	\(\Sigma p_1 q_0 = 1504\)	\(\Sigma p_0 q_0 = 1135\)

The cost of living index number using the total expenditure method is:
\(I = \frac{\Sigma p_1 q_0}{\Sigma p_0 q_0} \times 100\)
\(I = \frac{1504}{1135} \times 100\)
\(I = 1.3251 \times 100\)
\(I = 132.51\)
In simple words: First, we made sure all the units for quantity and price were the same. Then, we calculated the total cost of all items for 2014 (\(p_0 q_0\)) and 2015 (\(p_1 q_0\)). By comparing these totals, we found that the index number for cost of living is 132.51, meaning expenses increased.

🎯 Exam Tip: Always check and standardize units for all items before starting calculations. Incorrect units are a common source of error in index number problems.

 

Question 4. Compute the general index number for the production using the following data:

Item	Cotton Cloth	Grains	Sugar	Steel	Copper	Cement
Weight	15	23	15	25	10	12
Index number of production	220	225	190	215	198	220

Answer: The calculation table is prepared as follows:

Item	Weight (W)	Index number of production (I)	IW
Cotton Cloth	15	220	3300
Grains	23	225	5175
Sugar	15	190	2850
Steel	25	215	5375
Copper	10	198	1980
Cement	12	220	2640
Total	\(\Sigma W = 100\)	-	\(\Sigma IW = 21320\)

The general index number of production is calculated as:
\(I = \frac{\Sigma IW}{\Sigma W}\)
\(I = \frac{21320}{100}\)
\(I = 213.20\)
In simple words: We calculated an overall production index number by multiplying each item's weight by its production index, then adding them up. Dividing this total by the sum of all weights gives us the general production index, which is 213.20.

🎯 Exam Tip: Ensure that the sum of weights \(\Sigma W\) is 100 or adjust the formula accordingly. Simple multiplication and addition errors are common here, so double-check your calculations.

 

Question 5. 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.

Item	Saree	Dhoti	Shirting	Other
Unit	Piece	Piece	Meter	Meter
Quantity in year 2010	5	8	20	15
Price in year 2010 (Rs.)	300	70	32.40	20.90
Price in year 2014 (Rs.)	400	100	38	23.80

Answer: Here, \(p_0\) is the price in 2010, \(q_0\) is the quantity in 2010, and \(p_1\) is the price in 2014.
The table for calculations, including both total expenditure and family budget methods, is prepared as follows:

Item	Unit	Year 2010		Year 2014	Total expenditure method		Family budget method
		\(q_0\)	\(p_0\)	\(p_1\)	\(p_1 q_0\)	\(p_0 q_0\)	\(W = p_0 q_0\)	Price relative \(I = \frac{p_1}{p_0} \times 100\)	IW
Saree	Pieces	5	300	400	2000	1500	1500	\(\frac{400}{300} \times 100 = 133.3\)	199950
Dhoti	Pieces	8	70	100	800	560	560	\(\frac{100}{70} \times 100 = 142.9\)	80024
Shirting	Meter	20	32.40	38	760	648	648	\(\frac{38}{32.40} \times 100 = 117.3\)	76010.4
Other	Meter	15	20.90	23.80	357	313.5	313.5	\(\frac{23.80}{20.90} \times 100 = 113.9\)	35707.65
Total	-	-	-	-	\(\Sigma p_1 q_0 = 3917\)	\(\Sigma p_0 q_0 = 3021.5\)	\(\Sigma W = 3021.5\)	-	\(\Sigma IW = 391692.05\)

Index number of clothing by total expenditure method:
\(I = \frac{\Sigma p_1 q_0}{\Sigma p_0 q_0} \times 100\)
\(I = \frac{3917}{3021.5} \times 100\)
\(I = 1.2964 \times 100 = 129.64\)
Index number of clothing by family budget method:
\(I = \frac{\Sigma IW}{\Sigma W}\)
\(I = \frac{391692.05}{3021.5}\)
\(I = 129.64\)
In simple words: We calculated the clothing index using two methods. The total expenditure method looks at the total cost of clothes in 2010 versus 2014. The family budget method weighs each clothing item by its importance in 2010 and then calculates the overall price change. Both methods show the clothing index as 129.64.

🎯 Exam Tip: Note that for the family budget method, the weight (W) is typically calculated as \(p_0 q_0\). Ensure you correctly calculate both \(\Sigma p_1 q_0\) and \(\Sigma p_0 q_0\) for the total expenditure method and \(\Sigma IW\) and \(\Sigma W\) for the family budget method.