GSEB Class 12 Statistics Solutions Chapter 1 Index Number Exercise 1.1

Gujarat Board Textbook Solutions Class 12 Statistics Part 1 Chapter 1 Index Number Ex 1.1

 

Question 1. The data about average daily wage of a group of workers employed in a factory in a city during the year 2008 to 2015 are as follows. Find the index number by (1) Fixed base method (taking base year 2008), (2) Chain base method and (3) Fixed base method by taking average of average daily wages of the years 2011 to 2013 as the wage for the base year.

Year	2008	2009	2010	2011	2012	2013	2014	2015
Average daily wage (Rs.)	275	284	289	293	297	313	328	345

Answer:The index numbers for daily wages have been calculated using three different methods, as shown in the table below.

Year	Average daily wage (Rs.)	(1)	(2)	(3)
		Fixed Base Method Base year: 2008 \( P_0 = 275 \) Index number \( = \frac{P_1}{P_0} \times 100 \)	Chain Base Method Index number \( = \frac{\text{Current year's average daily wage}}{\text{Previous year's average daily wage}} \times 100 \)	Index number based on average wage \( P_0 = \text{Average of daily wage from 2011 to 2013} \) \( = \frac{293 + 297 + 313}{3} = \frac{903}{3} = 301 \) Index number \( = \frac{P_1}{P_0} \times 100 \)
2008	275	\( = 100 \)	\( = 100 \)	\( \frac{275}{301} \times 100 = 91.36 \)
2009	284	\( \frac{284}{275} \times 100 = 103.27 \)	\( \frac{284}{275} \times 100 = 103.27 \)	\( \frac{284}{301} \times 100 = 94.35 \)
2010	289	\( \frac{289}{275} \times 100 = 105.09 \)	\( \frac{289}{284} \times 100 = 101.76 \)	\( \frac{289}{301} \times 100 = 96.01 \)
2011	293	\( \frac{293}{275} \times 100 = 106.55 \)	\( \frac{293}{289} \times 100 = 101.38 \)	\( \frac{293}{301} \times 100 = 97.34 \)
2012	297	\( \frac{297}{275} \times 100 = 108.00 \)	\( \frac{297}{293} \times 100 = 101.37 \)	\( \frac{297}{301} \times 100 = 98.67 \)
2013	313	\( \frac{313}{275} \times 100 = 113.82 \)	\( \frac{313}{297} \times 100 = 105.39 \)	\( \frac{313}{301} \times 100 = 103.99 \)
2014	328	\( \frac{328}{275} \times 100 = 119.27 \)	\( \frac{328}{313} \times 100 = 104.79 \)	\( \frac{328}{301} \times 100 = 108.97 \)
2015	345	\( \frac{345}{275} \times 100 = 125.45 \)	\( \frac{345}{328} \times 100 = 105.18 \)	\( \frac{345}{301} \times 100 = 114.62 \)

In simple words: This question shows three ways to calculate an index number for wages. It helps to see how wages change over time, either by comparing to a single base year, to the year just before, or to an average of a few base years.

🎯 Exam Tip: Remember to correctly identify the base year wage (\(P_0\)) for each method. Pay close attention to whether it's a fixed year, the previous year, or an average of specific years, as this is crucial for accurate calculations.

 

Question 2. From the following data about the retail prices of sugar in a city, find the index numbers of price of sugar by (1) Fixed base method with year 2008 as base year, (2) Chain base method and (3) Taking the average price of sugar for the year 2009 and 2010 as the base year price.

Year	2008	2009	2010	2011	2012	2013	2014	2015
Price of Sugar per kilogram (Rs.)	28	28.50	29.50	30	31	32	34	36

Answer:The price index numbers for sugar are found using three distinct methods, as detailed in the table below.

Year	Price of sugar per kg (Rs.)	(1)	(2)	(3)
		Fixed Base Method Base year: 2008 \( P_0 = 28 \) Index number \( = \frac{P_1}{P_0} \times 100 \)	Chain Base Method Index number \( = \frac{\text{Current year's price of sugar}}{\text{Previous year's price of sugar}} \times 100 \)	Index number based on average price of sugar \( P_0 = \text{Average price of 2009 and 2010} \) \( = \frac{28.50 + 29.50}{2} = \frac{58}{2} = 29 \) Index number \( = \frac{P_1}{P_0} \times 100 \)
2008	28	\( = 100 \)	\( = 100 \)	\( \frac{28}{29} \times 100 = 96.55 \)
2009	28.50	\( \frac{28.50}{28} \times 100 = 101.79 \)	\( \frac{28.50}{28} \times 100 = 101.79 \)	\( \frac{28.50}{29} \times 100 = 98.28 \)
2010	29.50	\( \frac{29.50}{28} \times 100 = 105.36 \)	\( \frac{29.50}{28.50} \times 100 = 103.51 \)	\( \frac{29.50}{29} \times 100 = 101.72 \)
2011	30	\( \frac{30}{28} \times 100 = 107.14 \)	\( \frac{30}{29.50} \times 100 = 101.69 \)	\( \frac{30}{29} \times 100 = 103.45 \)
2012	31	\( \frac{31}{28} \times 100 = 110.71 \)	\( \frac{31}{30} \times 100 = 103.33 \)	\( \frac{31}{29} \times 100 = 106.90 \)
2013	32	\( \frac{32}{28} \times 100 = 114.29 \)	\( \frac{32}{31} \times 100 = 103.23 \)	\( \frac{32}{29} \times 100 = 110.34 \)
2014	34	\( \frac{34}{28} \times 100 = 121.43 \)	\( \frac{34}{32} \times 100 = 106.25 \)	\( \frac{34}{29} \times 100 = 117.24 \)
2015	36	\( \frac{36}{28} \times 100 = 128.57 \)	\( \frac{36}{34} \times 100 = 105.88 \)	\( \frac{36}{29} \times 100 = 124.14 \)

In simple words: This question demonstrates how to compute index numbers for sugar prices using three methods: comparing to a fixed base year (2008), comparing to the price of the previous year (chain base), and comparing to an average price from specific base years (2009 and 2010).

🎯 Exam Tip: When dealing with average base prices, ensure you calculate the average correctly before using it as the denominator (\(P_0\)) for all subsequent years. This setup is key to avoiding errors.

 

Question 3. The following data are obtained about the annual average prices of wheat, rice and sugar in the wholesale market of a city. Find the general index number for three items by fixed base method with base year 2011 and by chain base method.

Year	2011	2012	2013	2014	2015
Item
Wheat	18	18.50	18.90	19	19.50
Rice	30	36	38	38	39
Sugar	30	31	32	34	36

Answer:

General Index Number of Price by Fixed Base Method:

Item	Price (Rs.)
	2011	2012	2013	2014	2015
Index number by Fixed Base method \( = \frac{\text{Current year's price}}{\text{Base year's price}} \times 100 \) Base year: 2011
Wheat	100	\( \frac{18.50}{18} \times 100 = 102.78 \)	\( \frac{18.90}{18} \times 100 = 105.00 \)	\( \frac{19}{18} \times 100 = 105.56 \)	\( \frac{19.50}{18} \times 100 = 108.33 \)
Rice	100	\( \frac{36}{30} \times 100 = 120.00 \)	\( \frac{38}{30} \times 100 = 126.67 \)	\( \frac{38}{30} \times 100 = 126.67 \)	\( \frac{39}{30} \times 100 = 130.00 \)
Sugar	100	\( \frac{31}{30} \times 100 = 103.33 \)	\( \frac{32}{30} \times 100 = 106.67 \)	\( \frac{34}{30} \times 100 = 113.33 \)	\( \frac{36}{30} \times 100 = 120.00 \)
Sum	300	326.11	338.34	345.56	358.33
General Index number of price \( = \frac{\text{Sum}}{3} \)	\( \frac{300}{3} = 100 \)	\( \frac{326.11}{3} = 108.70 \)	\( \frac{338.34}{3} = 112.78 \)	\( \frac{345.56}{3} = 115.19 \)	\( \frac{358.33}{3} = 119.44 \)

General Index Number of Price by Chain Base Method:

Item	Price (Rs.)
	2011	2012	2013	2014	2015
Index number by Chain Base method \( = \frac{\text{Current year's price}}{\text{Previous year's price}} \times 100 \)
Wheat	100	\( \frac{18.50}{18} \times 100 = 102.78 \)	\( \frac{18.90}{18.50} \times 100 = 102.16 \)	\( \frac{19}{18.90} \times 100 = 100.53 \)	\( \frac{19.50}{19} \times 100 = 102.63 \)
Rice	100	\( \frac{36}{30} \times 100 = 120.00 \)	\( \frac{38}{36} \times 100 = 105.56 \)	\( \frac{38}{38} \times 100 = 100.00 \)	\( \frac{39}{38} \times 100 = 102.63 \)
Sugar	100	\( \frac{31}{30} \times 100 = 103.33 \)	\( \frac{32}{31} \times 100 = 103.23 \)	\( \frac{34}{32} \times 100 = 106.25 \)	\( \frac{36}{34} \times 100 = 105.88 \)
Sum	300	326.11	310.95	306.78	311.14
General Index number of price \( = \frac{\text{Sum}}{3} \)	\( \frac{300}{3} = 100 \)	\( \frac{326.11}{3} = 108.70 \)	\( \frac{310.95}{3} = 103.65 \)	\( \frac{306.78}{3} = 102.26 \)	\( \frac{311.14}{3} = 103.71 \)

In simple words: This solution calculates the general index number for wheat, rice, and sugar using two methods. The fixed base method compares all prices to the base year 2011, while the chain base method compares each year's price to the previous year's price. Both methods help understand overall price changes over time for these items.

🎯 Exam Tip: When calculating general index numbers for multiple items, ensure individual item indices are correctly computed first. Then, sum these individual indices and divide by the number of items to get the aggregate index, carefully noting whether fixed or chain base method is required.

 

Question 4. The prices of five fuel related items in the years 2012 and 2014 are as follows. Calculate the general index number for five fuel items by taking the year 2012 as the base year and state the overall increase in the prices of fuel items.

Item	Unit	Electricity	Gas	Match Box	Kerosene	Wood
Unit		Unit	Cylinder	Box	Litre	Kilogram
Price in 2012 (Rs.)		3	345	1.00	15	12
Price in 2014 (Rs.)		3.5	370	1.50	20	15

Answer:

Item	Unit	Price in 2012 (Rs.) \( P_0 \)	Price in 2014 (Rs.) \( P_1 \)	Price relative \( = \frac{P_1}{P_0} \)
Electricity	Unit	3	3.5	\( \frac{3.5}{3} = 1.1667 \)
Gas	Cylinder	345	370	\( \frac{370}{345} = 1.0725 \)
Match Box	Box	1.00	1.50	\( \frac{1.50}{1} = 1.5000 \)
Kerosene	Litre	15	20	\( \frac{20}{15} = 1.3333 \)
Wood	Kg	12	15	\( \frac{15}{12} = 1.2500 \)
Total			\( n = 5 \)	\( \sum \frac{P_1}{P_0} = 6.3225 \)

General index number of price for five fuel items
\[ = \frac{\sum\left(\frac{p_{1}}{p_{0}}\right)}{n} \times 100 \]
\[ = \frac{6.3225}{5} \times 100 \]
\[ = 126.45 \] Compared to the year 2012, the general price index number for five fuel items in 2014 is 126.45. This shows an overall price increase for these fuel items. To find the exact increase, we subtract the base year index (100) from the calculated index.
Hence, the overall increase in the prices of fuel items is \( (126.45 - 100) = 26.45\% \).In simple words: This problem finds out how much fuel prices have increased from 2012 to 2014. By comparing the prices of five fuel items, we calculate a single index number. This number tells us that, on average, fuel prices went up by 26.45% in 2014 compared to 2012.

🎯 Exam Tip: Remember to calculate the price relative (\(P_1/P_0\)) for each item individually first. Sum these ratios and then divide by the number of items before multiplying by 100 to get the general index number. Clearly state the percentage increase by subtracting 100 from the final index.