OP Malhotra Class 11 Maths Solutions Chapter 30 Index Numbers Chapter Test

Question 1. Construct the consumer price index number for 1990, taking 1989 as the base year and using simple average of price relative method for the following data :

Answer: First, we construct a table to organize the given prices for each commodity. Then, we calculate the price relative for each item by dividing its 1990 price (\( P_1 \)) by its 1989 price (\( P_0 \)) and multiplying the result by 100. The sum of all these price relatives is 380. We use this to find the Consumer Price Index for 1990 by dividing the total sum of price relatives by the number of commodities, which is 4. This calculation gives us an index number of 95. An index number provides a clear picture of how prices have changed over a period.

Using the simple average of price relatives method:
Required index number \( P_{01} = \frac{1}{N} \Sigma \left(\frac{P_1}{P_0} \times 100\right) \)
\( \implies P_{01} = \frac{1}{4} \times 380 \)
\( \implies P_{01} = 95 \)
In simple words: First, we make a table with the prices. Then, for each item, we find how much its price changed from 1989 to 1990. We add up all these changes and divide by the number of items to get the final index number, which is 95.

🎯 Exam Tip: Remember to clearly identify the base year and current year prices before calculating price relatives. Ensure you sum the correct column for the final average.

Question 2. A small industrial concern used three raw materials A, B and C in the manufacturing using 1995 as the base year, calculate a simple aggregate price index for 2005.

Answer: To calculate the simple aggregate price index for 2005 with 1995 as the base year, we first construct a table listing the commodities and their prices for both years. We then sum up all the prices for 1995 to get \( \Sigma P_0 = 100 \) and all prices for 2005 to get \( \Sigma P_1 = 104 \). Using the simple aggregate method, the required price index is calculated by dividing the total price of the current year by the total price of the base year, and multiplying by 100. This calculation results in a price index of 104. This shows a 4% increase in the aggregate prices from 1995 to 2005.

By simple aggregate method, we have
Required price index for 2005 \( P_{01} = \frac{\Sigma P_1}{\Sigma P_0} \times 100 \)
\( \implies P_{01} = \frac{104}{100} \times 100 \)
\( \implies P_{01} = 104 \)
In simple words: We list the prices of materials for 1995 and 2005. We add up prices for each year. Then, we divide the total price of 2005 by the total price of 1995 and multiply by 100. The answer is 104.

🎯 Exam Tip: For simple aggregate methods, accurately summing \( \Sigma P_1 \) and \( \Sigma P_0 \) is key. Always express the final index as a percentage for easy interpretation.

Question 3. Find the consumer price index number for the year 2010 as the base year by using method of weighted aggregates.

Answer: To find the consumer price index using the weighted aggregate method, we first build a comprehensive table. This table includes prices for the base year (2000, \( P_0 \)), current year (2010, \( P_1 \)), and the assigned weights (\( w \)) for each commodity. Next, we calculate the product of the base year price and its weight (\( P_0 w \)), and the product of the current year price and its weight (\( P_1 w \)) for each commodity. After summing these columns, we get \( \Sigma P_0 w = 1764.9 \) and \( \Sigma P_1 w = 2442.5 \). Finally, we apply the formula: divide \( \Sigma P_1 w \) by \( \Sigma P_0 w \) and multiply by 100. The calculated index number is approximately 138.39, which indicates a significant price increase considering the different importance of items.

Thus by weighted aggregate method,
Required index number \( = \frac{\Sigma P_1 w}{\Sigma P_0 w} \times 100 \)
\( \implies = \frac{2442.5}{1764.9} \times 100 \)
\( \implies = 138.39 \)
In simple words: We make a table with prices for two years and how important each item is (weight). We multiply each price by its weight for both years and then add these up. We divide the total for the current year by the total for the base year and multiply by 100. This gives us the index number, which is 138.39.

🎯 Exam Tip: Always double-check your calculations for \( P_0 w \) and \( P_1 w \) to avoid errors, as these products are crucial for the weighted aggregate formula.

Question 4. The price of six different commodities for years 2009 and year 2011 are as follows :

Answer: First, we construct a table with the commodity prices for 2009 (base year, \( p_0 \)) and 2011 (current year, \( p_1 \)), including the unknown values x and y. The problem states that the total price in 2009 is Rs 360. We sum the known 2009 prices (\( 35 + 80 + 25 + 30 + 80 = 250 \)) and add x to set up the equation: \( 250 + x = 360 \). Solving this gives us \( x = 110 \). Next, we use the given index number (125) and the simple aggregate method formula, substituting the total prices for 2009 (360) and 2011 (\( 390 + y \)). We then solve the equation \( 125 = \frac{390+y}{360} \times 100 \) to find y. This calculation leads to \( y = 60 \). Finding missing values like this is a key part of price index analysis.
In simple words: We have prices for items in 2009 and 2011, with some missing. We know the total price for 2009, so we find 'x' using that. Then, we use the given index number (125) and the formula to find the missing 'y' value. We get x = 110 and y = 60.

🎯 Exam Tip: When given a total sum, use it to find the missing component first. Then, apply the index number formula carefully to solve for any remaining unknowns.