GSEB Class 12 Statistics Solutions Chapter 1 Index Number Exercise 1.2

 

Question 1. The chain base index numbers of agricultural production of a state from the year 2008 to 2014 are as follows. Compute the fixed base index numbers. (Take 2007 as base year.)

Year	2008	2009	2010	2011	2012	2013	2014
Index number of agricultural production	100	110	95	108	120	106	110

Answer: The base year is 2007, so its index number is set to 100. To find the fixed base index number for each year, we use a special formula. We take the current year's chain base index number, multiply it by the fixed base index number from the previous year, and then divide the result by 100. This calculation is done step-by-step for each year from 2008 to 2014, converting the chain base numbers into fixed base numbers.

Year	Chain base index number of agricultural production	Fixed base index number = \(\frac{\text{Current year's chain base index number} \times \text{Previous year's fixed base index number}}{100}\)
2008	100	\(\frac{100 \times 100}{100} = 100.00\)
2009	110	\(\frac{110 \times 100}{100} = 110.00\)
2010	95	\(\frac{95 \times 110}{100} = 104.50\)
2011	108	\(\frac{108 \times 104.5}{100} = 112.86\)
2012	120	\(\frac{120 \times 112.86}{100} = 135.43\)
2013	106	\(\frac{106 \times 135.43}{100} = 143.56\)
2014	110	\(\frac{110 \times 143.56}{100} = 157.92\)

In simple words: To change chain base index numbers to fixed base index numbers, you always use the formula: (current year's chain index * previous year's fixed index) / 100. Start with the base year's fixed index as 100.

🎯 Exam Tip: Always remember to state the base year and its index value (usually 100) when calculating fixed base index numbers. Accuracy in multiplication and division is crucial.

 

Question 2. Obtain the chain base index number from the fixed base index numbers given below with the year 2007-08 as the base year for the wholesale prices of machines and equipments:

Year	2008-09	2009-10	2010-11	2011-12	2012-13	2013-14	2014-15
Index number of machines and equipments	117.4	118	121.3	125.1	128.4	131.6	134.6

Answer: To find the chain base index number from fixed base index numbers, we use a formula that compares the current year's index to the previous year's index. For the initial year (2008-09), since there's no previous year mentioned, we assume its chain base index number is the same as its fixed base index number. For all other years, we divide the current year's fixed base index number by the previous year's fixed base index number and then multiply by 100. This gives us the chain base index for each period.

Year	Fixed base index number of machines and equipments	Chain base index number = \(\frac{\text{Current year's fixed base index number}}{\text{Previous year's fixed base index number}} \times 100\)
2008-'09	117.4	\(\frac{117.4}{100} \times 100 = 117.40\)
2009-'10	118	\(\frac{118}{117.4} \times 100 = 100.51\)
2010-'11	121.3	\(\frac{121.3}{118} \times 100 = 102.80\)
2011-'12	125.1	\(\frac{125.1}{121.3} \times 100 = 103.13\)
2012-'13	128.4	\(\frac{128.4}{125.1} \times 100 = 102.64\)
2013-'14	131.6	\(\frac{131.6}{128.4} \times 100 = 102.49\)
2014-'15	134.6	\(\frac{134.6}{131.6} \times 100 = 102.28\)

In simple words: To convert fixed base index numbers to chain base index numbers, divide the current year's fixed index by the previous year's fixed index and multiply by 100. For the very first year given, its chain base index is the same as its fixed base index, assuming the base fixed index is 100.

🎯 Exam Tip: When converting fixed base to chain base, the formula is (current year's fixed index / previous year's fixed index) * 100. Remember that the first chain base index is typically the first fixed base index divided by 100 if the fixed base is calculated with base year 100.

 

Question 3. The fixed base index numbers of food from the month of January to October in the year 2015 for the industrial workers of Ahmedabad are as given below. Compute the chain base index numbers.

Month	January	February	March	April	May	June	July	August	September	October
Index number of food	271	270	268	268	278	283	283	293	293	299

Answer: To find the chain base index numbers from the given fixed base index numbers, we apply a specific formula. Since the base year (or month) is not explicitly stated, we treat the first month's fixed base index (January's 271) as the base for the chain index calculation, effectively making its chain base index 100. For every subsequent month, the chain base index is found by dividing the current month's fixed base index by the previous month's fixed base index and then multiplying by 100.

Month	Fixed base index number of food	Chain base index number = \(\frac{\text{Current year's fixed base index number}}{\text{Previous year's fixed base index number}} \times 100\)
January	271	\(= 100\)
February	270	\(\frac{270}{271} \times 100 = 99.63\)
March	268	\(\frac{268}{270} \times 100 = 99.26\)
April	268	\(\frac{268}{268} \times 100 = 100\)
May	278	\(\frac{278}{268} \times 100 = 103.73\)
June	283	\(\frac{283}{278} \times 100 = 101.80\)
July	283	\(\frac{283}{283} \times 100 = 100\)
August	293	\(\frac{293}{283} \times 100 = 103.53\)
September	293	\(\frac{293}{293} \times 100 = 100\)
October	299	\(\frac{299}{293} \times 100 = 102.05\)

In simple words: When the base is not given for chain index calculation from fixed base data, the first value's chain index is set to 100. Then, each new chain index is found by dividing the current fixed index by the previous month's fixed index and multiplying by 100.

🎯 Exam Tip: If the base period for chain base index is not specified, assume the first period in the given data series is the base, and its chain base index will be 100. Be careful with calculations as even small errors can propagate.

 

Question 4. The chain base index numbers for sales of a certain type of scooter from the year 2010 to 2015 are as follows. Find fixed base index numbers.

Year	2010	2011	2012	2013	2014	2015
Index number of sale	110	112	109	108	105	111

Answer: To compute the fixed base index numbers from the provided chain base index numbers, we begin with the first year, 2010. Its fixed base index number is simply its chain base index number (110). For the following years, we multiply the current year's chain base index number by the previous year's fixed base index number and then divide by 100. This method helps convert the chain series into a fixed series by using a constant base.

Year	Chain base index number of sale of scooter	Fixed base index number = \(\frac{\text{Current year's chain base index number} \times \text{Previous year's fixed base index number}}{100}\)
2010	110	\(= 110.00\)
2011	112	\(\frac{112 \times 110}{100} = 123.20\)
2012	109	\(\frac{109 \times 123.20}{100} = 134.29\)
2013	108	\(\frac{108 \times 134.29}{100} = 145.03\)
2014	105	\(\frac{105 \times 145.03}{100} = 152.28\)
2015	111	\(\frac{111 \times 152.28}{100} = 169.03\)

In simple words: To change chain base index numbers to fixed base index numbers, you always start by taking the first chain base index as the first fixed base index. Then, for each next year, you multiply its chain index by the previous year's fixed index and divide by 100.

🎯 Exam Tip: Remember that the first fixed base index number is always the same as the first chain base index number when no specific base period is mentioned for the fixed base calculation. Subsequent calculations depend on the previous year's computed fixed base index.