Maharashtra Board Class 12 Maths Part 2 Chapter 5 Index Numbers 5.3 Solutions

Calculate the cost of living index in problems 1 to 3.

Question 1.

Answer: Solution:

\(\Sigma P_1 q_0 = 17150, \Sigma P_0 q_0 = 12700\) Cost of living index is \[CLI = \frac{\Sigma P_1 q_0}{\Sigma P_0 q_0} \times 100\] \[CLI = \frac{17150}{12700} \times 100\] \( = 135.04\) In simple words: The Cost of Living Index (CLI) is calculated by summing the products of current year prices and base year quantities, dividing by the sum of products of base year prices and base year quantities, and then multiplying by 100. It measures the relative change in the cost of a fixed basket of goods and services.

🎯 Exam Tip: Ensure accurate calculation of \(\Sigma P_1 q_0\) and \(\Sigma P_0 q_0\) as these are critical for the correct determination of the Cost of Living Index. Present your answer with two decimal places.

Question 2.

Answer: Solution:

\(\Sigma P_1 q_0 = 4990\) \(\Sigma P_0 q_0 = 4310\) \[CLI = \frac{\Sigma P_1 q_0}{\Sigma P_0 q_0} \times 100\] \[CLI = \frac{4990}{4310} \times 100\] \( = 115.78\) In simple words: The Cost of Living Index for Question 2 is calculated using the same formula, which involves summing up the product of current prices and base quantities and dividing it by the sum of the product of base prices and base quantities, then multiplying by 100.

🎯 Exam Tip: Pay close attention to the quantity \((q_0)\) in the base year when calculating both \(\Sigma P_1 q_0\) and \(\Sigma P_0 q_0\). A common mistake is to use incorrect quantities.

Question 3.

Answer: Solution:

\(\Sigma P_1 q_0 = 11330, \Sigma P_0 q_0 = 10238\) \[CLI = \frac{\Sigma P_1 q_0}{\Sigma P_0 q_0} \times 100\] \[CLI = \frac{11330}{10238} \times 100\] \( = 110.67\) In simple words: Following the standard method, the Cost of Living Index for the third problem is computed by dividing the total value of current prices at base year quantities by the total value of base prices at base year quantities, then multiplying by 100.

🎯 Exam Tip: Double-check all arithmetic operations, especially in summation, to avoid errors. Ensure the correct application of the Laspeyres' method for Cost of Living Index calculation.

Base year weights (W) and current year price relatives (I) are given in problems 4 to 8. Calculate the cost of living index in each case.

Question 4.

Answer: Solution:

\(\Sigma W = 20, \Sigma IW = 1540\) \[CLI = \frac{\Sigma IW}{\Sigma W}\] \[ = \frac{1540}{20}\] \( = 77\) In simple words: The Cost of Living Index for problems 4-8 is calculated using the weighted average of price relatives, where price relatives (I) are multiplied by their respective weights (W) and then summed up, finally divided by the sum of weights.

🎯 Exam Tip: For the weighted average method, clearly identify the price relatives (I) and weights (W). Calculate the product IW for each group accurately before summing them up.

Question 5.

Answer: Solution:

\(\Sigma W = 17, \Sigma IW = 3500\) \[CLI = \frac{\Sigma IW}{\Sigma W}\] \[ = \frac{3500}{17}\] \( = 205.88\) In simple words: The Cost of Living Index in this case is determined by summing the products of individual price relatives (I) and their corresponding weights (W), and then dividing this sum by the total sum of the weights.

🎯 Exam Tip: Ensure precise calculation of the sum of IW and the sum of W. Rounding errors should be minimized until the final result is obtained, typically to two decimal places.

Question 6.

Answer: Solution:

\(\Sigma W = 150, \Sigma IW = 25400\) \[CLI = \frac{\Sigma IW}{\Sigma W}\] \[ = \frac{25400}{150}\] \( = 169.33\) In simple words: The Cost of Living Index is calculated by first finding the sum of (Price Relative × Weight) for all groups, and then dividing this sum by the sum of all weights.

🎯 Exam Tip: When dealing with larger numbers, use a calculator carefully to avoid calculation errors. Always verify your summations before proceeding to the final division.

Question 7.

Answer: Solution:

\(\Sigma W = x + 18, \Sigma IW = 100x + 3600\) CLI = 150 \(\frac{\Sigma IW}{\Sigma W} = 150\)
\( \implies \frac{100x + 3600}{x + 18} = 150\)
\( \implies 100x + 3600 = 150x + 2700\)
\( \implies 50x = 900\)
\( \implies x = 18\) In simple words: When a weight is unknown but the Cost of Living Index is given, we can set up an equation using the formula for CLI. By substituting the known values and solving the algebraic equation, the value of the unknown weight can be found.

🎯 Exam Tip: Remember to correctly form the equation when an unknown variable is involved. Be careful with algebraic manipulation and cross-multiplication to solve for the unknown.

Question 8. Find y if the cost of living index is 200

Answer: Solution:

\(\Sigma W = y + 14, \Sigma IW = 300y + 2200\) CLI = 200 \(\frac{\Sigma IW}{\Sigma W} = 200\)
\( \implies \frac{300y + 2200}{y + 14} = 200\)
\( \implies 300y + 2200 = 200y + 2800\)
\( \implies 100y = 600\)
\( \implies y = 6\) In simple words: Given the Cost of Living Index and an unknown weight, we apply the weighted average formula, creating an algebraic equation. Solving this equation by isolating the unknown variable will yield its value.

🎯 Exam Tip: Practice solving algebraic equations involving CLI. Ensure all terms are correctly transposed and combined to avoid errors in finding the unknown weight.

Question 9. The cost of living Index numbers for years 1995 and 1999 are 140 and 200 respectively. A person earns Rs. 11,200 per month in the year 1995. What should be his monthly earning in the year 1999 in order to maintain his standard of living as in the year 1995?
Answer: Solution: CLI (1995) = 140 CLI (1999) = 200 Income (1995) = Rs. 11200 Income (1999) = ? For year 1995 Real Income = \(\frac{\text{Income}}{\text{CLI}} \times 100\) \[ = \frac{11200}{140} \times 100\] \( = \text{Rs. } 8000\) For year 1999 Real Income = \(\frac{\text{Income}}{\text{CLI}} \times 100\) \[8000 = \frac{\text{Income}}{200} \times 100\]
\( \implies \text{Income} = 16000\) Income in 1999 = Rs. 16000 In simple words: To maintain the same standard of living, real income must remain constant. By calculating the real income in the base year, we can then determine the nominal income required in a future year by adjusting for the change in the Cost of Living Index.

🎯 Exam Tip: Understand the concept of real income and its relationship with the Cost of Living Index. Remember that "maintaining the standard of living" implies keeping real income constant across different time periods.