GSEB Class 12 Statistics Solutions Chapter 1 Index Number

Answer The Following Questions By Selecting A Correct Option From The Given Options:

Question 1. Which method is useful to compare the long term variations in the values of the variable?
(a) Chain base method
(b) Laspeyre's method
(c) Fixed base method
(d) Paasche's method
Answer: (c) Fixed base method
In simple words: To see changes over a long period, the fixed base method is best because it always compares to the same starting point.
🎯 Exam Tip: Understanding when to use each index number method is key. For long-term comparisons, fixed base is generally preferred as it provides a consistent reference point.

 

Question 2. Which consumption is used in the calculation of Laspeyre's index number?
(a) Consumption of base year
(b) Consumption of current year
(c) Consumption of average year
(d) Consumption of any year
Answer: (a) Consumption of base year
In simple words: Laspeyre's index number uses the amount of goods consumed in the initial (base) year to calculate changes.
🎯 Exam Tip: Remember that Laspeyre's index uses base year quantities, while Paasche's uses current year quantities. This distinction is crucial for calculations.

 

Question 3. Which prices are considered in the construction of the cost of living index number ?
(a) Market price
(b) Wholesale price
(c) Average price
(d) Retail price
Answer: (d) Retail price
In simple words: When calculating how much it costs for people to live, the prices they pay in stores (retail prices) are used.
🎯 Exam Tip: The cost of living index directly impacts individuals, so retail prices (what consumers actually pay) are the most relevant.

 

Question 4. Which expenditure of items is assigned as weights in the method of family budget?
(a) Expenditure of selected year
(b) Average annual expenditure
(c) Expenditure of base year
(d) Expenditure of current year
Answer: (c) Expenditure of base year
In simple words: In the family budget method, the spending on items from the starting year is used to show their importance.
🎯 Exam Tip: Weights in the family budget method are usually derived from the base year's consumption patterns to reflect initial spending habits.

 

Question 5. Which average is considered as the best average in construction of the index number ?
(a) Harmonic mean
(b) Arithmetic mean
(c) Weighted mean
(d) Geometric mean
Answer: (d) Geometric mean
In simple words: The geometric mean is seen as the best type of average to use when building an index number.
🎯 Exam Tip: The geometric mean is often preferred for averaging ratios or rates of change, making it ideal for index number construction, particularly for Fisher's ideal index.

 

Question 6. Which index number gives idea of the standard of living of people ?
(a) Index number of industrial production
(c) Fisher's index number
(d) Cost of living index number
Answer: (d) Cost of living index number
In simple words: The cost of living index number helps us understand how the standard of living of people changes over time.
🎯 Exam Tip: The Cost of Living Index is a direct measure of how price changes affect the average household's purchasing power and well-being.

 

Question 7. The price of an item increased by 4.5 times in the current year as compared to the base year. What will be the price index number?
(a) 45
(b) 450
(c) 550
(d) 350
Answer: (c) 550
In simple words: If a price becomes 4.5 times bigger, it means it increased by 3.5 times its original value, so adding the base 100% gives 450%, plus the base price, so 100 (original) + 450 (increase) = 550.
🎯 Exam Tip: An increase "by x times" means the new value is (x+1) times the original, if starting from 1. If an item increased by 4.5 times, its new value is 1 + 4.5 = 5.5 times the base. So, the index is 5.5 * 100 = 550.

 

Question 8. If the purchasing power of money is 0.75 in the year 2016 with respect to the base year 2015, then what will be the price index number for the year 2016?
(a) 750
(b) 175
(c) 133.33
(d) 275
Answer: (c) 133.33
In simple words: If money buys less (0.75 of what it used to), prices must have gone up. The price index is found by dividing 1 by the purchasing power and multiplying by 100.
🎯 Exam Tip: Purchasing power of money and the price index number are inversely related. The formula is: Price Index = (1 / Purchasing Power) * 100.

 

Question 9. If the cost of living index number for the people of a class is 200 for the year 2016 with respect to the year 2015, which of the following statements is true ?
(a) There is an average 200 per cent rise in 'the current year prices of the items consumed by that class.
(b) There is an average 100 per cent decrease in the current year prices of the items consumed by that class.
(c) Purchasing power of money is. Rs. 0.5.
(d) The current year prices of the items consumed by that class are stable.
Answer: (c) Purchasing power of money is. Rs. 0.5.
In simple words: If the cost of living index is 200, it means prices have doubled. So, Rs. 1 can buy only half of what it used to, making the purchasing power 0.5.
🎯 Exam Tip: An index number of 200 implies a 100% increase from the base (100). Purchasing power is 100 / Index Number, so 100/200 = 0.5.

 

Question 10. If \( I_P = I_F \), which of the following statements is true ?
(a) \( I_P = 2I_L \)
(b) \( I_F = \frac{I_L}{2} \)
(C) \( I_F = I_P = I_L \)
(d) \( 4I_F = I_L \)
Answer: (C) \( I_F = I_P = I_L \)
In simple words: Fisher's index is the geometric mean of Laspeyre's and Paasche's. If Fisher's and Paasche's index numbers are the same, then all three (Laspeyre's, Paasche's, and Fisher's) must be equal.
🎯 Exam Tip: This question tests your understanding of the relationship between Laspeyre's (\(I_L\)), Paasche's (\(I_P\)), and Fisher's (\(I_F\)) index numbers, where \( I_F = \sqrt{I_L \times I_P} \). If \(I_P = I_F\), then \(I_F = \sqrt{I_L \times I_F}\), which means \(I_F^2 = I_L \times I_F\), simplifying to \(I_F = I_L\). Therefore, \(I_F = I_P = I_L\).

 

Question 11. If the average disposable income of families of a class is Rs. 20000 in the year 2013 and if the cost of living index number of that class for the year 2015 with the base year 2013 is 130, what should be the average disposable income of the families of this class in the year 2015?
(a) Rs. 26000
(b) 20130
(c) 20000
(d) Rs. 14000
Answer: (a) Rs. 26000
In simple words: If income was Rs. 20000 and the cost of living index went up by 30% (from 100 to 130), then the income also needs to go up by 30% to keep the same buying power. So, the new income should be Rs. 20000 plus 30% of Rs. 20000.
🎯 Exam Tip: To maintain the same standard of living, disposable income must increase proportionally to the cost of living index. The calculation is: (Base Year Income / Base Year Index) * Current Year Index.

 

Question 12. What weight is assigned as expenditure to the price relatives \( \frac{p_{1}}{p_{0}} \) of the items to obtain the formula for Paasche's index number ?
(a) \( p_0q_0 \)
(b) \( P_1q_1 \)
(c) \( P_0q_1 \)
(d) \( P_1q_0 \)
Answer: (c) \( P_0q_1 \)
In simple words: For Paasche's index, the importance (weight) of each item's price change is based on the quantity consumed in the current year, multiplied by its base year price.
🎯 Exam Tip: Remember the quantity used for weighting: Laspeyre uses base year quantities (\(q_0\)), while Paasche uses current year quantities (\(q_1\)).

 

Section B

Answer The Following Questions In One Sentence:

Question 1. What is a price relative ?
Answer: A price relative is the ratio of an item's current year price to its base year price.
In simple words: A price relative simply shows how much an item's price has changed from an old year to a new year.
🎯 Exam Tip: Price relatives are foundational for understanding price changes and constructing various index numbers. It's a simple ratio: current price divided by base price.

 

Question 2. Which method is more suitable to compare the changes in a variable at two different time periods?
Answer: The method of ratio is more suitable for comparing changes in a variable across two different time periods.
In simple words: Using a ratio is better to see how something changes between two different times.
🎯 Exam Tip: Ratio-based comparisons are fundamental in statistics for analyzing relative changes, often expressed as percentages in index numbers.

 

Question 3. If the quantity index number of an item for a certain year is 130, interpret it.
Answer: A quantity index number of 130 for an item in a specific year means that the quantity of that item has increased by 30% compared to the base year.
In simple words: An index of 130 means there is 30% more of that item now than before.
🎯 Exam Tip: An index number of 100 represents the base value. Any value above 100 indicates an increase, and any value below 100 indicates a decrease, with the difference from 100 being the percentage change.

 

Question 4. What is a base year ?
Answer: A base year is a fixed period used for comparison when evaluating the value of a variable in the current period.
In simple words: A base year is an old year chosen to compare prices or quantities to, helping us see changes.
🎯 Exam Tip: The base year serves as the reference point (index 100) for all comparisons in index number calculations. Its selection is critical.

 

Question 5. Write the formula for the conversion of chain base index number into fixed base index number.
Answer: The formula for converting a chain base index number into a fixed base index number is as follows:
\[ \text{Fixed base index number} = \frac{\text{Current year's chain base index number} \times \text{Previous year's fixed base index number}}{100} \]
In simple words: To change a chain base index to a fixed base index, you multiply the current year's chain index by the previous year's fixed index and then divide by 100.
🎯 Exam Tip: This formula helps link short-term (chain base) changes to a consistent long-term (fixed base) perspective. Make sure to apply the previous year's *fixed base* index in the numerator.

 

Question 6. Define the index number.
Answer: An index number measures the percentage change in the value of a variable associated with an item for the current period, compared to its value in a fixed base period.
In simple words: An index number tells us how much something (like a price) has changed in percentage from a past year to the current year.
🎯 Exam Tip: An index number quantifies relative changes, providing a standardized way to compare economic variables over time.

 

Question 7. Define the cost of living index number.
Answer: The cost of living index number shows the percentage of relative changes in the cost of living for a specific part of society in the current year, compared to a base year.
In simple words: The cost of living index number shows how much more or less money people need to buy the same things today compared to a past year.
🎯 Exam Tip: This index is vital for economic analysis and policy-making, as it reflects the impact of price changes on consumer welfare.

 

Question 8. What is weight?
Answer: Weight is a numerical value assigned to items, reflecting their importance or proportion in relation to others.
In simple words: A "weight" is a number that shows how important or big something is compared to other things.
🎯 Exam Tip: Weights are crucial in composite index numbers, as they ensure that items with greater economic significance contribute more to the overall index value.

 

Question 9. What is implicit weight?
Answer: Implicit weight is the weight determined by the number of different varieties of items chosen for an index number's construction.
In simple words: Implicit weight is when an item's importance is shown by how many different kinds of that item are included in a calculation.
🎯 Exam Tip: Implicit weights are not explicitly stated but are built into the selection process, often through the number of representative items chosen.

 

Question 10. Name the important basic tests of index numbers.
Answer: The important basic tests for index numbers are:

• Time reversal test, and
• Factor reversal test.

In simple words: The main tests for index numbers are the time reversal test and the factor reversal test, which check if the index works correctly forwards and backward, and when price and quantity are swapped.
🎯 Exam Tip: These tests are fundamental criteria for evaluating the consistency and reliability of an index number formula, with Fisher's index being notable for satisfying both.

 

Question 11. What is a chain base index number?
Answer: A chain base index number is calculated by comparing the current period with the immediately preceding period. Its formula is as follows:
\[ \text{Chain base index number} = \frac{\text{Value of the variable in the current period}}{\text{Value of the variable in the preceding period}} \times 100 \]
In simple words: A chain base index number compares how something changed this year compared to only the year right before it, not a fixed old year.
🎯 Exam Tip: Chain base index numbers are useful for capturing recent changes and allowing for the introduction of new items or the removal of old ones in the index basket.

 

Question 12. 'The price index number of oil is Rs. 500.' State whether this statement is true or false and if false, correct and rewrite it.
Answer: The statement 'The price index number of oil is Rs. 500.' is false.
Correct statement: 'The price index number of oil is 500'.
In simple words: An index number is a pure number, not a currency value, so it should not have a "Rs." symbol.
🎯 Exam Tip: Index numbers are dimensionless figures, representing relative change, not absolute values. Therefore, they should never be expressed with units like currency symbols.

 

Question 13. Which index number is used to find the rate of inflation ? Write the formula to find the rate of inflation.
Answer: The wholesale price index number is used to calculate the rate of inflation. Its formula is as follows:
\[ \text{Rate of inflation} = \frac{\left( \text{Wholesale price index number of current year} \right) - \left( \text{Wholesale price index number of preceding year} \right)}{\text{Wholesale price index number of preceding year }} \times 100 \]
In simple words: To find how fast prices are rising (inflation), we use the wholesale price index. We subtract last year's index from this year's, divide by last year's index, and multiply by 100.
🎯 Exam Tip: The Wholesale Price Index (WPI) is a primary indicator of inflation in many economies because it tracks prices at the producer or wholesale level before they reach consumers.

 

Question 14. Which index number is used in India to find the rate of dearness allowance ?
Answer: In India, the cost of living index number is used to determine the rate of dearness allowance.
In simple words: The cost of living index number helps decide how much extra money (dearness allowance) workers get in India to deal with rising prices.
🎯 Exam Tip: Dearness Allowance (DA) is a component of salary paid to government employees and pensioners to offset the impact of inflation, directly linked to the cost of living index.

 

Question 15. State the main difference between fixed base method and chain base method.
Answer: The main difference between the fixed base method and the chain base method is that in the fixed base method, the base year remains constant throughout the comparison period, whereas in the chain base method, the preceding year is used as the base year for any given time.
In simple words: The fixed base method always compares to the same starting year, but the chain base method always compares to the year right before it.
🎯 Exam Tip: Understanding this fundamental difference is crucial for choosing the appropriate index number method based on the analytical objective (long-term trend vs. short-term changes).

 

Question 16. In which method does the base year change each year ?
Answer: In the chain base method, the base year changes annually.
In simple words: The chain base method uses a different base year for each new calculation.
🎯 Exam Tip: The changing base year in the chain base method helps reflect more recent economic conditions, but it can make long-term comparisons more complex.

 

Question 17. Which is the appropriate average for the construction of index number?
Answer: The weighted average is the appropriate average for constructing index numbers.
In simple words: When making an index number, the weighted average is the best type of average to use.
🎯 Exam Tip: Weighted averages account for the varying importance of different items, leading to a more accurate and representative index number.

 

Question 18. How should be the base year in the calculation of index number ?
Answer: For calculating an index number, the base year should be a standard or normal year, free from natural disasters and unusual man-made events.
In simple words: The base year should be a normal year without big problems like floods or wars.
🎯 Exam Tip: Choosing a "normal" base year ensures that the index number reflects typical economic conditions and that subsequent comparisons are not distorted by unusual events.

 

Section C

Answer The Following Questions As Required:

Question 1. What is a base year ? Which points should be considered while choosing it?
Answer: A base year is a fixed period used to compare the value of a variable in the current period. The following points should be considered when selecting a base year:

• The base year should be standard or normal.
• It should be free from natural calamities like floods, droughts, earthquakes, and abnormal man-made events such as war, revolt, riots, strikes, agitations, political events, or economic disturbances.
• It should not be from a distant past.
• It should be chosen to accurately reflect the current situation.
• If a single period isn't normal, then an average of several past periods should be used.

In simple words: A base year is a reference year. It should be a normal year without big problems, not too long ago, and truly represent the usual situation. If no single year is normal, use an average of a few past years.
🎯 Exam Tip: The proper selection of a base year is fundamental for the reliability and validity of any index number series. A distorted base year can lead to misleading interpretations of economic trends.

 

Question 2. State the characteristics of index number.
Answer: The characteristics of an index number are as follows :

• An index number is a relative measure that shows percentage changes in a variable's quantity.
• It does not depend on specific units of measurement.
• It is a comparative tool.
• It compares situations at two different points in time using a ratio.
• It is an average measure, so it has all the properties of an average.
• It is a weighted average.

In simple words: An index number shows how much something changes in percentage. It doesn't use units, it compares things, it's a type of average, and it gives more importance to some items (weighted average).
🎯 Exam Tip: Knowing these characteristics helps in understanding why index numbers are widely used in economics and statistics for various types of analysis.

 

Question 3. What is quantity index number ?
Answer: For a group of 'n' related items, a quantity index number is found by using the average of the relative changes in the quantity of each item in that group. The formula for the quantity index number is:
\[ \text{Quantity index number} = \frac{\sum\left(\frac{q_{1i}}{q_{0i}}\right)}{n} \times 100 \]
Where, \( q_{1i} \) = Current year's quantity for the \( i^{th} \) item (i = 1, 2, 3, ............. n)
\( q_{0i} \) = Base year's quantity for the \( i^{th} \) item (i = 1, 2, 3, ............. n)
\( n \) = Number of items
In simple words: A quantity index number measures how the total amount of different goods produced or consumed changes over time, using an average of individual quantity changes.
🎯 Exam Tip: Quantity index numbers are used to track changes in production volumes or consumption levels, similar to how price index numbers track price changes. The formula averages the ratios of current quantities to base quantities.

 

Question 4. What is weight in the construction of index numbers ? State the types of weight.
Answer: In constructing index numbers, 'weight' refers to a number assigned to each item based on its relative importance, as not all items have the same significance. The items are weighted in two types:
1. Implicit weight
2. Explicit weight.
In simple words: In index numbers, a "weight" is a number given to each item to show how important it is. There are two kinds: implicit (hidden) and explicit (clear).
🎯 Exam Tip: Weights are crucial for an index to accurately reflect changes in the overall group. Misassigned weights can lead to a biased index.

 

Question 5. Why is Fisher's index number called an ideal number?
Answer: Fisher's index number is considered ideal for the following reasons:

• It accounts for both base year and current year prices and quantities in its construction.
• It satisfies two crucial fundamental tests: the time reversal test and the factor reversal test.
• The geometric mean, which is the best average for constructing index numbers, is used in its calculation.
• It balances the weaknesses of both Laspeyre's and Paasche's index numbers.

In simple words: Fisher's index is called "ideal" because it uses prices and quantities from both the old and new years, passes important math tests, uses the best type of average (geometric mean), and fixes problems found in other index numbers.
🎯 Exam Tip: Fisher's Ideal Index provides a robust measure by mitigating the upward bias of Laspeyre's and the downward bias of Paasche's, offering a more balanced estimate of price or quantity change.

 

Question 6. State the main difference between explicit weight and implicit weight.
Answer: The main differences between explicit weight and implicit weight are as follows:

Explicit weight	Implicit weight
1. Explicit weights are determined based on the importance of items. For example, if wheat is twice as important as rice in food items, wheat gets an explicit weight of two, and rice gets one.	1. Implicit weight is suggested by the selection of items. The weight is considered based on the varieties of items chosen. For example, if four varieties of wheat are selected in food items, wheat gets an implicit weight of four.
2. Explicit weight can be directly expressed in numbers.	2. Implicit weight cannot be expressed in numbers directly.
3. This is a direct way of assigning weight. Thus, it is consciously decided.	3. This is an indirect way of assigning weight. Therefore, there is no other method to determine it.

In simple words: Explicit weight is a clear number given to an item based on its importance, like saying wheat is twice as important as rice. Implicit weight is not a clear number but is shown by how many different types of an item are picked, like choosing four types of wheat.
🎯 Exam Tip: Differentiating between explicit and implicit weights is important for understanding how different index number construction methods account for item importance. Explicit weights offer greater transparency in the weighting scheme.

 

Question 7. The cost of living index number increased from 280 to 340 during a certain time period and the wage increased from Rs. 13500 to Rs. 14750. Find the real gain or loss of the worker.
Answer:
When the cost of living index number is 280, the wage is Rs. 13500.
To maintain the same living standard when the cost of living index number rises to 340, the required wage would be:
Required wage = \( \frac{13500 \times 340}{280} \)
Required wage = \( \frac{4590000}{280} \)
Required wage = Rs. 16392.86
However, the worker's actual wage increased to only Rs. 14750.
So, the loss for the worker = Rs. (16392.86 - 14750) = Rs. 1642.86
In simple words: The cost of living went up, so the worker needed more money to live the same way. The worker's pay also went up, but not enough to match the higher costs. This means the worker lost money in real terms.
🎯 Exam Tip: To calculate real gain or loss, always compare the actual wage increase with the wage increase *required* to offset the rise in the cost of living index. A worker experiences a real loss if their nominal wage increase is less than the percentage increase in the cost of living.

 

Question 8. The cost of living index numbers and average monthly wage from the year 2010 to 2013 are given as follows. Find the real wage for each year.

Year	2010	2011	2012	2013
Average monthly wage (Rs.)	35,000	40,000	42,000	50,000
Cost of living index number	120	150	130	160

Answer:

Year	Average monthly wage (Rs.)	Cost of living index number I	Real wage = \( \frac{\text{Average monthly wage}}{\text{Cost of living index number}} \times 100 \)
2010	35000	120	\( \frac{35000}{120} \times 100 = 29166.67 \)
2011	40000	150	\( \frac{40000}{150} \times 100 = 26666.67 \)
2012	42000	130	\( \frac{42000}{130} \times 100 = 32307.69 \)
2013	50000	160	\( \frac{50000}{160} \times 100 = 31250.00 \)

In simple words: Real wage shows how much you can actually buy with your money. To find it, divide your monthly wage by the cost of living index for that year and multiply by 100.
🎯 Exam Tip: Real wage calculations are essential for assessing changes in purchasing power over time. A higher nominal wage doesn't necessarily mean a higher real wage if the cost of living has increased more.

 

Question 9. Wholesale price index numbers for the year 2014 and 2015 are found to be 177.6 and 181.2 respectively. Find the rate of inflation using index numbers of both the years.
Answer:
Wholesale price index number for 2014 = 177.6
Wholesale price index number for 2015 = 181.2
Rate of inflation = \( \frac{\left( \text{Wholesale price index number for the year 2015} \right) - \left( \text{Wholesale price index number for the year 2014} \right)}{\text{Wholesale price index number for the year 2014}} \times 100 \)
Rate of inflation = \( \frac{181.2 - 177.6}{177.6} \times 100 \)
Rate of inflation = \( \frac{3.6}{177.6} \times 100 \)
Rate of inflation = \( 0.0203 \times 100 \)
Rate of inflation = 2.03%
In simple words: To find the inflation rate, we look at how much the wholesale price index increased from one year to the next, then express that increase as a percentage of the earlier year's index.
🎯 Exam Tip: The rate of inflation is calculated as the percentage change in the price index from one period to the next. It's a key economic indicator.

 

Question 10. The percentage increase in the price relatives of three items are 315, 328 and 390 respectively. If the importance of these items has ratio 5:7:8, find the general price index number.
Answer:

Item	Percentage increase in the price	Index number \( I = 100 + \text{increase in percentage} \)	W	IW
1	315	\( 100 + 315 = 415 \)	5	2075
2	328	\( 100 + 328 = 428 \)	7	2996
3	390	\( 100 + 390 = 490 \)	8	3920
Total	-	-	\( \Sigma W = 20 \)	\( \Sigma IW = 8991 \)

General price index number = \( \frac{\Sigma IW}{\Sigma W} \)
General price index number = \( \frac{8991}{20} \)
General price index number = 449.55
In simple words: We find the index for each item, multiply it by its importance (weight), add these up, and then divide by the total importance to get the overall index.
🎯 Exam Tip: This calculation uses a weighted average of price relatives. Ensure correct individual index number calculation (100 + percentage increase) and accurate multiplication with weights.

 

Question 11. If the average disposable income of family of a class is Rs. 25000 in the year 2014 and if the cost of living index number of that class for the year 2016 with the base year 2014 is 120, estimate the average disposable income of the family of that class in the year 2016.
Answer:
The average disposable income of a family in 2014 = Rs. 25000.
The cost of living index number for 2014 (base year) = 100.
The cost of living index number for 2016 = 120.
If the index number is 100 (2014), then the average disposable income is Rs. 25000.
Therefore, if the index number is 120 (2016), the average disposable income will be:
Average disposable income = \( \frac{25000 \times 120}{100} \)
Average disposable income = Rs. 30000
In simple words: If the cost of living goes up by 20% (from 100 to 120), then income also needs to go up by 20% to keep the same buying power. So, the income of Rs. 25000 becomes Rs. 30000.
🎯 Exam Tip: This question demonstrates how to adjust nominal income to maintain real purchasing power in the face of changing living costs. The direct proportionality between income and the cost of living index is crucial.

 

Question 12. The average monthly income of a worker was Rs. 16000 in the year 2015 and it increased to Rs. 20000 in the year 2016. Find the index number of income for the year 2016 in comparison to the year 2015.
Answer:
Average monthly income of a worker in 2015 = Rs. 16000
Average monthly income of a worker in 2016 = Rs. 20000
The index number of income for 2016 (with 2015 as base) = \( \frac{\text{Average monthly income in 2016}}{\text{Average monthly income in 2015}} \times 100 \)
\( = \frac{20000}{16000} \times 100 \)
\( = 1.25 \times 100 \)
\( = 125 \)
In simple words: To find the income index number, we divide the new income by the old income and multiply by 100.
🎯 Exam Tip: Income index numbers show the relative change in income over time. It's calculated similarly to a price index, but with income values instead of prices.

 

Question 13. If the production of an item increased by \( \frac{9}{5} \) times in the year 2016 as compared to the base year, find the index number of production for the year 2016.
Answer:
Let's assume the production in the base year = 1.
The production in 2016 increased by \( \frac{9}{5} \) times, meaning the increase is \( 1 \times \frac{9}{5} \).
So, the new production in 2016 = Base production + increase
\( = 1 + (1 \times \frac{9}{5}) = 1 + \frac{9}{5} = \frac{5}{5} + \frac{9}{5} = \frac{14}{5} \)
The index number of production for 2016 = \( \frac{\text{Production in 2016}}{\text{Production in base year}} \times 100 \)
\( = \frac{\frac{14}{5}}{1} \times 100 \)
\( = \frac{14}{5} \times 100 \)
\( = 2.8 \times 100 \)
\( = 280 \)
In simple words: If production grows by \( \frac{9}{5} \) of its original amount, the new production is \( 1 + \frac{9}{5} \) times the original. Multiply this by 100 to get the index number.
🎯 Exam Tip: "Increased by x times" means the new quantity is \( 1 + x \) times the base quantity. Be careful not to confuse "increased to x times" with "increased by x times."

 

Question 14. If \( I_L = 221.5 \) and \( I_F = 222 \), find \( I_P \).
Answer:
Given:
\( I_L = 221.5 \)
\( I_F = 222 \)
\( I_P = ? \)
The formula for Fisher's index number is:
\( I_F = \sqrt{I_L \times I_P} \)
Substitute the given values:
\( 222 = \sqrt{221.5 \times I_P} \)
Square both sides:
\( 222^2 = 221.5 \times I_P \)
\( 49284 = 221.5 \times I_P \)

\( \implies I_P = \frac{49284}{221.5} \)
\( \implies I_P = 222.5 \)
In simple words: Fisher's index is the square root of Laspeyre's index multiplied by Paasche's index. We can use this to find Paasche's index if the other two are known.
🎯 Exam Tip: Remember the relationship \( I_F = \sqrt{I_L \times I_P} \). To find an unknown index, square both sides to remove the square root and then solve algebraically.

 

Section D

Answer The Following Questions As Required:

Question 1. State the merits and limitations of fixed base method.
Answer: The merits and limitations of the fixed base method are as follows:
Merits:

• The base year remains constant throughout the entire comparison period.
• Because the base year is fixed, uniformity is maintained when comparing relative changes in variable values.
• This method is useful for comparing long-term changes in variable values.
• It is easy to understand and simple to compute.

Limitations:

• Over time, consumer tastes, habits, and fashion change, leading to changes in the items consumed. This method cannot remove outdated items or include new ones.
• It is not always possible to find a standard base year with normal conditions, making base year selection difficult.
• If the base year is not chosen correctly, the reliability of the index number decreases.
• It is not suitable for comparing short-term changes in the variable's value.
• It does not allow for necessary changes in determining item weights due to changes in item quality.
• If the base year is from a very distant past, comparing relative changes in the variable's value may not be appropriate.

In simple words: The fixed base method is good because it always compares to the same starting year, making it easy to see long-term changes. But it's bad because it can't add new items or remove old ones over time, and picking a perfect base year can be hard.
🎯 Exam Tip: When evaluating index number methods, consider the trade-offs between consistency (fixed base) and adaptability to changing consumption patterns (chain base). Fixed base is better for showing trends from a singular historical point.

Merits And Limitations Of Chain Base Method

Question 2. State the merits and limitations of chain base method.
Answer: The chain base method has several advantages and disadvantages, listed below:
Merits:

• In this method, the problem of choosing a base year does not come up.
• For this method, the year right before the current year is always used as the base year.
• Because the comparison is made with the immediate previous year, new items can be added based on consumer preferences, and old, unused items can be removed.
• Since the current period's value is compared with the recent past, this method is useful in economics, trade, and commerce.
• The index numbers for production and sales found using this method are mostly free from seasonal and repeating changes.
• Short-term changes in prices and production of goods can be compared easily using this index.

Limitations:

• This method is not very good for comparing things over a long time.
• If there is a mistake in calculating an index number, that error will affect all future index numbers.
• There is no consistent way to compare index numbers found with this method.
• If data for a particular year is missing, then the index number for the next year cannot be calculated.

In simple words: The chain base method is good for short-term comparisons because it always uses the previous year as a base, allowing for adjustments in items. However, any calculation error can spread, and it's not ideal for long-term trends or when data is missing.

🎯 Exam Tip: When discussing merits and limitations, provide clear, concise points for both aspects. For this topic, highlight how the changing base year impacts its applicability for short vs. long-term analysis and error propagation.

 

Question 3. Differentiate between fixed base and chain base methods.
Answer: The main differences between the fixed base method and the chain base method are:

Fixed base method	Chain base method
1. The base year is a normal year.	1. The year right before the current year is the base year.
2. The base year stays the same for all calculations.	2. The base year changes for each period.
3. Since the base year is fixed, comparisons of changes in variable quantities are consistent.	3. Since the base year changes, consistency in comparing changes in variable quantities is not maintained.
4. New items cannot be added, and old, unused items cannot be removed.	4. New items can be added, and old, unused items can be removed.
5. Choosing a base year is hard.	5. Choosing a base year is easy because it is automatic.
6. This method is useful for comparing long-term changes in a variable.	6. This method is useful for comparing short-term changes in a variable.
7. It is easy to understand and calculate.	7. If there is a mistake in one year's calculation, it affects all future years.

In simple words: The fixed base method uses one constant year for all comparisons, making it easy to understand long-term trends but difficult to update items. The chain base method uses the previous year as the base, which is flexible for including new items and better for short-term changes, but errors can accumulate over time.

🎯 Exam Tip: For differentiation questions, presenting information in a clear table format is highly effective. Focus on contrasting key characteristics like base year constancy, applicability for short/long term, and flexibility with item inclusion.

 

Question 4. Give the meaning of cost of living index number and state the points to be considered for its construction.
Answer: Meaning of cost of living index number: This number shows how much the cost of living for a specific group of people has changed in the current year compared to a past base year, expressed as a percentage.

The following points should be considered when building a cost of living index number:
1. Purpose of construction: First, clearly define why the index number is being created. Determine which group of people it is for. This helps decide which goods to include, how many, and in what quantities.
2. Family budget inquiry: To understand family spending, a random group of people from the target class is chosen, and their living expenses are studied.
3. Classification of commodities: Items found during the family budget study are grouped into five main categories:
(1) Food items,
(2) Clothing,
(3) House rent,
(4) Fuel and Light,
(5) Miscellaneous expenditure
4. Availability of prices of items: Prices of items consumed by families are collected from government-approved stores in their area. Sometimes, an average of retail prices from different stores over different times is used. Prices are collected using standard units (e.g., Rs. 10 per kg, not Rs. 1 per 100 gm), based on weekly, monthly, or yearly needs.
5. Selection of base year: The base year chosen should be a normal year. Price relatives for all selected items are then calculated based on this year:
\( \text{Price relative (I)} = \frac{\text{Retail price of the current year}}{\text{Retail price of the base year}} \times 100 \)
6. Selection of average: To get a single price relative from different item price relatives, a suitable average should be used. While the geometric mean is ideal for index numbers, it's hard to calculate. So, a weighted average is often used.
7. Selection of weighted method: The importance (weight) of each group and item is decided based on total spending. Cost of living index numbers are then made using two methods:
(1) Total expenditure method and
(2) Family budget method.
In simple words: A cost of living index number shows how much prices change for a specific group. To make it, you must first know its purpose, study family budgets, group items like food and housing, gather accurate retail prices, pick a normal base year, use a weighted average, and apply either the total expenditure or family budget method.

🎯 Exam Tip: When defining key terms, ensure clarity and simplicity. For construction points, a structured, numbered list with brief explanations for each step will help score well, covering all relevant statistical and practical considerations.

 

Question 5. State the uses of cost of living index number.
Answer: The cost of living index number is used for several important purposes:

• It gives a true picture of the economic state of different groups of people. This helps in deciding wages, dearness allowance, bonuses, and other payments for these groups.
• It guides the government on which items to control and which to keep free from control.
• It helps the government plan tax policies. If a tax is put on an item, this index number helps understand its possible impact on people's standard of living.
• It is used to measure changes in the purchasing power of money and to understand a person's real earning ability.
• It helps the government and other public organizations decide what facilities and benefits to offer different groups of people to improve their living conditions.

In simple words: The cost of living index number helps us understand how much prices affect people's money. It is used to set wages, allowances, and taxes, and helps the government make decisions about controlling goods and offering benefits to improve people's lives.

🎯 Exam Tip: List the uses clearly and concisely. Focus on the practical applications of the index number, such as its role in economic policy, wage adjustments, and understanding purchasing power.

 

Question 6. State the limitations of cost of living index number.
Answer: The cost of living index number has certain limitations:

• A single cost of living index number cannot be made for all groups in society. Different groups have different spending patterns.
• It shows average percentage changes in the cost of living for a group, but it cannot analyze changes for an individual person.
• Different cost of living index numbers must be created for different types of people in various regions.
• An index number made for one group of people cannot be used for the same group in another region.
• The spending of a group depends on family size, lifestyle, hobbies, and habits. So, an average family cannot be seen as a standard family.
• The calculation assumes that "base year expenditure stays the same," which is often not true. Over time, people's choices, hobbies, and habits change. Therefore, regular family budget surveys are needed to make necessary adjustments.
• People in the same group may consume different amounts of items. This makes it difficult to assign accurate weights to different items.

In simple words: A single cost of living index cannot fit everyone because people, families, and regions have different spending habits. It only shows average changes, not individual ones. Also, it wrongly assumes that spending habits from the base year stay the same over time, even though people's tastes change. This makes it hard to assign correct importance to items.

🎯 Exam Tip: Focus on aspects like variability among different groups, individual vs. aggregate data, and the static assumption of consumption patterns over time when listing limitations.

 

Question 7. The prices of three items among five fuel items increased by 50%, 90% and 110% in the year 2015 as compared to the base year 2014. The prices of other two items decreased by 5% and 2% respectively. If the ratio of importance of these five items is 5: 4:3 : 2 : 1, find the index number of fuel prices for the year 2015.
Answer:

Item	Percentage increase (+) / Decrease (-)	Index number \( \text{I} = \begin{cases} 100 + \text{Increase} \\ 100 - \text{Decrease} \end{cases} \)	Weight W	IW
A	+ 50	\( 100 + 50 = 150 \)	5	750
B	+ 90	\( 100 + 90 = 190 \)	4	760
C	+ 110	\( 100 + 110 = 210 \)	3	630
D	- 05	\( 100 - 05 = 95 \)	2	190
E	- 02	\( 100 - 02 = 98 \)	1	98
Total	-	-	\( \Sigma \text{W} = 15 \)	\( \Sigma \text{IW} = 2428 \)

Index number of fuel prices \( = \frac{\Sigma \text{IW}}{\Sigma \text{W}} \)
\( \implies = \frac{2428}{15} \)
\( \implies = 161.87 \)
In simple words: We calculated the new prices for five fuel items by adding or subtracting their percentage changes from 100. Then, we multiplied these new price index values by their importance weights. Finally, we summed up the weighted indices and divided by the sum of weights to get an overall fuel price index of 161.87.

🎯 Exam Tip: For problems involving percentage changes and weights, always calculate the individual index numbers first (100 + increase / 100 - decrease). Organize your calculations in a table, clearly showing each step to avoid errors and ensure full marks.

 

Question 8. Find the fixed base index numbers from the following data about average annual income of workers in a company from the year 2008 to the year 2014: (Take base year as 2008.)

Year	2008	2009	2010	2011	2012	2013	2014
Average annual income (Rs.)	36,000	40,000	48,000	52,000	60,000	80,000	95,000

Answer: Here, the base year is 2008, so the base year income (p₀) is Rs. 36,000.

Year	Average annual income (Rs. 10,000) (p1)	Index Number \( = \frac{\text{Current year's price}}{\text{Base year's price}} \times 100 \)
2008	36	\( = \frac{36}{36} \times 100 = 100 \)
2009	40	\( = \frac{40}{36} \times 100 = 111.11 \)
2010	48	\( = \frac{48}{36} \times 100 = 133.33 \)
2011	52	\( = \frac{52}{36} \times 100 = 144.44 \)
2012	60	\( = \frac{60}{36} \times 100 = 166.67 \)
2013	80	\( = \frac{80}{36} \times 100 = 222.22 \)
2014	95	\( = \frac{95}{36} \times 100 = 263.89 \)

In simple words: To find the fixed base index numbers, we compared each year's average income to the income of the base year (2008) and multiplied by 100. This shows how much the income has changed relative to 2008.

🎯 Exam Tip: Remember that for fixed base index numbers, the denominator (base year value) remains constant for all calculations. Clearly state the base year and its value at the beginning of your answer.

 

Question 9. The average closing prices of shares of a certain company in different January 2014 are as follows. Find the chain base index numbers.

Month	January '14	February '14	March '14	April '14	May '14	June '14
Fixed base index number	100	104	105	108	109	127

Answer:

Month	Fixed base index number	Chain base index number \( = \frac{\text{Fixed base index number of current year}}{\text{Fixed base index number of preceding year}} \times 100 \)
January '14	100	\( = 100 \)
February '14	104	\( = \frac{104}{100} \times 100 = 104 \)
March '14	105	\( = \frac{105}{104} \times 100 \approx 100.96 \)
April '14	108	\( = \frac{108}{105} \times 100 \approx 102.86 \)
May '14	109	\( = \frac{109}{108} \times 100 \approx 100.93 \)
June '14	127	\( = \frac{127}{109} \times 100 \approx 116.51 \)

In simple words: To find the chain base index numbers from fixed base index numbers, we took the current month's fixed base index and divided it by the previous month's fixed base index, then multiplied by 100. For the first month, it's 100.

🎯 Exam Tip: When converting fixed base to chain base, the formula uses the fixed base index of the current period divided by the fixed base index of the *preceding* period, multiplied by 100. The first period's chain base index is always 100.

 

Question 10. Find the fixed base index numbers from the chain base index numbers given below:

Year	2011	2012	2013	2014
Index Number	120	90	140	125

Answer:

Year	Chain base index number	Fixed base index number \( = \frac{\text{Current year's chain base index number} \times \text{Preceding year's fixed base index number}}{100} \)
2011	120	\( = 120 \)
2012	90	\( = \frac{90 \times 120}{100} = 108 \)
2013	140	\( = \frac{140 \times 108}{100} = 151.20 \)
2014	125	\( = \frac{125 \times 151.20}{100} = 189 \)

In simple words: To convert chain base index numbers to fixed base, we start with the first year's chain base index as its fixed base index. For each next year, we multiply its chain base index by the previous year's fixed base index and divide by 100.

🎯 Exam Tip: The conversion from chain base to fixed base involves multiplying the current year's chain base index by the *previous* year's fixed base index and dividing by 100. The first year's fixed base index is simply its chain base index.

 

Question 11. Find the chain base index numbers from the following data regarding the price of an item:

Year	2009	2010	2011	2012	2013	2014
Price (Rs.)	40	45	48	55	60	70

Answer:

Year	Price (Rs.)	Chain base index number \( = \frac{\text{Current year's price}}{\text{Preceding year's price}} \times 100 \)
2009	40	\( = 100 \)
2010	45	\( = \frac{45}{40} \times 100 = 112.50 \)
2011	48	\( = \frac{48}{45} \times 100 \approx 106.67 \)
2012	55	\( = \frac{55}{48} \times 100 \approx 114.58 \)
2013	60	\( = \frac{60}{55} \times 100 \approx 109.09 \)
2014	70	\( = \frac{70}{60} \times 100 \approx 116.67 \)

In simple words: To find the chain base index for each year, we compare its price to the price of the year just before it. We divide the current year's price by the previous year's price and multiply by 100. For the first year, the index is 100.

🎯 Exam Tip: For chain base index numbers, the "base" is always the immediate preceding period. Ensure your calculations correctly use the previous year's price in the denominator for each step, and remember the first year's index is 100 by definition.

 

Question 12. Find the cost of living index number from the given information for the month of April, 2015 regarding group index numbers and weights of items of living of industrial workers.

Group	A	B	C	D	E	F
Index Number	247	167	259	196	212	253
Weight	44	20	16	6	10	4

Answer:

Group	Index number I	Weight W	IW
A	247	44	10868
B	167	20	3340
C	259	16	4144
D	196	6	1176
E	212	10	2120
F	253	4	1012
Total	-	\( \Sigma \text{W} = 100 \)	\( \Sigma \text{IW} = 22660 \)

Cost of living index number \( = \frac{\Sigma \text{IW}}{\Sigma \text{W}} \)
\( \implies = \frac{22660}{100} \)
\( \implies = 226.6 \)
In simple words: To find the cost of living index number, we multiplied each group's index number by its weight, then added all these products together. Finally, we divided this sum by the total sum of all weights to get the final index number.

🎯 Exam Tip: The cost of living index number is a weighted average of group index numbers. Always use the formula \( \frac{\Sigma \text{IW}}{\Sigma \text{W}} \) and ensure correct multiplication before summation.

 

Question 13. If \( \Sigma p_1 q_0 : \Sigma p_0 q_0 = 5:3 \) and \( \Sigma p_1 q_1 : \Sigma p_0 q_1 = 3:2 \), compute the Laspeyre's, Paasche's and Fisher's index numbers.
Answer: Given:
\( \frac{\Sigma p_1 q_0}{\Sigma p_0 q_0} = \frac{5}{3} \)
\( \frac{\Sigma p_1 q_1}{\Sigma p_0 q_1} = \frac{3}{2} \)

Now,
Laspeyre's index number:
\( I_L = \frac{\Sigma p_1 q_0}{\Sigma p_0 q_0} \times 100 \)
\( \implies I_L = \frac{5}{3} \times 100 \)
\( \implies I_L = 166.67 \)

Paasche's index number:
\( I_P = \frac{\Sigma p_1 q_1}{\Sigma p_0 q_1} \times 100 \)
\( \implies I_P = \frac{3}{2} \times 100 \)
\( \implies I_P = 150 \)

Fisher's index number:
\( I_F = \sqrt{I_L \times I_P} \)
\( \implies I_F = \sqrt{166.67 \times 150} \)
\( \implies I_F = \sqrt{25000.5} \)
\( \implies I_F \approx 158.12 \)
In simple words: We used the given ratios to directly calculate Laspeyre's and Paasche's index numbers by multiplying them by 100. Then, for Fisher's index, we found the square root of the product of Laspeyre's and Paasche's index numbers.

🎯 Exam Tip: Know the formulas for Laspeyre's, Paasche's, and Fisher's index numbers by heart. For Fisher's, ensure you calculate the square root of the product of L and P, not their sum.

 

Question 14. If the ratio of Laspeyre's and Paasche's index number is 4:5 and Fisher's index number is 150, find Paasche's index number.
Answer: Here, given:
\( I_L : I_P = 4 : 5 \)
\( I_F = 150 \)
We need to find \( I_P \).

From the ratio, we can write:
\( \frac{I_L}{I_P} = \frac{4}{5} \)
\( \implies I_L = \frac{4}{5} I_P \)

Now, using Fisher's index formula:
\( I_F = \sqrt{I_L \times I_P} \)
Substitute the value of \( I_F \) and \( I_L \):
\( 150 = \sqrt{\frac{4}{5} I_P \times I_P} \)
\( 150 = \sqrt{\frac{4}{5} I_P^2} \)
Square both sides:
\( 150^2 = \frac{4}{5} I_P^2 \)
\( 22500 = \frac{4}{5} I_P^2 \)
\( I_P^2 = \frac{22500 \times 5}{4} \)
\( I_P^2 = \frac{112500}{4} \)
\( I_P^2 = 28125 \)
\( I_P = \sqrt{28125} \)
\( I_P \approx 167.71 \)
In simple words: We used the given ratio of Laspeyre's to Paasche's index to write Laspeyre's in terms of Paasche's. Then, using Fisher's index formula, we substituted this relationship and the given Fisher's index value. By squaring both sides and solving the equation, we found the value of Paasche's index number.

🎯 Exam Tip: When given a ratio of index numbers (like \( I_L : I_P \)) and Fisher's index, express one index in terms of the other. Substitute this into Fisher's formula and solve for the unknown. Always double-check your algebraic steps, especially squaring and square roots.

 

Section E

Solve The Following:

 

Question 1. Find the general index number using the following data about prices of different items in the year 2012 by taking the base year 2010.

Item	Unit	Price of year 2010 (Rs.)	Price of year 2012 (Rs.)
A	Quintal	110	120
B	Kilogram	50	70
C	Dozen	40	60
D	Meter	80	90
E	Litre	20	20

Answer:

Item	Unit	Price of year 2010 (Rs.) p0	Price of year 2012 (Rs.) p1	Price relative \( \frac{p_1}{p_0} \)
A	Quintal	110	120	\( \frac{120}{110} \approx 1.0909 \)
B	Kilogram	50	70	\( \frac{70}{50} = 1.4 \)
C	Dozen	40	60	\( \frac{60}{40} = 1.5 \)
D	Meter	80	90	\( \frac{90}{80} = 1.125 \)
E	Litre	20	20	\( \frac{20}{20} = 1 \)
Total	n = 5	-	-	\( \Sigma \frac{p_1}{p_0} = 6.1159 \)

General Index number \( = \frac{\Sigma \left(\frac{p_1}{p_0}\right)}{n} \times 100 \)
\( \implies = \frac{6.1159}{5} \times 100 \)
\( \implies = 1.22318 \times 100 \)
\( \implies = 122.32 \)
In simple words: To find the general index number, we first calculated the price relative for each item by dividing its 2012 price by its 2010 price. We then summed these price relatives, divided by the number of items (5), and multiplied by 100 to get the overall index.

🎯 Exam Tip: When calculating a simple aggregate price index, remember to find the price relative for each item, sum them up, divide by the count of items, and then multiply by 100. Ensure currency symbols are normalized to "Rs.". Always clearly label your columns in tables.

 

Question 2. Compute the index number for the year 2015 with the base year 2010 by the method of total expenditure using the following data:

Item	Price of year 2010 (Rs.)	Price of year 2015 (Rs.)	Quantity in year 2010
A	10	14	8
B	30	42	4
C	40	80	4
D	20	26	16

Answer: Here, the base year is 2010 and the current year is 2015.
Therefore, p₀ = Price in 2010; q₀ = Quantity in 2010 and p₁ = Price in 2015.
The table for calculation is prepared as follows:

Item	Price of year 2010 (Rs.) p0	Price of year 2015 (Rs.) p1	Quantity in the year 2010 q0	p1q0	p0q0
A	10	14	8	112	80
B	30	42	4	168	120
C	40	80	4	320	160
D	20	26	16	416	320
Total	-	-	-	\( \Sigma p_1 q_0 = 1016 \)	\( \Sigma p_0 q_0 = 680 \)

Index number by total expenditure method \( = \frac{\Sigma p_1 q_0}{\Sigma p_0 q_0} \times 100 \)
\( \implies = \frac{1016}{680} \times 100 \)
\( \implies = 1.4941 \times 100 \)
\( \implies = 149.41 \)
In simple words: Using the total expenditure method, we calculated the total cost of base-year quantities at both current (2015) and base-year (2010) prices. We then divided the current year's total cost by the base year's total cost and multiplied by 100 to find the index number.

🎯 Exam Tip: For the total expenditure method, always use the *base year quantities* (q₀) with both current (p₁) and base year (p₀) prices. Ensure clear labeling of columns \( p_0, p_1, q_0, p_1 q_0, p_0 q_0 \) in your table.

 

Question 3. Compute the index number by the method of total expenditure for the year 2014 by taking the base year 2013 using the following data:

Item	Consumption (Quantity)	Year 2013 Price (Rs.)	Year 2014 Price (Rs.)
Wheat	15 kg	20	24
Rice	10 kg	40	45
Bajri	5 kg	16	20
Tuver Dal	3 kg	80	90

Answer: Here, the base year is 2013 and the current year is 2014.
Therefore, p₀ = Price in 2013; p₁ = Price in 2014 and q₁ = Quantity in 2014 (since consumption is given for 2014, and the question uses "quantity consumed" in the original image implying q₁).
The table for calculation is prepared as follows:

Item	q1	p1	p0	p1q1	p0q1
Wheat	15	24	20	360	300
Rice	10	45	40	450	400
Bajri	5	20	16	100	80
Tuver Dal	3	90	80	270	240
Total	-	-	-	\( \Sigma p_1 q_1 = 1180 \)	\( \Sigma p_0 q_1 = 1020 \)

Index number by total expenditure method \( = \frac{\Sigma p_1 q_1}{\Sigma p_0 q_1} \times 100 \)
\( \implies = \frac{1180}{1020} \times 100 \)
\( \implies = 1.15686 \times 100 \)
\( \implies = 115.69 \)
In simple words: We calculated the total cost of quantities consumed in 2014 using both 2014 prices and 2013 prices. Then, we divided the total cost at 2014 prices by the total cost at 2013 prices and multiplied by 100 to get the index number.

🎯 Exam Tip: When given "consumption (quantity)" for the current year, use it as q₁. The total expenditure method formula is \( \frac{\Sigma p_1 q_1}{\Sigma p_0 q_1} \times 100 \) for Paasche's method where \( q_1 \) is the current year quantity.

 

Question 4. Use the following information to find (i) fixed base index numbers with the year 2008 as the base year and (ii) the index numbers by taking the average price of the years 2008 and 2009 as the base year price:

Year	2008	2009	2010	2011	2012	2013	2014
Price (Rs.)	32	38	40	42	45	60	65

Answer:
(i) Base year = 2008
Base year price (p₀) = Rs. 32

Year	Price (Rs.)	Fixed base index number \( = \frac{\text{Current year's price}}{\text{Base year's price}} \times 100 \)
2008	32	\( = 100 \)
2009	38	\( = \frac{38}{32} \times 100 = 118.75 \)
2010	40	\( = \frac{40}{32} \times 100 = 125 \)
2011	42	\( = \frac{42}{32} \times 100 = 131.25 \)
2012	45	\( = \frac{45}{32} \times 100 = 140.63 \)
2013	60	\( = \frac{60}{32} \times 100 = 187.5 \)
2014	65	\( = \frac{65}{32} \times 100 = 203.13 \)

(ii) Base year = Average of prices of the year 2008 and 2009:
Average price of 2008 and 2009 (p₀) \( = \frac{\text{Price of 2008} + \text{Price of 2009}}{2} \)
\( \implies = \frac{32 + 38}{2} \)
\( \implies = \frac{70}{2} = 35 \)

Year	Price (Rs.) p1	Index number by average price \( = \frac{\text{Current year's price}}{\text{Average price}} \times 100 \)
2008	32	\( = \frac{32}{35} \times 100 \approx 91.43 \)
2009	38	\( = \frac{38}{35} \times 100 \approx 108.57 \)
2010	40	\( = \frac{40}{35} \times 100 \approx 114.29 \)
2011	42	\( = \frac{42}{35} \times 100 = 120 \)
2012	45	\( = \frac{45}{35} \times 100 \approx 128.57 \)
2013	60	\( = \frac{60}{35} \times 100 \approx 171.43 \)
2014	65	\( = \frac{65}{35} \times 100 \approx 185.71 \)

In simple words: First, for a fixed base, we compared each year's price to the 2008 price. Second, we found the average price of 2008 and 2009 and used that as the new base price. Then, we compared each year's price to this average base price to get new index numbers.

🎯 Exam Tip: When asked for multiple base year scenarios (e.g., a single year vs. an average of years), clearly separate your calculations for each part. Ensure correct identification of the base price (denominator) for each index number calculation.

 

Question 5. The index numbers of different groups of industrial output of a city and the weights of these groups are given below. Find the index number of the industrial production.

Group	Index Number	Weight
Iron	390.2	30
Textile	247.6	31
Chemical	510.2	18
Engineering goods	403.3	17
Cement	624.4	4

Answer:

Group	Index number I	Weight W	IW
Iron	390.2	30	11706.0
Textile	247.6	31	7675.6
Chemical	510.2	18	9183.6
Engineering goods	403.3	17	6856.1
Cement	624.4	4	2497.6
Total	-	\( \Sigma \text{W} = 100 \)	\( \Sigma \text{IW} = 37918.9 \)

Index number of industrial production \( I = \frac{\Sigma \text{IW}}{\Sigma \text{W}} \)
\( \implies = \frac{37918.9}{100} \)
\( \implies = 379.19 \)
In simple words: To find the industrial production index, we multiplied each group's index number by its weight. Then, we summed up these weighted values and divided by the total sum of all weights to get the final index number.

🎯 Exam Tip: This is a weighted aggregate index calculation. Ensure you correctly multiply each group's index by its corresponding weight and sum the results before dividing by the total weight. Accuracy in arithmetic is crucial here.

 

Question 6. The price of wheat increased by 70 % and price of rice increased by 40 % in the year 2015 with respect to the year 2010. The price of bajri decreased by 25 %. The price of oil increased by 40 % and the price of ghee decreased by 5 %. If the importance of oil is three times and of rice is double that of ghee and the importance of each of wheat and bajri is double that of rice, find price index number of the group of these five items and interpret it.
Answer:
First, let's determine the weights based on importance:
Suppose the importance of ghee is 1.
Importance of oil is \( 3 \times 1 = 3 \).
Importance of rice is double that of ghee, so \( 2 \times 1 = 2 \).
Importance of wheat and bajri each is double that of rice, so \( 2 \times 2 = 4 \) for wheat and \( 2 \times 2 = 4 \) for bajri.

The index number for an item is \( 100 + \% \text{ increase} \) or \( 100 - \% \text{ decrease} \).

Item	Percentage increase (+) / decrease (-)	Index number \( \text{I} = \begin{cases} 100 + \text{increase} \\ 100 - \text{decrease} \end{cases} \)	Weight W	IW
Wheat	+ 70	\( 100 + 70 = 170 \)	4	680
Rice	+ 40	\( 100 + 40 = 140 \)	2	280
Bajri	- 25	\( 100 - 25 = 75 \)	4	300
Oil	+ 40	\( 100 + 40 = 140 \)	3	420
Ghee	- 5	\( 100 - 5 = 95 \)	1	95
Total	-	-	\( \Sigma \text{W} = 14 \)	\( \Sigma \text{IW} = 1775 \)

General index number of price \( I = \frac{\Sigma \text{IW}}{\Sigma \text{W}} \)
\( \implies = \frac{1775}{14} \)
\( \implies = 126.79 \)
Hence, the price index for the group of five items is 126.79.
Interpretation: There is a \( (126.79 - 100 =) 26.79\% \) increase in the prices of these five items for the group.
In simple words: We first set the importance (weight) for each item based on the given ratios. Then, we calculated an index number for each item based on its price change. We multiplied each item's index by its weight, added these up, and divided by the total weight to get the overall price index. The result means prices increased by 26.79%.

🎯 Exam Tip: Start by correctly assigning weights based on the given relative importance. Then, calculate each item's individual index (100 + % change or 100 - % change). Finally, compute the weighted average and clearly interpret the overall percentage change relative to the base (100).

 

Question 7. Calculate the real wages of a worker class from the following data about their monthly wages. Find the purchasing power of money in the year 2015 considering the year 2008 as the base year.
Answer:

Year	Average monthly wage (Rs.)	Cost of living index number	Real wage \( = \frac{\text{Average monthly wage}}{\text{Cost of living index number}} \times 100 \)
2010	15000	120	\( \frac{15000}{120} \times 100 = 12500 \)
2011	18000	180	\( \frac{18000}{180} \times 100 = 10000 \)
2012	19000	205	\( \frac{19000}{205} \times 100 = 9268.29 \)
2013	20000	220	\( \frac{20000}{220} \times 100 = 9090.91 \)
2014	22000	235	\( \frac{22000}{235} \times 100 = 9361.70 \)
2015	25000	260	\( \frac{25000}{260} \times 100 = 9615.38 \)

Purchasing power of money:
Base year 2008, cost of living index number for the year 2015 is 260
Therefore, the purchasing power of money in the year 2015
\( = \frac{1}{\text{Cost of living index number}} \times 100 \)
\( = \frac{1}{260} \times 100 \)
\( = \frac{100}{260} = 0.38 \)
In simple words: This solution calculates how much a worker's money is worth each year by dividing their monthly wage by the cost of living index. It also shows that the purchasing power of money in 2015 was 0.38, meaning money could buy less compared to the base year.

🎯 Exam Tip: Remember to clearly show the formula used for real wage and purchasing power of money, and ensure accurate calculations for each year. Pay attention to the base year specified for the index number.

Section F

Question 1. Find the Laspeyre's and Paasche's index numbers 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.
Answer:
Here, the price of items and their total expenditure are given. We find the consumption (quantity) of items using the formula:
Consumption \( = \frac{\text{Total expenditure}}{\text{Price per unit}} \)
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 below shows the calculations:

Item	Year 2014			Year 2015			\( p_1q_0 \)	\( p_0q_0 \)	\( p_1q_1 \)	\( p_0q_1 \)
	\( p_0 \)	Total expenditure	\( q_0 = \frac{\text{Total expenditure}}{p_0} \)	\( p_1 \)	Total expenditure	\( q_1 = \frac{\text{Total expenditure}}{p_1} \)
Wheat	16	224	\( \frac{224}{16} = 14 \)	18	270	\( \frac{270}{18} = 15 \)	\( 18 \times 14 = 252 \)	\( 16 \times 14 = 224 \)	\( 18 \times 15 = 270 \)	\( 16 \times 15 = 240 \)
Rice	35	140	\( \frac{140}{35} = 4 \)	40	200	\( \frac{200}{40} = 5 \)	\( 40 \times 4 = 160 \)	\( 35 \times 4 = 140 \)	\( 40 \times 5 = 200 \)	\( 35 \times 5 = 175 \)
Tuver Dal	100	200	\( \frac{200}{100} = 2 \)	120	360	\( \frac{360}{120} = 3 \)	\( 120 \times 2 = 240 \)	\( 100 \times 2 = 200 \)	\( 120 \times 3 = 360 \)	\( 100 \times 3 = 300 \)
Oil	108	432	\( \frac{432}{108} = 4 \)	120	600	\( \frac{600}{120} = 5 \)	\( 120 \times 4 = 480 \)	\( 108 \times 4 = 432 \)	\( 120 \times 5 = 600 \)	\( 108 \times 5 = 540 \)
Total	-	-	-	-	-	-	\( \Sigma p_1q_0 = 1132 \)	\( \Sigma p_0q_0 = 996 \)	\( \Sigma p_1q_1 = 1430 \)	\( \Sigma p_0q_1 = 1255 \)

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:
\( I_F = \sqrt{I_L \times I_P} \)
\( = \sqrt{113.65 \times 113.94} \)
\( = \sqrt{12949.281} \)
\( = 113.79 \)
In simple words: We calculated three different index numbers (Laspeyre's, Paasche's, and Fisher's) to see how prices changed from 2014 to 2015 for various items. Laspeyre's shows a 13.65% increase, Paasche's shows a 13.94% increase, and Fisher's, which is considered ideal, shows an overall increase of 13.79%. This means prices generally went up by about 13.79% in 2015 compared to 2014.

🎯 Exam Tip: When calculating Laspeyre's, Paasche's, and Fisher's index numbers, ensure you correctly identify the base year and current year quantities and prices. Remember that Laspeyre's uses base year quantities, Paasche's uses current year quantities, and Fisher's is the geometric mean of the two.

 

Question 2. The quantity consumed and total expenditure of four different items are as given below. Find Paasche's and Fisher's index numbers for the year 2015 with respect to the year 2013.
Answer:
Here, the quantity (consumption) of items and their total expenditure are given. We find 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, \( q_0 \) = Quantity in 2013 and \( q_1 \) = Quantity in 2015.
The table below shows the calculations:

Item	Year 2013			Year 2015			\( p_1q_0 \)	\( p_0q_0 \)	\( p_1q_1 \)	\( p_0q_1 \)
	Total expenditure	\( q_0 \)	\( p_0 = \frac{\text{Total expenditure}}{q_0} \)	Total expenditure	\( q_1 \)	\( p_1 = \frac{\text{Total expenditure}}{q_1} \)
A	360	60 kg	\( \frac{360}{60} = 6 \)	375	25 kg	\( \frac{375}{25} = 15 \)	\( 15 \times 60 = 900 \)	\( 6 \times 60 = 360 \)	\( 15 \times 25 = 375 \)	\( 6 \times 25 = 150 \)
B	160	10 litre	\( \frac{160}{10} = 16 \)	416	20 litre	\( \frac{416}{20} = 20.8 \)	\( 20.8 \times 10 = 208 \)	\( 16 \times 10 = 160 \)	\( 20.8 \times 20 = 416 \)	\( 16 \times 20 = 320 \)
C	480	15 kg	\( \frac{480}{15} = 32 \)	613.2	6 kg	\( \frac{613.2}{6} = 102.2 \)	\( 102.2 \times 15 = 1533 \)	\( 32 \times 15 = 480 \)	\( 102.2 \times 6 = 613.2 \)	\( 32 \times 6 = 192 \)
D	336	3 kg	\( \frac{336}{3} = 112 \)	400	2.5 kg	\( \frac{400}{2.5} = 160 \)	\( 160 \times 3 = 480 \)	\( 112 \times 3 = 336 \)	\( 160 \times 2.5 = 400 \)	\( 112 \times 2.5 = 280 \)
Total	-	-	-	-	-	-	\( \Sigma p_1q_0 = 3121 \)	\( \Sigma p_0q_0 = 1336 \)	\( \Sigma p_1q_1 = 1804.2 \)	\( \Sigma p_0q_1 = 942 \)

Paasche's index number:
\( I_P = \frac{\Sigma p_1 q_1}{\Sigma p_0 q_1} \times 100 \)
\( = \frac{1804.2}{942} \times 100 \)
\( = 191.53 \)

Fisher's index number:
\( I_F = \sqrt{\frac{\Sigma p_1 q_0}{\Sigma p_0 q_0} \times \frac{\Sigma p_1 q_1}{\Sigma p_0 q_1}} \times 100 \)
\( = \sqrt{\frac{3121}{1336} \times \frac{1804.2}{942}} \times 100 \)
\( = \sqrt{2.336 \times 1.915} \times 100 \)
\( = \sqrt{4.4742} \times 100 \)
\( = 2.1152 \times 100 \)
\( = 211.52 \)
In simple words: This problem asks us to find the price changes from 2013 to 2015 for different items using Paasche's and Fisher's index numbers. We first figure out the price per unit for each item in both years. Paasche's index shows a 91.53% price increase, and Fisher's, which is a blend of different measures, shows a 111.52% price increase.

🎯 Exam Tip: When given total expenditure and quantity, always calculate the price per unit first. Ensure clear separation of base year and current year data. Double-check all multiplications and divisions in the summations for accuracy.

 

Question 3. Compute the Fisher's index number from the data given below about six different items:
Answer:
Here, the base year is 2013 and the current year is 2015.
So, \( p_0 \) = 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 items A, B, C and F are not equal. We need to make them equal before calculation.

Explanation of unit changes:

Item	Change	Year 2013	Year 2015
A	Price per kg	\( \frac{600}{20} = 30 \) Rs.	\( \frac{880}{20} = 44 \) Rs.
B	Price per kg	\( \frac{1600}{100} = 16 \) Rs.	\( \frac{2400}{100} = 24 \) Rs.
C	Quantity in kg	\( \frac{1200}{1000} = 1.2 \) kg	\( \frac{2000}{1000} = 2 \) kg
F	Price per piece	\( \frac{30}{12} = 2.5 \) Rs.	\( \frac{36}{12} = 3 \) Rs.

The table for calculation is prepared as follows:

Item	Unit	Base year 2013		Current year 2015		\( p_1q_0 \)	\( p_0q_0 \)	\( p_1q_1 \)	\( p_0q_1 \)
		\( q_0 \)	\( p_0 \)	\( q_1 \)	\( p_1 \)
A	1 kg	5	30	12	44	\( 44 \times 5 = 220 \)	\( 30 \times 5 = 150 \)	\( 44 \times 12 = 528 \)	\( 30 \times 12 = 360 \)
B	1 kg	10	16	12	24	\( 24 \times 10 = 240 \)	\( 16 \times 10 = 160 \)	\( 24 \times 12 = 288 \)	\( 16 \times 12 = 192 \)
C	1 kg	1.2	60	2	75	\( 75 \times 1.2 = 90 \)	\( 60 \times 1.2 = 72 \)	\( 75 \times 2 = 150 \)	\( 60 \times 2 = 120 \)
D	1 Litre	30	52	36	32	\( 32 \times 30 = 960 \)	\( 52 \times 30 = 1560 \)	\( 32 \times 36 = 1152 \)	\( 52 \times 36 = 1872 \)
E	1 Meter	12	8	20	12	\( 12 \times 12 = 144 \)	\( 8 \times 12 = 96 \)	\( 12 \times 20 = 240 \)	\( 8 \times 20 = 160 \)
F	1 Piece	20	2.5	16	3	\( 3 \times 20 = 60 \)	\( 2.5 \times 20 = 50 \)	\( 3 \times 16 = 48 \)	\( 2.5 \times 16 = 40 \)
Total	-	-	-	-	-	\( \Sigma p_1q_0 = 1714 \)	\( \Sigma p_0q_0 = 2088 \)	\( \Sigma p_1q_1 = 2406 \)	\( \Sigma p_0q_1 = 2744 \)

Fisher's index number:
\( I_F = \sqrt{\frac{\Sigma p_1 q_0}{\Sigma p_0 q_0} \times \frac{\Sigma p_1 q_1}{\Sigma p_0 q_1}} \times 100 \)
\( = \sqrt{\frac{1714}{2088} \times \frac{2406}{2744}} \times 100 \)
\( = \sqrt{0.8208 \times 0.8761} \times 100 \)
\( = \sqrt{0.7198} \times 100 \)
\( = 0.8484 \times 100 \)
\( = 84.84 \)
In simple words: To find Fisher's index number, we first ensure all units are the same. Then, we calculate the total value of items using base year and current year prices and quantities. The Fisher's index number of 84.84 means there was a decrease in prices by 15.16% in 2015 compared to 2013 for these items.

🎯 Exam Tip: Always check and standardize units for all items before performing calculations to avoid errors. The intermediate sums (\( \Sigma p_1 q_0, \Sigma p_0 q_0, \Sigma p_1 q_1, \Sigma p_0 q_1 \)) are critical for correctly applying the Fisher's formula.

 

Question 4. data given below:
Answer:
Here, the base year is 2014 and the current year is 2015.
So, \( p_0 \) = Price in 2014, \( q_0 \) = Quantity in 2014, \( p_1 \) = Price in 2015 and \( q_1 \) = Quantity in 2015.
The table for calculation is prepared as follows:

Item	Base year 2014		Current year 2015		\( p_1q_0 \)	\( p_0q_0 \)	\( p_1q_1 \)	\( p_0q_1 \)
	\( q_0 \)	\( p_0 \)	\( q_1 \)	\( p_1 \)
A	25 kg	42	32 kg	45	\( 45 \times 25 = 1125 \)	\( 42 \times 25 = 1050 \)	\( 45 \times 32 = 1440 \)	\( 42 \times 32 = 1344 \)
B	15 litre	28	20 litre	30	\( 30 \times 15 = 450 \)	\( 28 \times 15 = 420 \)	\( 30 \times 20 = 600 \)	\( 28 \times 20 = 560 \)
C	10 pieces	30	20 pieces	36	\( 36 \times 10 = 360 \)	\( 30 \times 10 = 300 \)	\( 36 \times 20 = 720 \)	\( 30 \times 20 = 600 \)
D	8 meter	20	15 meter	25	\( 25 \times 8 = 200 \)	\( 20 \times 8 = 160 \)	\( 25 \times 15 = 375 \)	\( 20 \times 15 = 300 \)
E	30 litre	60	36 litre	65	\( 65 \times 30 = 1950 \)	\( 60 \times 30 = 1800 \)	\( 65 \times 36 = 2340 \)	\( 60 \times 36 = 2160 \)
Total	-	-	-	-	-	\( \Sigma p_1q_0 = 4085 \)	\( \Sigma p_0q_0 = 3730 \)	\( \Sigma p_1q_1 = 5475 \)	\( \Sigma p_0q_1 = 4964 \)

Laspeyre's index number:
\( I_L = \frac{\Sigma p_1 q_0}{\Sigma p_0 q_0} \times 100 \)
\( = \frac{4085}{3730} \times 100 \)
\( = 109.52 \)

Paasche's index number:
\( I_P = \frac{\Sigma p_1 q_1}{\Sigma p_0 q_1} \times 100 \)
\( = \frac{5475}{4964} \times 100 \)
\( = 110.29 \)

Fisher's index number:
\( I_F = \sqrt{I_L \times I_P} \)
\( = \sqrt{109.52 \times 110.29} \)
\( = \sqrt{12078.96} \)
\( = 109.90 \)
In simple words: We calculated three different index numbers for given data. Laspeyre's index shows a 9.52% price rise, Paasche's shows a 10.29% price rise, and Fisher's index shows an average price increase of 9.90% from 2014 to 2015.

🎯 Exam Tip: Accurately calculating the four summations (\( \Sigma p_1q_0, \Sigma p_0q_0, \Sigma p_1q_1, \Sigma p_0q_1 \)) is fundamental. Any error in these sums will propagate through all three index numbers. Ensure proper unit consistency before starting the calculations.

 

Question 5. 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:
Answer:
Here, the base year is 2013 and the current year is 2015.
So, \( p_0 \) = 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 items like wheat, rice, and pulses are not equal. We need to convert them to be equal.
We will calculate the index number for the year 2015 using both the total expenditure method and the family budget method after making the units of these items equal.

Explanation of unit changes:

Item	Change	Year 2013	Year 2015
Wheat	Price per kg	\( \frac{1800}{100} = 18 \) Rs.	\( \frac{2400}{100} = 24 \) Rs.
Rice	Price per kg	\( \frac{700}{20} = 35 \) Rs.	\( \frac{800}{20} = 40 \) Rs.
Pulses	Price per kg	\( \frac{2000}{20} = 100 \) Rs.	\( \frac{2400}{20} = 120 \) Rs.

The table for calculation is prepared as follows:

Item	Unit	Year 2013		Year 2015		Total expenditure method		Family budget method
		\( q_0 \)	\( p_0 \)	\( p_1 \)	\( I = \frac{p_1}{p_0} \times 100 \)	\( p_1q_0 \)	\( p_0q_0 \)	\( W = p_0q_0 \)	\( IW \)
Wheat	1 kg	100	18	24	\( \frac{24}{18} \times 100 = 133.33 \)	\( 24 \times 100 = 2400 \)	\( 18 \times 100 = 1800 \)	1800	239994
Rice	1 kg	40	35	40	\( \frac{40}{35} \times 100 = 114.29 \)	\( 40 \times 40 = 1600 \)	\( 35 \times 40 = 1400 \)	1400	160006
Sugar	1 kg	40	30	36	\( \frac{36}{30} \times 100 = 120.00 \)	\( 36 \times 40 = 1440 \)	\( 30 \times 40 = 1200 \)	1200	144000
Oil	1 kg	60	108	120	\( \frac{120}{108} \times 100 = 111.11 \)	\( 120 \times 60 = 7200 \)	\( 108 \times 60 = 6480 \)	6480	719992.8
Pulses	1 kg	40	100	120	\( \frac{120}{100} \times 100 = 120.00 \)	\( 120 \times 40 = 4800 \)	\( 100 \times 40 = 4000 \)	4000	480000
Ghee	1 kg	36	400	480	\( \frac{480}{400} \times 100 = 120.00 \)	\( 480 \times 36 = 17280 \)	\( 400 \times 36 = 14400 \)	14400	1728000
Total	-	-	-	-	-	\( \Sigma p_1q_0 = 34720 \)	\( \Sigma p_0q_0 = 29280 \)	\( \Sigma W = 29280 \)	\( \Sigma IW = 3471992.8 \)

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
\( = \frac{\Sigma IW}{\Sigma W} \)
\( = \frac{3471992.8}{29280} \)
\( = 118.58 \)
Thus, the index numbers obtained by the total expenditure method and family budget method are equal.
In simple words: This problem asks us to find the price index for 2015 using two methods: total expenditure and family budget. First, we convert all units to be consistent. Then, we calculate the total value of items and the weighted average of price relatives. Both methods show that the index number is 118.58, meaning there's an 18.58% increase in prices compared to the base year 2013.

🎯 Exam Tip: When using both total expenditure and family budget methods, remember to convert units first. The total expenditure method directly uses sums of \( p_1q_0 \) and \( p_0q_0 \), while the family budget method requires calculating individual price relatives (\( I \)) and weights (\( W = p_0q_0 \)). Both should yield the same result if calculated correctly.

 

Question 6. The data about index numbers and weights for groups of items for the living of industrial workers in Ahmedabad city in the years 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 price in the year 2015 ?
Answer:
The table for calculation is prepared as follows:

Group	Weight W	Index number of 2014 \( I_1 \)	Index number of 2015 \( I_2 \)	\( I_1W \)	\( I_2W \)
Food	31	270	281	\( 270 \times 31 = 8370 \)	\( 281 \times 31 = 8711 \)
Fuel and Electricity	14	168	178	\( 168 \times 14 = 2352 \)	\( 178 \times 14 = 2492 \)
Housing	22	205	210	\( 205 \times 22 = 4510 \)	\( 210 \times 22 = 4620 \)
Clothing	10	174	177	\( 174 \times 10 = 1740 \)	\( 177 \times 10 = 1770 \)
Miscellaneous expense	23	303	337	\( 303 \times 23 = 6969 \)	\( 337 \times 23 = 7751 \)
Total	\( \Sigma W = 100 \)	-	-	\( \Sigma I_1W = 23941 \)	\( \Sigma I_2W = 25344 \)

Cost of living index number for the year 2014 \( = \frac{\Sigma I_1 W}{\Sigma W} = \frac{23941}{100} = 239.41 \)

Cost of living index number for the year 2015 \( = \frac{\Sigma I_2 W}{\Sigma W} = \frac{25344}{100} = 253.44 \)

Percentage increase in the cost of living expenditure in 2015 as compared to 2014:
\( = \frac{\text{Increase in cost of living expenditure in the year 2015}}{\text{Cost of living index number for the year 2014}} \times 100 \)
\( = \frac{253.44 - 239.41}{239.41} \times 100 \)
\( = \frac{14.03}{239.41} \times 100 \)
\( = 0.0586 \times 100 \)
\( = 5.86 \% \)

The wages of workers increased by 5% in the year 2015.
Since the increase in wages (5%) is less than the increase in the cost of living expenditure (5.86%), the wage increase is not enough to cover the rise in prices. The workers are short by \( (5.86 - 5) = 0.86 \% \).
In simple words: We calculated the cost of living index for industrial workers in Ahmedabad for 2014 and 2015. The index increased from 239.41 to 253.44, meaning living costs rose by 5.86%. Workers' wages increased by 5%. This 5% wage hike is not enough because the cost of living went up by 5.86%, leaving workers short by 0.86%.

🎯 Exam Tip: To interpret the sufficiency of wage increases, always compare the percentage increase in wages with the percentage increase in the cost of living index. A positive difference means the wage increase is insufficient, while a negative or zero difference indicates it's sufficient or more than sufficient.

 

Question 7. 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 Rs. 6,000, what should be the monthly salary in the current year 2014 to maintain the same standard of living?
Answer:

Group	Price index of 2014 \( I \)	Weight W	\( IW \)
Food	255	42	\( 255 \times 42 = 10710 \)
Fuel and Electricity	174	8	\( 174 \times 8 = 1392 \)
Housing	234	12	\( 234 \times 12 = 2808 \)
Clothing	153	18	\( 153 \times 18 = 2754 \)
Miscellaneous expense	274	20	\( 274 \times 20 = 5480 \)
Total	-	\( \Sigma W = 100 \)	\( \Sigma IW = 23144 \)

Cost of living index number for industrial workers \( I = \frac{\Sigma IW}{\Sigma W} \)
\( = \frac{23144}{100} \)
\( = 231.44 \)

Base year = 2012
The percentage increase in the cost of living expenditure in 2014 as compared to the base year 2012 is \( (231.44 - 100) = 131.44 \% \).

The average monthly salary of the workers in the year 2012 is Rs. 6,000.
To maintain the same standard of living, the average monthly salary in the current year 2014 should be:
\( = \frac{\text{Current year's (i.e. 2014) cost of living index number}}{\text{Base year's (i.e. 2012) cost of living index number}} \times \text{Average salary in the year 2012} \)
\( = \frac{231.44}{100} \times 6000 \)
\( = 13886.40 \)
In simple words: First, we find the cost of living index number for 2014, which is 231.44. This means living costs increased by 131.44% since 2012. To keep the same living standard, a worker earning Rs. 6,000 in 2012 would need to earn Rs. 13,886.40 in 2014.

🎯 Exam Tip: When asked to determine the required salary to maintain the same standard of living, use the ratio of the current year's cost of living index to the base year's index, multiplied by the base year's salary. Clearly state the base year and current year indices used.

 

Question 8. 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.
Answer:

Industry	Unit	Production in the year 2013 \( p_0 \)	Production in the year 2015 \( p_1 \)	\( I = \frac{p_1}{p_0} \times 100 \)	Weight W	\( IW \)
Mine	Lakh tons	10	15	\( \frac{15}{10} \times 100 = 150 \)	4	\( 150 \times 4 = 600 \)
Textile	Crore meters	20	25	\( \frac{25}{20} \times 100 = 125 \)	6	\( 125 \times 6 = 750 \)
Engineering	Lakh tons	30	25	\( \frac{25}{30} \times 100 = 83.33 \)	30	\( 83.33 \times 30 = 2500 \)
Chemicals	Hundred tons	40	50	\( \frac{50}{40} \times 100 = 125 \)	3	\( 125 \times 3 = 375 \)
Food	Lakh tons	50	60	\( \frac{60}{50} \times 100 = 120 \)	4	\( 120 \times 4 = 480 \)
Total	-	-	-	-	\( \Sigma W = 47 \)	\( \Sigma IW = 4705 \)

Index number of industrial production \( I = \frac{\Sigma IW}{\Sigma W} \)
\( = \frac{4705}{47} \)
\( = 100.10 \)
Interpretation: The index number of industrial production for 2015 is 100.10. This shows a small increase of \( (100.10 - 100 =) 0.10 \% \) in industrial production in 2015 compared to the base year 2013.
In simple words: We calculated the industrial production index for 2015 using data from 2013 as the base year. The index came out to be 100.10. This means that industrial production had a very slight increase of 0.10% in 2015 compared to 2013.

🎯 Exam Tip: Industrial production index numbers use quantities as the variable. Always calculate the individual index for each item (\( I \)) by dividing the current year's quantity by the base year's quantity and multiplying by 100. Then, apply the weighted average formula using the given weights.

 

Question 9. The data about per unit price and weights of four different items in the year 2014 and 2015 are as follows. Compute the index number of the year 2015.
Answer:

Item	Weight W	Price per unit in the year 2014 \( p_0 \)	Price per unit in the year 2015 \( p_1 \)	\( I = \frac{p_1}{p_0} \times 100 \)	\( IW \)
A	40	32	40	\( \frac{40}{32} \times 100 = 125 \)	\( 125 \times 40 = 5000 \)
B	25	80	100	\( \frac{100}{80} \times 100 = 125 \)	\( 125 \times 25 = 3125 \)
C	20	24	30	\( \frac{30}{24} \times 100 = 125 \)	\( 125 \times 20 = 2500 \)
D	15	4	6	\( \frac{6}{4} \times 100 = 150 \)	\( 150 \times 15 = 2250 \)
Total	\( \Sigma W = 100 \)	-	-	-	\( \Sigma IW = 12875 \)

Index number of the year 2015 \( I = \frac{\Sigma IW}{\Sigma W} \)
\( = \frac{12875}{100} \)
\( = 128.75 \)
In simple words: To find the index number for 2015, we first calculate how much each item's price changed compared to 2014. Then, we multiply these changes by their importance (weight) and add them up. The final index number is 128.75, which means prices have increased by 28.75% in 2015 compared to 2014.

🎯 Exam Tip: For weighted index numbers, correctly calculate each item's price relative (I) and then multiply it by its weight (W). The sum of IW divided by the sum of W gives the overall index number.

 

Question 10. The index numbers 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. 4000 to Rs. 6000 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:
Given:
Index number of food = 150
Index number of clothing = 224.7

Index number of fuel = 100 + (220% increase) = 100 + 220 = 320

Index number of rent = \( \frac{\text{Current year's expense}}{\text{Base year's expense}} \times 100 \)
\( = \frac{6000}{4000} \times 100 = 150 \)

Index number of miscellaneous expenses = 100 + (100 \( \times \) 1.75) = 100 + 175 = 275

Expenditure percentages (weights):
Food = 40%
Clothing = 18%
Fuel = 12%
Rent = 20%
Miscellaneous expenses = 100 - (40 + 18 + 12 + 20) = 100 - 90 = 10%

The table for calculation is prepared as follows:

Group	Index number \( I \)	Percentage of expenditure W	\( IW \)
Food	150	40	\( 150 \times 40 = 6000 \)
Clothing	224.7	18	\( 224.7 \times 18 = 4044.6 \)
Fuel	320	12	\( 320 \times 12 = 3840 \)
Rent	150	20	\( 150 \times 20 = 3000 \)
Miscellaneous expense	275	10	\( 275 \times 10 = 2750 \)
Total	-	\( \Sigma W = 100 \)	\( \Sigma IW = 19634.6 \)

Cost of living index number for the year 2015 \( I = \frac{\Sigma IW}{\Sigma W} \)
\( = \frac{19634.6}{100} \)
\( = 196.35 \)
Interpretation: The cost of living index number for 2015 is 196.35. This means there is a \( (196.35 - 100 =) 96.35 \% \) increase in the cost of living expenditure of the year 2015 as compared to the base year.
In simple words: We calculated the cost of living index for 2015 by finding index numbers for different categories like food, fuel, rent, and miscellaneous expenses, and then weighing them by their expenditure percentages. The final index is 196.35, which means the cost of living has gone up by 96.35% in 2015 compared to the base year.

🎯 Exam Tip: Carefully calculate individual index numbers for items where direct values are not given (e.g., fuel increase, rent change, multiple increase for miscellaneous). Ensure that the sum of weights (expenditure percentages) always equals 100%.

 

Question 10.The index numbers 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. 4000 to Rs. 6000 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:First, calculate the individual index numbers and weights for each group.
Index number of fuel = 100 + 220% increase = 320
Index number of rent = \( \frac{6000}{4000} \times 100 = 150 \)
Index number of miscellaneous expenses = 100 + (100 \( \times \) 1.75) = 100 + 175 = 275
The remaining percentage expenditure for miscellaneous items is 100 - (40 + 18 + 12 + 20) = 100 - 90 = 10%.

Group	Index number I	Percentage of expenditure W	IW
Food	150	40	6000
Clothing	224.7	18	4044.6
Fuel	320	12	3840
Rent	150	20	3000
Miscellaneous expense	275	10	2750
Total	-	\( \Sigma\text{W} = 100 \)	\( \Sigma\text{IW} = 19634.6 \)

The cost of living index number for the year 2015 is calculated as follows:
\( \text{I} = \frac{\Sigma \text{IW}}{\Sigma \text{W}} \)
\( \implies \frac{19634.6}{100} \)
\( = 196.35 \)
Interpretation: The cost of living expenditure in 2015 shows a 96.35% increase compared to the base year (196.35 - 100 = 96.35%).In simple words: The cost of living index number for 2015 is 196.35. This means that living expenses increased by 96.35% in 2015 compared to the base year.

🎯 Exam Tip: Remember to calculate each component's index and weight accurately before summing them. The interpretation of the final index number is crucial and must clearly state the percentage change from the base period.