OP Malhotra Class 11 Maths Solutions Chapter 30 Index Numbers Exercise 30 (A)

 

Question 1.

Answer: To find the price index, we first organize the given prices into a table, clearly showing the base year prices (\( P_0 \)) and the current year prices (\( P_1 \)). This helps in systematically calculating the sums needed for the formula.Using the simple aggregate method, the price index \( P_{01} \) is calculated as: \[ P_{01} = \frac{\Sigma P_1}{\Sigma P_0} \times 100 \]
\( \implies \) \( P_{01} = \frac{176}{107} \times 100 \)
\( \implies \) \( P_{01} = 164.486 \)In simple words: First, we add up all the prices from 1993 (\( P_0 \)) and all the prices from 1995 (\( P_1 \)). Then, we divide the total 1995 price by the total 1993 price and multiply by 100. This tells us how much prices have changed from 1993 to 1995.

🎯 Exam Tip: Always clearly label \( P_0 \) for the base year and \( P_1 \) for the current year. Double-check your sums (\( \Sigma P_0 \) and \( \Sigma P_1 \)) before applying the formula to avoid calculation errors.

 

Question 2.

Answer: To find the price index using the simple aggregate method, we first organize the given price data into a table. The prices from 1990 are considered the base year prices (\( P_0 \)), and the prices from 1993 are the current year prices (\( P_1 \)).The required price index number \( P_{01} \) is calculated using the formula: \[ P_{01} = \frac{\Sigma P_1}{\Sigma P_0} \times 100 \]
\( \implies \) \( P_{01} = \frac{410}{350} \times 100 \)
\( \implies \) \( P_{01} = 117.143 \)In simple words: First, sum up all the prices from 1990 and 1993 separately. Then, divide the total of 1993 prices by the total of 1990 prices, and multiply by 100. This shows how much prices have generally changed between the two years.

🎯 Exam Tip: Remember that the base year's sum (\( \Sigma P_0 \)) always goes in the denominator, and the current year's sum (\( \Sigma P_1 \)) goes in the numerator for the simple aggregate method.

 

Question 3.

Answer: We begin by constructing a table to organize the prices. Here, 1997 is the base year, so its prices are \( P_0 \), and 1998 is the current year, with its prices as \( P_1 \). This clear distinction is key for correct calculation.Using the simple aggregate method, the Price Index \( P_{01} \) is: \[ P_{01} = \frac{\Sigma P_1}{\Sigma P_0} \times 100 \]
\( \implies \) \( P_{01} = \frac{300}{250} \times 100 \)
\( \implies \) \( P_{01} = 120 \)In simple words: Add up all prices for 1997 and get a total. Do the same for 1998 prices. Divide the 1998 total by the 1997 total, then multiply by 100 to get the overall price change.

🎯 Exam Tip: The simple aggregate method is straightforward: sum the current prices, sum the base prices, then divide and multiply by 100. Always ensure you are summing the correct prices for the respective years.

 

Question 4. Using 2005 as base year, the index numbers for the price of a commodity in 2006 and 2007 are 118 and 125. Calculate the index numbers for 2005 and 2007 if 2006 is taken as the base year.
Answer: Let \( P_0, P_1, \) and \( P_2 \) be the prices in 2005, 2006, and 2007 respectively. We are given the following information: Index number for 2006 with 2005 as base year: \( \frac{P_1}{P_0} \times 100 = 118 \) ... (1) Index number for 2007 with 2005 as base year: \( \frac{P_2}{P_0} \times 100 = 125 \) ... (2) Now, we need to calculate new index numbers using 2006 as the new base year. This means 2006 prices become \( P_1 \). Index number for 2005 with 2006 as base year: \[ = \frac{P_0}{P_1} \times 100 \]
\( \implies \) We know from (1) that \( \frac{P_1}{P_0} \times 100 = 118 \), so \( \frac{P_0}{P_1} \times 100 = \frac{100}{118} \times 100 \)
\( \implies \) \( = 84.745 \) Index number for 2007 with 2006 as base year: \[ = \frac{P_2}{P_1} \times 100 \]
\( \implies \) We can find this by dividing equation (2) by equation (1):
\( \implies \) \( \frac{\frac{P_2}{P_0} \times 100}{\frac{P_1}{P_0} \times 100} \times 100 = \frac{125}{118} \times 100 \)
\( \implies \) \( = 105.93 \)In simple words: We know how prices changed using 2005 as the starting point. Now, we want to see how prices changed if 2006 is the new starting point. For 2005, we just flip the earlier calculation. For 2007, we divide its price index relative to 2005 by the 2006 price index relative to 2005, then multiply by 100. This helps us understand price changes from a different perspective.

🎯 Exam Tip: When shifting the base year, remember that if \( \frac{P_1}{P_0} \times 100 \) is the index for year 1 with base year 0, then the index for year 0 with base year 1 is \( \frac{P_0}{P_1} \times 100 = \frac{100}{\text{Index}_{01}} \times 100 \). For comparing two other years, divide their original index numbers if they share the same base.

 

Question 5. Compute a price index for the following by using price relative method.

Answer: To compute the price index using the price relative method, we first construct a table. For each commodity, we calculate its price relative by dividing its current price (\( P_1 \)) by its base price (\( P_0 \)) and multiplying by 100. Then we sum up these individual price relatives.Using the price relative method, the price index is calculated by dividing the sum of the price relatives by the number of commodities (\( n \)): \[ \text{Price index} = \frac{\Sigma (\frac{P_1}{P_0} \times 100)}{n} \]
\( \implies \) \( = \frac{796.67}{5} \)
\( \implies \) \( = 159.334 \)In simple words: First, for each item, calculate how much its price has changed by dividing the new price by the old price and multiplying by 100. Then, add up all these individual percentages. Finally, divide this total by the number of items to get the average price index.

🎯 Exam Tip: In the price relative method, it's crucial to calculate each commodity's relative price change first. Ensure you sum these individual relatives before dividing by the total number of commodities to find the average index.

 

Question 6.

Answer: We construct a table to find the price index using the price relative method. The prices from 1969 are \( P_0 \) (base year), and prices from 1970 are \( P_1 \) (current year). We calculate the price relative (\( \frac{P_1}{P_0} \times 100 \)) for each commodity and then sum them up.Using the price relative method, the price index is: \[ \text{Price index} = \frac{\Sigma (\frac{P_1}{P_0} \times 100)}{n} \]
\( \implies \) \( = \frac{630.53}{4} \)
\( \implies \) \( = 157.6325 \)In simple words: We calculate how much each item's price changed from 1969 to 1970 as a percentage. Then, we add up all these percentages and divide by the number of items (4) to find the average price change.

🎯 Exam Tip: Be careful with decimal numbers in price relatives. Rounding off too early can lead to inaccuracies. Always carry sufficient decimal places during intermediate calculations.

 

Question 7. The index number for the following data for the year 2008, taking 2004 as base year was found to be 116. The simple aggregate method was used for calculation. Find the numerical value of x and y if the sum of the prices in the year 2008 is Rs 203.

Answer: First, we set up a table with the given prices. The 2004 prices are \( P_0 \) (base year), and 2008 prices are \( P_1 \) (current year). We calculate the sums of prices for both years, including the unknown values x and y.We are given that the sum of prices in 2008 is Rs 203. So, \( \Sigma P_1 = 150 + y = 203 \)
\( \implies \) \( y = 203 - 150 \)
\( \implies \) \( y = 53 \) The index number for 2008 (with 2004 as base) is 116, using the simple aggregate method. The formula is: \( \text{Index Number} = \frac{\Sigma P_1}{\Sigma P_0} \times 100 \)
\( \implies \) \( 116 = \frac{150 + y}{135 + x} \times 100 \) Substitute the value of \( y = 53 \):
\( \implies \) \( 116 = \frac{150 + 53}{135 + x} \times 100 \)
\( \implies \) \( 116 = \frac{203}{135 + x} \times 100 \) Now, solve for \( x \):
\( \implies \) \( 135 + x = \frac{203 \times 100}{116} \)
\( \implies \) \( 135 + x = 175 \)
\( \implies \) \( x = 175 - 135 \)
\( \implies \) \( x = 40 \) So, the numerical value of \( x \) is 40 and \( y \) is 53.In simple words: We are given the total price for 2008 and the overall price index. First, we use the given total to find the missing price 'y'. Then, we use the price index formula with all the sums, including the calculated 'y' and the unknown 'x', to find 'x'. This helps us fill in the missing data points.

🎯 Exam Tip: Always solve for the simpler unknown first (here, \( y \)) if enough information is given directly. When dealing with index numbers, make sure to use the correct sums for the numerator (\( \Sigma P_1 \)) and denominator (\( \Sigma P_0 \)) in the formula.

 

Question 8. Construct index numbers by the simple average of relative method for 1990 and 1991 with 1989 as the base year.

Answer: To construct the index numbers by the simple average of relative method, we create a table that includes the base year prices (\( P_0 \)) from 1989, current prices for 1990 (\( P_1 \)), and 1991 (\( P_2 \)). Then, we calculate the price relatives for 1990 (with 1989 as base) and 1991 (with 1989 as base).Thus, for 1990 with 1989 as base year: \[ \text{Price index} = \frac{\Sigma (\frac{P_1}{P_0} \times 100)}{n} \]
\( \implies \) \( = \frac{579.17}{5} \)
\( \implies \) \( = 115.834 \) For 1991 with 1989 as base year: \[ \text{Price index} = \frac{\Sigma (\frac{P_2}{P_0} \times 100)}{n} \]
\( \implies \) \( = \frac{715}{5} \)
\( \implies \) \( = 143 \)In simple words: For each item, we first find how its price changed compared to the 1989 base year, expressed as a percentage. We do this for both 1990 and 1991. Then, we add up all these percentages for each year and divide by the total number of items to get the average price index for 1990 and 1991.

🎯 Exam Tip: When calculating index numbers for multiple years with the same base, create separate price relative columns for each year to keep calculations clear and organized. Double-check your summations and division by 'n' (number of commodities).

 

Question 9. Construct the index number for 1991 taking 1990 as the base year from the following data by simple average of price relative method.

Answer: To find the index number using the simple average of price relative method, we first create a table. In this table, 1990 prices are \( P_0 \) (base year), and 1991 prices are \( P_1 \) (current year). We calculate the price relative (\( \frac{P_1}{P_0} \times 100 \)) for each commodity and then sum them up.By the simple average of price relative method, the price index for 1991 with 1990 as base year is: \[ \text{Price index} = \frac{\Sigma (\frac{P_1}{P_0} \times 100)}{n} \]
\( \implies \) \( = \frac{611.591}{5} \)
\( \implies \) \( = 122.3182 \)In simple words: We calculate how much each item's price changed from 1990 to 1991 as a percentage. After summing up all these individual percentages, we divide the total by the number of items (5) to get the average price index for 1991.

🎯 Exam Tip: Ensure that 'n' in the formula represents the total number of commodities for which price relatives have been calculated. A common mistake is to miscount 'n' or forget to divide by it.

 

Question 10. Construct index number from the following data for 1991 and 1992 taking 1990 as base by using the method of simple average of price relatives :

Answer: We construct a table to calculate the price relatives for 1991 and 1992, using 1990 as the base year (\( P_0 \)). This involves dividing each year's price by the 1990 price and multiplying by 100 for each commodity.By the method of simple average of price relatives, we have: Price Index for 1991: \[ = \frac{\Sigma (\frac{P_1}{P_0} \times 100)}{n} \]
\( \implies \) \( = \frac{512.5}{4} \)
\( \implies \) \( = 128.125 \) Price Index for 1992: \[ = \frac{\Sigma (\frac{P_2}{P_0} \times 100)}{n} \]
\( \implies \) \( = \frac{456}{4} \)
\( \implies \) \( = 114 \)In simple words: We calculate how much each item's price changed compared to the 1990 base year, showing it as a percentage. We do this for both 1991 and 1992. Then, for each year, we add up all these percentages and divide by the number of items (4) to get the average price index.

🎯 Exam Tip: When computing multiple index numbers from the same base, organize your table with separate columns for each current year's price relative to streamline calculations. Be careful to use the correct 'n' for the number of commodities.

 

Question 11. The following data relate to the price of rice in different years. Find out price relatives (i) 1988 as base; (ii) 1992 as base ; (iii) taking average of 1988,1989 and 1990 as base.

Answer: We construct a table to calculate price relatives using different base years as requested. The price relative for any year is calculated as \( \frac{\text{Price in Current Year}}{\text{Price in Base Year}} \times 100 \). For part (iii), the base price (\( P_0 \)) is the average of 1988, 1989, and 1990 prices, which is \( \frac{6+7+7}{3} = \frac{20}{3} \).In simple words: A price relative shows how much a price has changed from a base year to a current year, expressed as a percentage. We calculate this for each year using three different base prices: first, the price from 1988; second, the price from 1992; and third, the average price of 1988, 1989, and 1990.

🎯 Exam Tip: When calculating price relatives with a non-integer base (like an average), it's often easier to keep it as a fraction in your calculations to maintain precision until the final step. Always make sure the correct base price is in the denominator.

 

Question 12. Compute a price index for the following by (i) simple aggregate and (ii) average of price relative method.

Answer: We construct a table to organize the prices for 1994 (\( P_0 \), base year) and 1999 (\( P_1 \), current year). We also calculate the price relative for each commodity to use in the average of price relative method. There are 6 commodities (n=6).(i) By simple aggregate method, the price index is: \[ \text{Price index} = \frac{\Sigma P_1}{\Sigma P_0} \times 100 \]
\( \implies \) \( = \frac{205}{175} \times 100 \)
\( \implies \) \( = 117.143 \) (ii) By average of price relative method, the price index is: \[ \text{Price index} = \frac{\Sigma (\frac{P_1}{P_0} \times 100)}{n} \]
\( \implies \) \( = \frac{737.5}{6} \)
\( \implies \) \( = 122.92 \)In simple words: For the simple aggregate method, we sum all prices from 1994 and 1999 separately, then divide the 1999 total by the 1994 total and multiply by 100. For the average of price relative method, we find each item's price change as a percentage, add all those percentages, and then divide by the total number of items.

🎯 Exam Tip: Remember to use the correct method as specified in the question. The simple aggregate method only requires sums of prices, while the average of price relative method requires individual price relatives to be calculated first.

 

Question 13. Construct an index for 1998 taking 1997 as base by Average of Relatives.

Answer: We construct a table to calculate the price index for 1998, using 1997 as the base year (\( P_0 \)). We calculate the price relative (\( \frac{P_1}{P_0} \times 100 \)) for each commodity. Then we sum up these price relatives. There are 5 commodities (n=5).Then by average of relative method, the Price index for 1998 is: \[ \text{Price index} = \frac{\Sigma (\frac{P_1}{P_0} \times 100)}{n} \]
\( \implies \) \( = \frac{611.591}{5} \)
\( \implies \) \( = 122.3182 \)In simple words: For each commodity, we first figure out how its price changed from 1997 to 1998 as a percentage. Then, we add up all these percentages and divide by the number of commodities (5) to get the overall average price index for 1998.

🎯 Exam Tip: Double-check that you're using the correct input data for each question. If a question asks for specific years, ensure your tables and calculations reflect those years, even if the source material has a repeated table from a previous problem.

 

Question 14. 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: We construct a table to calculate the consumer price index using the simple average of price relative method. The prices from 1989 are \( P_0 \) (base year), and prices from 1990 are \( P_1 \) (current year). We calculate the price relative (\( \frac{P_1}{P_0} \times 100 \)) for each commodity. There are 4 commodities (n=4).Using the simple average of price relatives method, the required index number \( P_{01} \) is: \[ P_{01} = \frac{\Sigma (\frac{P_1}{P_0} \times 100)}{N} \]
\( \implies \) \( P_{01} = \frac{380}{4} \)
\( \implies \) \( P_{01} = 95 \)In simple words: For each food item, we find how much its price changed from 1989 to 1990 as a percentage. We then add up all these percentages and divide by the number of food items (4) to get the average consumer price index for 1990.

🎯 Exam Tip: The simple average of price relatives method is useful for understanding overall price changes when individual commodity prices vary. Make sure all individual price relatives are computed correctly before summing them up and averaging.

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