Maharashtra Board Class 12 Maths Part 2 Chapter 5 Index Numbers 5.1 Solutions

Question 1. Use 1995 as the base year in the following problem.

Answer:

\[ P_{01} = \frac{\Sigma P_1}{\Sigma P_0} \times 100 \] \[ = \frac{182}{109} \times 100 \] \[ = 166.97 \]
In simple words: The price index number for 2000, with 1995 as the base year, is 166.97, indicating a 66.97% increase in prices over this period.

🎯 Exam Tip: When calculating index numbers, ensure you correctly identify the base year and current year values, and sum them accurately before applying the formula. Precision in calculation is key.

Question 2. Use 1995 as the base year in the following problem.

Answer:

\[ P = \frac{\Sigma P_1}{\Sigma P_0} \times 100 \] \[ = \frac{439}{316} \times 100 \] \[ = 138.92 \]
In simple words: The price index for 2005, with 1995 as the base year, is 138.92, showing an approximate 38.92% increase in aggregate prices over the decade.

🎯 Exam Tip: Double-check your summation of base year prices (\(\Sigma P_0\)) and current year prices (\(\Sigma P_1\)). A small error in these sums will lead to an incorrect final index number.

Question 3.

Answer:

\[ P_{01} = \frac{\Sigma P_1}{\Sigma P_0} \times 100 \] \[ = \frac{313}{243} \times 100 \] \[ = 128.81 \]
In simple words: The price index number is 128.81, indicating that the overall price level of these commodities in the current year is 28.81% higher than in the base year.

🎯 Exam Tip: Always ensure the correct units are maintained for each commodity when presenting the data, although they do not affect the calculation of the simple aggregate price index.

Question 4. Use 2000 as the base year in the following problem.

Answer:

\[ P_{01} = \frac{\Sigma P_1}{\Sigma P_0} \times 100 \] \[ = \frac{13315}{7760} \times 100 \] \[ = 171.59 \]
In simple words: The price index number for 2006, with 2000 as the base year, is 171.59, indicating a 71.59% increase in the aggregate prices of these items over the six-year period.

🎯 Exam Tip: Pay close attention to the base year and current year specified in the question to ensure correct assignment of \(P_0\) and \(P_1\) values. This is fundamental for accurate index calculation.

Question 5. Use 1990 as the base year in the following problem.

Answer:

\[ P_{01} = \frac{\Sigma P_1}{\Sigma P_0} \times 100 \] \[ = \frac{468}{366} \times 100 \] \[ = 127.87 \]
In simple words: The price index for 1997, using 1990 as the base year, is 127.87, indicating a 27.87% increase in the collective prices of these food items.

🎯 Exam Tip: Always present your calculations clearly, showing the summation steps for both base and current year prices. This helps in verifying the accuracy of your final index number.

Question 6. Assume 2000 to be a base year in the following problem.

Answer:

\[ P_{01} = \frac{\Sigma P_1}{\Sigma P_0} \times 100 \] \[ = \frac{634}{483} \times 100 \] \[ = 131.26 \]
In simple words: The price index for 2007, with 2000 as the base year, is 131.26, indicating an increase of 31.26% in the aggregate prices of these fruits.

🎯 Exam Tip: For problems involving various units, ensure that prices for similar items (e.g., 'doz' vs. 'kg') are added correctly within their respective years. Units do not impact the calculation, but consistency is important.

Find the Quantity Index Number using the Simple Aggregate Method in each of the following examples.

Question 8.

Answer:

\[ Q_{01} = \frac{\Sigma q_1}{\Sigma q_0} \times 100 \] \[ = \frac{585}{785} \times 100 \] \[ = 74.52 \]
In simple words: The quantity index number is 74.52, indicating that the total quantity of commodities produced or consumed in the current year is about 25.48% less than in the base year.

🎯 Exam Tip: For quantity index numbers, ensure you sum the base year quantities (\(\Sigma q_0\)) and current year quantities (\(\Sigma q_1\)) correctly. The formula structure is analogous to the price index number, but with quantities instead of prices.

Question 9.

Answer:

\[ Q_{01} = \frac{\Sigma q_1}{\Sigma q_0} \times 100 \] \[ = \frac{1740}{1400} \times 100 \] \[ = 124.29 \]
In simple words: The quantity index number is 124.29, indicating a 24.29% increase in the aggregate quantity of these commodities from the base year to the current year.

🎯 Exam Tip: Maintain accuracy in summing the quantities. The interpretation of a quantity index number (increase or decrease in quantity) is directly derived from its value relative to 100.

Find the value Index Number using the Simple Aggregate Method in each of the following examples.

Question 10.

Answer:

\[ V_{01} = \frac{\Sigma P_1q_1}{\Sigma P_0q_0} \times 100 \] \[ = \frac{3660}{2840} \times 100 \] \[ = 128.87 \]
In simple words: The value index number is 128.87, indicating that the total monetary value of these commodities has increased by 28.87% from the base year to the current year.

🎯 Exam Tip: For value index numbers, it's crucial to correctly calculate the product of price and quantity for both the base year (\(P_0q_0\)) and current year (\(P_1q_1\)) before summing them up. Accuracy in multiplication and addition is vital.

Question 11.

Answer:

\[ V_{01} = \frac{\Sigma P_1q_1}{\Sigma P_0q_0} \times 100 \] \[ = \frac{10870}{6940} \times 100 \] \[ = 156.63 \]
In simple words: The value index number is 156.63, indicating a significant 56.63% increase in the total value of transactions for these commodities from the base year to the current year.

🎯 Exam Tip: Be methodical in calculating \(P_0q_0\) and \(P_1q_1\) for each commodity. Organizing your work in a clear table helps prevent errors in these intermediate steps before final summation.

Question 12. Find x if the Price Index Number by Simple Aggregate Method is 125

Answer:

\[ P_{01} = \frac{\Sigma P_1}{\Sigma P_0} \times 100 \] Given \( P_{01} = 125 \), \[ 125 = \frac{x + 80}{76} \times 100 \] \[ \frac{125 \times 76}{100} = x + 80 \] \[ 95 = x + 80 \] \[ x = 95 - 80 \] \[ x = 15 \]
In simple words: Given the overall price index is 125, the missing current year price for commodity R, denoted by 'x', is calculated to be 15 Rs.

🎯 Exam Tip: When a missing value is involved, set up the index number formula with the unknown variable, then solve the algebraic equation systematically. Ensure all summations are correct before isolating the variable.

Question 13. Find y is the Price Index Number by Simple Aggregate Method is 120, taking 1995 as the base year.

Answer:

\[ P_{01} = \frac{\Sigma P_1}{\Sigma P_0} \times 100 \] Given \( P_{01} = 120 \), \[ 120 = \frac{324}{y + 210} \times 100 \] \[ 120 \times (y + 210) = 324 \times 100 \] \[ 120(y + 210) = 32400 \] \[ y + 210 = \frac{32400}{120} \] \[ y + 210 = 270 \] \[ y = 270 - 210 \] \[ y = 60 \]
In simple words: With a given price index of 120, the missing base year price for commodity B, represented by 'y', is calculated to be 60 Rs.

🎯 Exam Tip: When the unknown variable is in the denominator (like \(P_0\)), carefully cross-multiply and rearrange the equation to solve for the variable. Algebraic accuracy is crucial to arrive at the correct base year price.