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

S Chand Class 11 ICSE Maths Solutions Chapter 30 Index Numbers Ex 30(B)

 

Question 1. Explain briefly, what is meant by a “Weighted average.” Calculate a cost of living index from the following table of prices and weights.

Answer: To find the cost of living index, we first create a table to calculate the product of price index \( (\mathrm{I}) \) and weight \( (\mathrm{w}) \).Using the weighted average method for price relatives, we calculate the Cost of living index. This method gives more importance to items that are consumed more.
Cost of living index \( = \frac{\Sigma \mathrm{I} \mathrm{w}}{\Sigma \mathrm{w}} \)
\( = \frac{10441.5}{100} \)
\( = 104.415 \)
In simple words: A weighted average gives different importance (weights) to different items. We multiply each item's price index by its weight, add them up, and then divide by the total weight. This tells us how much more or less expensive things are compared to a base year.

🎯 Exam Tip: When calculating a weighted average, always ensure you multiply each value by its corresponding weight before summing them up. Double-check your column totals for accuracy.

 

Question 2. Taking 1975 as the base year with an index number 100, calculate an index number for 1985 based on weighted average of price relatives derived from the table given below :

Answer: We construct a table to calculate the price relatives and their weighted sums. This helps compare prices from two different years.Now, using the weighted average of price relatives method, we find the Price Index. This method accounts for the different importance (weights) of each commodity.
Price Index \( = \frac{\Sigma \mathrm{w} \mathrm{x}}{\Sigma \mathrm{w}} \)
\( = \frac{21250}{100} \)
\( = 212.50 \)
In simple words: We calculate how much each item's price changed (its price relative) and then average these changes, considering how important each item is (its weight). This gives us a single number to show the overall price change.

🎯 Exam Tip: Remember to correctly identify \( \mathrm{P}_0 \) (base year price) and \( \mathrm{P}_1 \) (current year price) for each commodity before calculating price relatives. Any mistake here will affect the final index number.

 

Question 3. Calculate the index number for the year 1979 with 1970 as base from the following data using weighted average of price relatives.

Answer: We will create a table to calculate the price relative \( (\mathrm{I}) \) for each commodity and then find \( \mathrm{Iw} \). This helps us apply weights to different items when calculating the index.Now, using the weighted average of price relative method, the required index number is calculated. This method shows how prices have changed overall for a group of items, considering their importance.
Required index number \( = \frac{\Sigma \mathrm{I} \mathrm{w}}{\Sigma \mathrm{w}} \)
\( = \frac{17123.846}{100} \)
\( = 171.24 \)
In simple words: We find the percentage price change for each item and multiply it by how important that item is. Then, we add all these weighted changes and divide by the total importance to get the overall index.

🎯 Exam Tip: Always use the correct base year prices for \( \mathrm{P}_0 \) and current year prices for \( \mathrm{P}_1 \). Ensure accurate calculations for price relatives and their products with weights to avoid errors in the final index.

 

Question 4. Construct a composite index number, as a weighted mean from the following data :

Answer: We understand that a composite index number is an average of index numbers for different groups of variables. We will construct a table to find the product of each index number \( (\mathrm{I}) \) and its weight \( (\mathrm{w}) \). This table helps us combine individual indices into one overall index.Thus, by the weighted average of price relative method, the required index number is determined. This composite index gives a single value that represents the overall change across all the groups, taking into account their individual importance.
Required price index \( = \frac{\Sigma \mathrm{I} \mathrm{w}}{\Sigma \mathrm{w}} \)
\( = \frac{3048}{25} \)
\( = 121.92 \)
In simple words: We take several different index numbers and combine them into one. We multiply each index by its importance (weight), add these up, and divide by the total importance. This gives us an overall index that considers all the different parts.

🎯 Exam Tip: When constructing a composite index, ensure each individual index number is correctly multiplied by its corresponding weight. Errors in this step will lead to an incorrect overall index.

 

Question 5. Construct a composite index number from the following index numbers and weights :

Answer: We need to calculate a composite index number, which is a combined average of the given index numbers based on their weights. We will create a table to help us organize the data and perform the necessary calculations.Using the weighted average method for index numbers, we find the required Price Index. This index shows an overall trend, giving more significance to items with higher weights.
Required price Index \( = \frac{\Sigma \mathrm{I} \mathrm{w}}{\Sigma \mathrm{w}} \)
\( = \frac{3713}{26} \)
\( = 142.81 \)
In simple words: To get an overall index, we multiply each index number by its weight, add them all up, and then divide by the total weight. This gives us one number that represents all the different items, accounting for how important each one is.

🎯 Exam Tip: Always set up a clear table to calculate \( \mathrm{Iw} \) and \( \Sigma\mathrm{w} \) accurately. Double-check your multiplication and summation to avoid errors in the final index computation.

 

Question 6. A small industrial concern used three raw materials A, B and C in its manufacturing process. The price, in £ per kg, of these materials are shown below :

Using 1957 as the base year, calculate for 1967.
(i) a simple aggregate price index.
(ii) price relatives for the three materials and hence a simple average of relatives index. Does either index suffer from any disadvantage ? If the number of kg's of A, B and C used per year are 30,5 and 10 respectively, calculate a weighted aggregate price index for 1967 using 1957 as the base year.
Answer: We construct a table to organize the prices, weights, and calculated values for each material. This allows us to systematically compute different types of price indexes.(i) By the simple aggregate method, we calculate the price index by summing up the prices for each year and comparing them. This provides a quick, unweighted overview of price changes.
Required price index for 1967 \( = \frac{\Sigma \mathrm{P}_1}{\Sigma \mathrm{P}_0} \times 100 \)
\( = \frac{104}{100} \times 100 \)
\( = 104 \)
(ii) By the simple average of price relative method, we take the average of individual price relatives.
Required price index for 1967 \( = \frac{\Sigma(\frac{\mathrm{P}_1}{\mathrm{P}_0} \times 100)}{\mathrm{N}} \)
\( = \frac{336.67}{3} \)
\( = 112.22 \)
By the weighted aggregate method, where weights are quantities used:
Required price index for 1967 \( = \frac{\Sigma \mathrm{P}_1 \mathrm{w}}{\Sigma \mathrm{P}_0 \mathrm{w}} \times 100 \)
\( = \frac{855}{780} \times 100 \)
\( \approx 109.62 \)
By the weighted average of price relative method:
Required price index for 1967 \( = \frac{\Sigma \mathrm{I} \mathrm{w}}{\Sigma \mathrm{w}} \)
\( = \frac{5391.7}{45} \)
\( = 119.81 \)
The first two indices (simple aggregate and simple average of relatives) do not use weights. This is a disadvantage because they treat all materials as equally important, which may not reflect actual changes in production cost. For example, if a company uses a lot of material A and very little of material B, a simple average might not show the true impact of price changes. Since the fourth index number \( (119.81) \) is greater than 100, it indicates that the cost of production has gone up. The weighted methods give a more realistic picture of cost changes.
In simple words: We calculated different ways to see how prices changed. The first two ways just added up prices or price changes without thinking about how much of each material was used. This can be misleading. The last two ways, called "weighted" methods, use how much of each material is actually used, which gives a better idea of how the total cost of making things has really changed. If the index is above 100, costs have increased.

🎯 Exam Tip: When faced with multiple methods, clearly label each calculation. For questions about disadvantages, focus on why unweighted methods might be less accurate than weighted methods in real-world scenarios, particularly when quantities of items differ significantly.

 

Question 7. A manufacturer uses 4 raw materials A, B, C, D in the production of a certain commodity. Masses of raw materials used in manufacturing are in the ratio 2 : 3 : 4 : 1. The prices, in Rs., of the materials per kilogram in the years 1978,1980 are given in the following table :

Calculate the index number for the total cost of the raw materials used for the manufacture of the commodity in 1980, using 1978 as the base year. If the commodity is sold for Rs. 5.75 in 1978, calculate the selling price in 1980, on the assumption that selling prices are directly proportional to the cost of raw material.
Answer: We will construct a table to calculate the products of prices and weights for both the base year \( (\mathrm{P}_0\mathrm{w}) \) and the current year \( (\mathrm{P}_1\mathrm{w}) \). This helps in determining how the total cost has changed.Using the weighted aggregate method, we can find the required index number. This method is good for showing how the total cost of a basket of goods has changed over time.
Required index number \( = \frac{\Sigma \mathrm{P}_1 \mathrm{w}}{\Sigma \mathrm{P}_0 \mathrm{w}} \times 100 \)
\( = \frac{108}{94} \times 100 \)
\( = 114.89 \) (rounded)
Given selling price of commodity in 1978 \( = \) Rs. 5.75
Since selling prices are directly proportional to the cost of raw material, the ratio of selling prices will be the same as the ratio of total raw material costs.
Required selling price of commodity in 1980 \( = \frac{\Sigma \mathrm{P}_1 \mathrm{w}}{\Sigma \mathrm{P}_0 \mathrm{w}} \times 5.75 \)
\( = \frac{108 \times 5.75}{94} \)
\( = \frac{621}{94} \)
\( = 6.61 \) (rounded to two decimal places)
So, the selling price in 1980 would be Rs. 6.61.
In simple words: First, we calculated an index number to see how much more expensive the raw materials became from 1978 to 1980, considering how much of each material was used. Then, because the selling price changes exactly like the cost of raw materials, we used this index to figure out the new selling price for 1980.

🎯 Exam Tip: Pay close attention to the definition of weights and ensure you multiply prices by the correct weights (quantities) for both the base and current years. Remember to use the index number to scale the base year selling price for the current year, if direct proportionality is stated.

 

Question 8. The table shows the averages prices of coffee, sugar and milk in 1979 and 1980, and the weights used to calculate the cost of making a cup of coffee.

Calculate, correct to one decimal place, the index number for the cost of a cup of coffee in 1980 using :
(i) weighted price relatives, taking the index number for 1979 as 100 in each case
Answer: We construct a table to calculate the price relatives (\(\mathrm{I}\)), the product of price relative and weight (\(\mathrm{Iw}\)), and the products of prices and weights for both the base year (\(\mathrm{P}_0\mathrm{w}\)) and the current year (\(\mathrm{P}_1\mathrm{w}\)). This table helps us find the index number using different methods.(i) By the weighted price relative method:
Required index number for 1980 \( = \frac{\Sigma \mathrm{I} \mathrm{w}}{\Sigma \mathrm{w}} \)
\( = \frac{1433.34}{9} \)
\( = 159.3 \) (correct to one decimal place)
(ii) By the weighted aggregated method:
Required index number for 1980 \( = \frac{\Sigma \mathrm{P}_1 \mathrm{w}}{\Sigma \mathrm{P}_0 \mathrm{w}} \times 100 \)
\( = \frac{275}{201} \times 100 \)
\( = 136.8 \) (correct to one decimal place)
In simple words: We calculated the index number using two ways to see how the cost of making coffee changed. The first way looked at how much each ingredient changed in price, weighted by its importance. The second way looked at the total cost of all ingredients in both years. Both show an increase in cost.

🎯 Exam Tip: When a question asks for a specific number of decimal places, always round your final answer accordingly. Be ready to calculate index numbers using different methods like weighted price relatives and weighted aggregate, as each provides a slightly different perspective on price changes.

 

Question 9. An enquiry into the budget of the middle class families in a city in England gave the following information :

What changes in cost of living figures of 1928 as compared with that of 1929 are seen ?
Answer: We will construct a table to calculate the price relatives (\( \mathrm{x} \)) for each commodity, where 1928 is the base year (\( \mathrm{P}_0 \)) and 1929 is the current year (\( \mathrm{P}_1 \)). We will also find the product of weight and price relative (\( \mathrm{wx} \)). This helps us determine the overall change in the cost of living.Using the weighted average of price relative method, the required index number is found. This index helps us understand if living costs have become cheaper or more expensive.
Required index number \( = \frac{\Sigma \mathrm{w} \mathrm{x}}{\Sigma \mathrm{w}} \)
\( = \frac{9786.85}{100} \)
\( = 97.8685 \)
Since the index number is less than 100, living in 1929 was cheaper compared to living in 1928. This means that, on average, the costs for these essential items decreased.
In simple words: We checked how prices changed for different household expenses between 1928 and 1929, giving more importance to bigger expenses. The total index number came out to be less than 100, which means that living in 1929 was cheaper than in 1928.

🎯 Exam Tip: When comparing costs between two years, an index number below 100 indicates a decrease in cost (or deflation), while an index above 100 indicates an increase (or inflation). Always clearly state your conclusion based on this comparison.

 

Question 10. Calculate the cost of living index number from the following group data :

Answer: We construct a table to calculate the product of each group's weight \( (\mathrm{w}) \) and its group index number \( (\mathrm{I}) \). This sum of these products, divided by the total weight, gives us the overall cost of living index.Therefore, the cost of living index is calculated using the weighted average method. This index represents the overall change in the cost of a standard basket of goods and services, giving an idea of inflation or deflation.
Cost of living Index \( = \frac{\Sigma \mathrm{I} \mathrm{w}}{\Sigma \mathrm{w}} \)
\( = \frac{20576}{89} \)
\( = 231.19 \) (rounded to two decimal places)
In simple words: To find the cost of living index, we multiply each group's importance (weight) by its index number, add all these results, and then divide by the total importance. This gives us one overall number that shows how much the cost of living has changed.

🎯 Exam Tip: When using group data, remember that the "Group Index No." already reflects the price change for that category. Your task is to combine these group indices using their respective weights to get the overall cost of living index.

 

Question 11. The following commodities have the given price indices relative to a base of 100. The weights are also given.

Calculate the new index for this set of commodities.
Answer: We construct a table to calculate the product of each commodity's relative index \( (\mathrm{I}) \) and its weight \( (\mathrm{w}) \). This total sum, divided by the sum of weights, will give us the new overall index.Using the weighted average of pure relative method, we find the required Index number. This index reflects the combined price changes of these commodities, giving more importance to those with higher weights.
Required Index number \( = \frac{\Sigma \mathrm{I} \mathrm{w}}{\Sigma \mathrm{w}} \)
\( = \frac{3510}{26} \)
\( = 135 \)
In simple words: We want to find an overall price index for these items. We do this by multiplying each item's price index by its importance (weight), adding all these products together, and then dividing by the total importance. This gives us one number that shows the average price change.

🎯 Exam Tip: Always remember that "relative to a base of 100" means the given price indices are already in percentage form. Simply multiply these indices by their weights and sum them up for the numerator of the weighted average formula.

 

Question 12. Calculate as index number for the second year, taking the first year as base, taking into account the prices of the four commodities (in Rs. per kg) and the weights given here under :

Answer: We construct a table to calculate the products of prices and weights for both the first year \( (\mathrm{P}_0\mathrm{w}) \) and the second year \( (\mathrm{P}_1\mathrm{w}) \). This table will help us find the index number for the second year using the weighted aggregate method.Then, by the weighted aggregate method, we calculate the index number for the second year. This index helps us understand the overall price change for these commodities, taking into account their quantities.
Index no. \( = \frac{\Sigma \mathrm{P}_1 \mathrm{w}}{\Sigma \mathrm{P}_0 \mathrm{w}} \times 100 \)
\( = \frac{2818}{2028} \times 100 \)
\( = 138.95 \) (rounded to two decimal places)
In simple words: To find the index number for the second year compared to the first, we multiply the price of each item by its weight for both years. Then, we add up these weighted prices for each year and use them in a formula to get the overall index, which shows how much prices have gone up.

🎯 Exam Tip: When dealing with "first year as base" and "second year," clearly identify which prices correspond to \( \mathrm{P}_0 \) and \( \mathrm{P}_1 \). Ensure your sum of \( \mathrm{P}_0\mathrm{w} \) and \( \mathrm{P}_1\mathrm{w} \) are accurate, as these are critical for the weighted aggregate formula.

 

Question 13. Construct the consumer price index number for 1988 on basis of 1998 from the following data:

Answer: We construct a table to calculate the price relative \( (\mathrm{x}) \) for each commodity, using 1988 as the base year \( (\mathrm{P}_0) \) and 1998 as the current year \( (\mathrm{P}_1) \). We then find the product of weight and price relative \( (\mathrm{wx}) \). This table helps in finding the consumer price index.Then, by the weighted average method of price relatives, the index number is calculated. This consumer price index shows the average change in prices paid by consumers for a basket of goods and services.
Index No. \( = \frac{\Sigma \mathrm{w} \mathrm{x}}{\Sigma \mathrm{w}} \)
\( = \frac{12441.4}{100} \)
\( = 124.414 \)
In simple words: To find the consumer price index, we first calculate how much each item's price changed from 1988 to 1998. Then, we multiply each price change by how important that item is (its weight). Finally, we add up all these weighted changes and divide by the total weight to get one number that shows the overall price increase.

🎯 Exam Tip: When using the weighted average of price relatives method for a consumer price index, always remember that 'weight' represents the relative importance of each commodity in the consumer's budget. Ensure accurate calculation of price relatives and their weighted sums for a correct index.

 

Question 14. Calculate the index number for the year 2006 with 1996 as the base year by the weighted average of price relatives method from the following data.

Answer: To find the index number, we first create a table to calculate the price relatives and their weighted products.Using the weighted average method for price relatives, we find the required index number.
\( \implies \) Required Index number \( = \frac{\Sigma wx}{\Sigma w} \)
\( \implies \) Required Index number \( = \frac{13000}{100} \)
\( \implies \) Required Index number \( = 130 \) The index number shows how prices have changed over time.
In simple words: First, for each item, figure out how much its price changed from 1996 to 2006 as a percentage. Then, multiply this percentage by how important each item is (its weight). Add up all these weighted percentages and divide by the total weight to get the overall index number.

🎯 Exam Tip: Remember to correctly identify the base year price (\(P_0\)) and the current year price (\(P_1\)) when calculating the price relative for each commodity. Double-check your multiplication for \(wx\).

 

Question 15. Calculate the cost of living index for the following data :

Answer: We will construct a table to calculate the product of price relative and weight, then sum these values.Using the weighted average method for price relative, the required cost of living index is calculated as follows:
\( \implies \) Required cost of living index \( = \frac{\Sigma Iw}{\Sigma w} \)
\( \implies \) Required cost of living index \( = \frac{17122.9}{100} \)
\( \implies \) Required cost of living index \( = 171.229 \) This index helps understand how living costs change for different groups over time.
In simple words: When you have how much prices changed for each item (price relative) and how important each item is (weight), you multiply them together. Add up all these products and divide by the total importance. This gives you a single number showing how much the overall cost of living has changed.

🎯 Exam Tip: Ensure that all given price relatives are used in the calculation, and be precise with decimal points, especially when summing up the products for \( \Sigma Iw \).

 

Question 16. Find the consumer price index number for 1991 on the base of 1990 from the following data, using the method of weighted relatives :

Answer: We will construct a table to calculate the price relatives and their weighted products.By using the weighted average method of price relatives, the index number is calculated.
\( \implies \) Price Index or index number \( = \frac{\Sigma Iw}{\Sigma w} \)
\( \implies \) Price Index or index number \( = \frac{10570}{77} \)
\( \implies \) Price Index or index number \( = 137.27 \) This number tells us how much the cost of these items has changed between 1990 and 1991.
In simple words: For each item, find how its price changed from 1990 to 1991 as a percentage. Multiply this percentage by the item's quantity. Add up all these results and divide by the total quantity to get the final consumer price index.

🎯 Exam Tip: When using the weighted relatives method, ensure that the quantity (or weight) is correctly multiplied by the price relative for each item before summing them up. Watch out for unit consistency.

 

Question 17. From the following data compose price index by applying weighted average of price relatives method using arithmetic means :

Answer: We construct the table to calculate the price relatives and their weighted products.Thus by the weighted average of price relative method, the required price index is:
\( \implies \) Required price index \( = \frac{\Sigma Iw}{\Sigma w} \)
\( \implies \) Required price index \( = \frac{85000}{440} \)
\( \implies \) Required price index \( \approx 193.18 \) This index shows the overall change in prices for these commodities.
In simple words: First, calculate a "weight" for each item by multiplying its base year price by its base year quantity. Then, for each item, find how much its price changed as a percentage. Multiply this percentage by its weight. Add up all these results and divide by the total weight to get the final price index.

🎯 Exam Tip: When using the weighted average of price relatives, remember that the weights (\(w\)) are typically derived from the base year's price and quantity (\(P_0q_0\)), not just the quantity itself.

 

Question 18. The following table shows the prices per unit in 1980 and 1984 with weights of the commodities A, B, C, D:

Answer: We construct a table to calculate the price relatives and their weighted products.Thus by using the weighted average of price relative, the required index number is:
\( \implies \) Required index number \( = \frac{\Sigma Iw}{\Sigma w} \)
\( \implies \) Required index number \( = \frac{16250}{100} \)
\( \implies \) Required index number \( = 162.50 \) This number tells us how prices have changed from 1980 to 1984, weighted by the importance of each item.
In simple words: For each item, calculate how its price changed from 1980 to 1984 as a percentage. Then, multiply this percentage by the item's given weight. Add up all these weighted percentages and divide by the total weight to get the overall index number.

🎯 Exam Tip: Be careful to use the correct year for \(P_0\) (base year) and \(P_1\) (current year) in the price relative calculation. Always sum \(w\) and \(Iw\) accurately.

 

Question 19. The price quotations of four different commodities for 2001 to 2009 are as given below. Calculate the index number for 2009 with 2001 as the base year by using weighted average of price relative method.

Answer: We construct a table to calculate the price relatives and their weighted products.Then by the weighted average method of price relative, the index number is:
\( \implies \) Index number \( = \frac{\Sigma wx}{\Sigma w} \)
\( \implies \) Index number \( = \frac{50902}{99} \)
\( \implies \) Index number \( \approx 514.16 \) This index number reflects the overall change in prices for these commodities from 2001 to 2009.
In simple words: For each item, compare its price in 2009 to its price in 2001 to get a percentage change. Multiply this percentage by the item's weight. Add up these results and divide by the total weight to find the overall price index for 2009 compared to 2001.

🎯 Exam Tip: Always make sure your base year price (\(P_0\)) and current year price (\(P_1\)) are correctly identified according to the question, even if column labels in source data seem switched.