Maharashtra Board Class 12 Maths Part 2 Chapter 5 Index Numbers 5.2 Solutions

Question 1. Solution:

\(P_{01} (L) = \frac{\Sigma P_1q_0}{\Sigma P_0q_0} \times 100 = \frac{690}{420} \times 100 = 164.29\)
\(P_{01} (P) = \frac{\Sigma P_1q_1}{\Sigma P_0q_1} \times 100 = \frac{550}{335} \times 100 = 164.18\)
\(P_{01} (D-B) = \frac{P_{01} (L) + P_{01} (P)}{2} = \frac{164.29 + 164.18}{2} = 164.235\)
\(P_{01} (M-E) = \frac{\Sigma P_1q_0 + \Sigma P_1q_1}{\Sigma P_0q_0 + \Sigma P_0q_1} \times 100 = \frac{690+550}{420+335} \times 100 = 164.24\)
Answer: The calculated price index numbers are: Laspeyres' (L) = 164.29, Paasche's (P) = 164.18, Dorbish-Bowley's (D-B) = 164.235, and Marshall-Edgeworth's (M-E) = 164.24.
In simple words: This problem involves calculating four different types of price index numbers (Laspeyres', Paasche's, Dorbish-Bowley's, and Marshall-Edgeworth's) using given base year and current year price and quantity data. Each index uses a specific formula to measure the average change in prices over time.

🎯 Exam Tip: Remember the specific formulas for each index (Laspeyres', Paasche's, Dorbish-Bowley's, and Marshall-Edgeworth's) and practice accurately summing up the required products (\(\Sigma P_1q_0\), \(\Sigma P_0q_0\), etc.) from the given data tables to avoid calculation errors. The correct application of these formulas is key to scoring.

Question 2.
ℹ️ चित्र व्याख्या (Diagram Explanation): यह तालिका विभिन्न वस्तुओं (I, II, III, IV) के लिए आधार वर्ष और वर्तमान वर्ष के मूल्य (Price) और मात्रा (Quantity) डेटा को दर्शाती है। इसका उपयोग लासपेयर, पाश्चे, डोरबिश-बाउली और मार्शल-एजवर्थ जैसे सूचकांक संख्याओं की गणना के लिए किया जाएगा।

Solution:

\(P_{01} (L) = \frac{\Sigma P_1q_0}{\Sigma P_0q_0} \times 100 = \frac{1160}{720} \times 100 = 161.11\)
\(P_{01} (P) = \frac{\Sigma P_1q_1}{\Sigma P_0q_1} \times 100 = \frac{890}{550} \times 100 = 161.82\)
\(P_{01} (D-B) = \frac{P_{01}(L) + P_{01} (P)}{2} = \frac{161.11+161.82}{2} = 161.47\)
\(P_{01} (M-E) = \frac{\Sigma P_1q_0 + \Sigma P_1q_1}{\Sigma P_0q_0 + \Sigma P_0q_1} \times 100 = \frac{1160+890}{720+550} \times 100 = 161.42\)
Answer: The calculated price index numbers are: Laspeyres' (L) = 161.11, Paasche's (P) = 161.82, Dorbish-Bowley's (D-B) = 161.47, and Marshall-Edgeworth's (M-E) = 161.42.
In simple words: This question requires calculating the Laspeyres', Paasche's, Dorbish-Bowley's, and Marshall-Edgeworth's price index numbers based on provided base year and current year price and quantity data. Each index formula is applied to sum up the relevant price-quantity products and then multiplied by 100 to get the percentage change.

🎯 Exam Tip: Pay close attention to the subscripts in the formulas (e.g., \(P_0q_0\) vs. \(P_1q_0\)) to ensure you are multiplying the correct prices and quantities for each sum. Organize your calculations in a table to minimize errors, especially when dealing with multiple commodities.

Exercise 5.2 Solutions Commerce Maths

Question 3.
ℹ️ चित्र व्याख्या (Diagram Explanation): यह तालिका विभिन्न वस्तुओं (L, M, N) के लिए आधार वर्ष और वर्तमान वर्ष के मूल्य (Price) और मात्रा (Quantity) डेटा को दर्शाती है। इसका उपयोग वाल्श के मूल्य सूचकांक की गणना के लिए किया जाएगा।

Solution:

\(P_{01} (W) = \frac{\Sigma P_1\sqrt{q_0q_1}}{\Sigma P_0\sqrt{q_0q_1}} \times 100 = \frac{198}{208} \times 100 = 95.19\)
Answer: Walsh's Price Index Number is 95.19.
In simple words: This problem calculates Walsh's Price Index Number using the given base and current year prices and quantities. The formula involves summing the products of current prices with the geometric mean of base and current quantities, divided by the sum of base prices with the geometric mean of quantities, all multiplied by 100.

🎯 Exam Tip: When calculating Walsh's Index, remember to first calculate \(\sqrt{q_0q_1}\) for each commodity before multiplying by \(P_0\) and \(P_1\). Accuracy in square root calculation and summation is vital.

Question 4.
ℹ️ चित्र व्याख्या (Diagram Explanation): यह तालिका विभिन्न वस्तुओं (I, II, III, IV) के लिए आधार वर्ष और वर्तमान वर्ष के मूल्य (Price) और मात्रा (Quantity) डेटा को दर्शाती है। इसका उपयोग वाल्श के मूल्य सूचकांक की गणना के लिए किया जाएगा।

Solution:

\(P_{01}(W) = \frac{\Sigma P_1\sqrt{q_0q_1}}{\Sigma P_0\sqrt{q_0q_1}} \times 100 = \frac{2050}{1550} \times 100 = 132.26\)
Question asked was Walsh but answers given are \(P_{01} (L), P_{01} (P), P_{01} (D-B), P_{01} (M – E)\)

\(P_{01} (L) = \frac{\Sigma P_1q_0}{\Sigma P_0q_0} \times 100 = \frac{1165}{830} \times 100 = 140.36\)
\(P_{01} (P) = \frac{\Sigma P_1q_1}{\Sigma P_0q_1} \times 100 = \frac{4160}{3220} \times 100 = 129.19\)
\(P_{01} (D-B) = \frac{P_{01} (L) + P_{01} (P)}{2}\)
\( = \frac{140.36+129.19}{2}\)
\( = 134.78\)
\(P_{01} (M-E) = \frac{\Sigma P_1q_0 + \Sigma P_1q_1}{\Sigma P_0q_0 + \Sigma P_0q_1} \times 100\)
\( = \frac{1165+4160}{830+3220} \times 100\)
\( = \frac{5325}{4050} \times 100\)
\( = 131.48\)
Answer: Walsh's Price Index Number is 132.26. The other requested index numbers are: Laspeyres' (L) = 140.36, Paasche's (P) = 129.19, Dorbish-Bowley's (D-B) = 134.78, and Marshall-Edgeworth's (M-E) = 131.48.
In simple words: This problem involves calculating Walsh's Price Index and then provides calculations for Laspeyres', Paasche's, Dorbish-Bowley's, and Marshall-Edgeworth's price index numbers using the provided price and quantity data for the base and current years. Each index requires specific summations of price-quantity products.

🎯 Exam Tip: Be careful to identify which index number is being asked for. If multiple indices are requested (as in this case), ensure you calculate each one separately using its correct formula and sums. Double-check all calculations, especially those involving square roots and large sums.

Question 5. If \(P_{01}(L) = 90\), and \(P_{01}(P) = 40\), find \(P_{01}(D – B)\) and \(P_{01}(F)\) Solution: \(P_{01}(L) = 90\) \(P_{01}(P) = 40\)
\(P_{01} (D-B) = \frac{P_{01} (L) + P_{01} (P)}{2}\)
\( = \frac{90+40}{2}\)
\( = 65\)
\(P_{01} (F) = \sqrt{P_{01} (L) \times P_{01} (P)}\)
\( = \sqrt{90 \times 40}\)
\( = 60\)
Answer: \(P_{01}(D-B) = 65\) and \(P_{01}(F) = 60\).
In simple words: Given Laspeyres' and Paasche's price index numbers, this problem asks for the Dorbish-Bowley's and Fisher's index numbers. Dorbish-Bowley's is the arithmetic mean of Laspeyres' and Paasche's, while Fisher's is the geometric mean of the two.

🎯 Exam Tip: Remember that Dorbish-Bowley's index is the arithmetic average of Laspeyres' and Paasche's indices, and Fisher's index is their geometric average. These relationships are fundamental and frequently tested.

Question 6. If \(\Sigma p_0q_0 = 140\), \(\Sigma p_0q_1 = 200\), \(\Sigma p_1q_0 = 350\), \(\Sigma p_1q_1 = 460\), find Laspeyre's Paasche's Dorbish-Bowley's and Marshall- Edgeworth's Price Index Numbers. Solution: \(\Sigma p_0q_0 = 140\), \(\Sigma p_0q_1 = 200\), \(\Sigma p_1q_0 = 350\), \(\Sigma p_1q_1 = 460\)
\(P_{01} (L) = \frac{\Sigma P_1q_0}{\Sigma P_0q_0} \times 100 = \frac{350}{140} \times 100 = 250\)
\(P_{01} (P) = \frac{\Sigma P_1q_1}{\Sigma P_0q_1} \times 100 = \frac{460}{200} \times 100 = 230\)
\(P_{01} (D-B) = \frac{P_{01}(L) + P_{01} (P)}{2}\)
\( = \frac{250+230}{2}\)
\( = 240\)
\(P_{01} (M-E) = \frac{\Sigma P_1q_0 + \Sigma P_1q_1}{\Sigma P_0q_0 + \Sigma P_0q_1} \times 100\)
\( = \frac{350+460}{140+200} \times 100\)
\( = \frac{810}{340} \times 100 = 238.24\)
Answer: Laspeyre's (L) = 250, Paasche's (P) = 230, Dorbish-Bowley's (D-B) = 240, and Marshall-Edgeworth's (M-E) = 238.24.
In simple words: This problem provides the aggregate sums of price-quantity products for base and current years and asks to calculate four different price index numbers: Laspeyres', Paasche's, Dorbish-Bowley's, and Marshall-Edgeworth's. Each index uses a unique combination of these sums in its formula.

🎯 Exam Tip: When the sums (\(\Sigma p_0q_0\), etc.) are directly provided, substitute them carefully into the respective index formulas. This type of problem tests your knowledge of the formulas more than your calculation skills from raw data.

Question 7. Given that Laspeyre's and Dorbish-Bowley's Price Index Numbers are 160.32 and 164.18 respectively. Find Paasche's Price Index Number. Solution: \(P_{01} (L) = 160.32\) \(P_{01} (D-B) = 164.18\)
\(P_{01} (D-B) = \frac{P_{01}(L) + P_{01} (P)}{2}\)
\(164.18 = \frac{160.32 + P_{01} (P)}{2}\)
\(2 \times 164.18 = 160.32 + P_{01} (P)\)
\(328.36 = 160.32 + P_{01} (P)\)
\(P_{01} (P) = 328.36 - 160.32\)
\(P_{01} (P) = 168.04\)
Answer: Paasche's Price Index Number is 168.04.
In simple words: Given Laspeyres' and Dorbish-Bowley's index numbers, this problem requires finding Paasche's index number. This is achieved by using the relationship that Dorbish-Bowley's index is the arithmetic mean of Laspeyres' and Paasche's indices, and then solving for the unknown Paasche's index.

🎯 Exam Tip: Understand the relationship between Dorbish-Bowley's, Laspeyres', and Paasche's indices. If two are known, the third can be easily derived using the formula \(P_{01}(D-B) = (P_{01}(L) + P_{01}(P))/2\).

Question 8. Given that \(\Sigma p_0q_0 = 220\), \(\Sigma p_0q_1 = 380\), \(\Sigma p_1q_1 = 350\) is Marshall-Edgeworth's Price Index Number is 150, find Laspeyre's Price Index Number. Solution: \(\Sigma p_0q_0 = 220\), \(\Sigma p_0q_1 = 380\), \(\Sigma p_1q_1 = 350\) \(P_{01} (M-E) = 150\)
\(P_{01} (M-E) = \frac{\Sigma P_1q_0 + \Sigma P_1q_1}{\Sigma P_0q_0 + \Sigma P_0q_1} \times 100\)
\(150 = \frac{\Sigma P_1q_0 + 350}{220 + 380} \times 100\)
\(150 = \frac{\Sigma P_1q_0 + 350}{600} \times 100\)
\(150 \times 600 = (\Sigma P_1q_0 + 350) \times 100\)
\(90000 = (\Sigma P_1q_0 + 350) \times 100\)
\(900 = \Sigma P_1q_0 + 350\)
\(\Sigma P_1q_0 = 900 - 350\)
\(\Sigma P_1q_0 = 550\)
Now, \(P_{01} (L) = \frac{\Sigma P_1q_0}{\Sigma P_0q_0} \times 100\)
\( = \frac{550}{220} \times 100\)
\( = 250\)
Answer: Laspeyre's Price Index Number is 250.
In simple words: Given the sums of some price-quantity products and the Marshall-Edgeworth's index, this problem asks to find Laspeyre's index. First, we use the Marshall-Edgeworth formula to determine the missing sum (\(\Sigma P_1q_0\)), and then substitute that into Laspeyre's index formula.

🎯 Exam Tip: When an index value and partial sums are given, work backward using the index formula to find any missing sums. Once all necessary sums are found, calculating other indices becomes a straightforward application of their respective formulas.

Question 9. Find x in the following table if Laspeyres and Paasche's Price Index Numbers are equal.
ℹ️ चित्र व्याख्या (Diagram Explanation): यह तालिका विभिन्न वस्तुओं (A, B) के लिए आधार वर्ष और वर्तमान वर्ष के मूल्य (Price) और मात्रा (Quantity) डेटा को दर्शाती है, जिसमें वर्तमान वर्ष की वस्तु B की मात्रा (quantity) 'x' के रूप में अज्ञात है।

Solution:

\(\Sigma P_0q_0 = 30\), \(\Sigma P_0q_1 = 14\), \(\Sigma P_1q_0 = 5x + 20\), \(\Sigma P_1q_1 = 2x + 10\)
Given \(P_{01} (L) = P_{01} (P)\)
\(\frac{\Sigma P_1q_0}{\Sigma P_0q_0} \times 100 = \frac{\Sigma P_1q_1}{\Sigma P_0q_1} \times 100\)
\(\frac{5x+20}{30} = \frac{2x+10}{14}\)
\(14(5x+20) = 30(2x+10)\)
\(70x+280 = 60x + 300\)
\(70x - 60x = 300 - 280\)
\(10x = 20\)
\(x = 2\)
Answer: The value of x is 2.
In simple words: This problem asks to find the unknown quantity 'x' given that Laspeyres' and Paasche's price index numbers are equal. By setting their formulas equal to each other and substituting the sums derived from the provided table (which include 'x'), we can solve the resulting equation for 'x'.

🎯 Exam Tip: When solving for an unknown variable, ensure all sums are correctly expressed in terms of that variable. Carefully cross-multiply and simplify the equation. Algebraic precision is crucial for finding the correct value of 'x'.

Question 10. If Laspeyre's Price Index Number is four times Paasche's Price Index Number, then find the relation between Dorbish-Bowley's and Fisher's Price Index Numbers. Solution: Given \(P_{01}(L) = 4 \cdot P_{01} (P)\) Now \(P_{01} (D-B) = \frac{P_{01} (L) + P_{01} (P)}{2}\)
\( = \frac{4P_{01}(P) + P_{01} (P)}{2}\)
\( = \frac{5P_{01} (P)}{2}\) (i)
Also, \(P_{01} (F) = \sqrt{P_{01}(L) \times P_{01} (P)}\)
\( = \sqrt{4P_{01} (P) \times P_{01} (P)}\)
\( = \sqrt{4[P_{01} (P)]^2}\)
\( = 2P_{01} (P)\)
From (i), \(P_{01} (P) = \frac{2}{5} P_{01} (D-B)\)
Substituting this into the expression for \(P_{01}(F)\):
\(P_{01} (F) = 2 \left( \frac{2}{5} P_{01} (D-B) \right)\)
\(P_{01} (F) = \frac{4}{5} P_{01} (D-B)\)
\(5 \cdot P_{01} (F) = 4 \cdot P_{01} (D-B)\)
Answer: The relation between Dorbish-Bowley's and Fisher's Price Index Numbers is \(5 \cdot P_{01} (F) = 4 \cdot P_{01} (D-B)\).
In simple words: This problem establishes a relationship between Laspeyres' and Paasche's index numbers (Laspeyres' is four times Paasche's). Then, using the definitions of Dorbish-Bowley's (arithmetic mean) and Fisher's (geometric mean) indices, we derive a direct relationship between these two indices in terms of the given condition.

🎯 Exam Tip: Clearly state the given conditions and the formulas for each index. Substitute the given relationship (\(P_{01}(L) = 4P_{01}(P)\)) into the Dorbish-Bowley's and Fisher's index formulas and simplify to find the desired relationship.

Question 11. If Dorbish-Bowley's and Fisher's Price Index Numbers are 5 and 4, respectively, then find Laspeyres and Paasche's Price Index Numbers. Solution: \(P_{01}(D-B) = 5\) \(P_{01}(F) = 4\)
We know:
\(P_{01}(D-B) = \frac{P_{01}(L) + P_{01} (P)}{2}\)
\(5 = \frac{P_{01}(L) + P_{01} (P)}{2}\)
\(P_{01}(L) + P_{01} (P) = 10\) (i)
And:
\(P_{01}(F) = \sqrt{P_{01}(L) \cdot P_{01} (P)}\)
\(4 = \sqrt{P_{01}(L) \cdot P_{01} (P)}\)
Squaring both sides:
\(16 = P_{01}(L) \cdot P_{01} (P)\) (ii)
From (i), \(P_{01}(P) = 10 - P_{01}(L)\)
Substitute into (ii):
\(16 = P_{01}(L) \cdot (10 - P_{01}(L))\)
\(16 = 10P_{01}(L) - [P_{01}(L)]^2\)
\([P_{01}(L)]^2 - 10P_{01}(L) + 16 = 0\)
This is a quadratic equation. Factorizing:
\([P_{01}(L) - 2] [P_{01}(L) - 8] = 0\)
So, \(P_{01}(L) = 2\) or \(P_{01}(L) = 8\)
Case 1: If \(P_{01}(L) = 2\)
From (ii), \(16 = 2 \cdot P_{01}(P) \implies P_{01}(P) = 8\)
Case 2: If \(P_{01}(L) = 8\)
From (ii), \(16 = 8 \cdot P_{01}(P) \implies P_{01}(P) = 2\)
We know that for price indices, usually the Arithmetic Mean (\(P_{01}(D-B)\)) is greater than or equal to the Geometric Mean (\(P_{01}(F)\)) if all values are positive. This is satisfied here since 5 > 4.
However, usually, Laspeyres' index is greater than Paasche's for an upward price trend, and vice versa. Without further context, both solutions are mathematically valid. However, in many practical scenarios, the AM > GM principle implies that the values tend to be distributed such that it often aligns with Laspeyres being higher if price trends are positive. But here, let's consider the problem as a purely mathematical one. Both (L=2, P=8) and (L=8, P=2) satisfy the equations.
Answer: There are two possible solutions:
(a) Laspeyres' Price Index Number = 2, and Paasche's Price Index Number = 8.
(b) Laspeyres' Price Index Number = 8, and Paasche's Price Index Number = 2.
In simple words: Given the Dorbish-Bowley's and Fisher's index numbers, this problem requires finding Laspeyres' and Paasche's index numbers. We use the formulas for Dorbish-Bowley's (arithmetic mean) and Fisher's (geometric mean) in terms of Laspeyres' and Paasche's indices to form a system of two equations. Solving this system yields a quadratic equation, leading to two possible pairs of values for Laspeyres' and Paasche's indices.

🎯 Exam Tip: This problem involves solving a system of equations derived from the relationships between Dorbish-Bowley's, Fisher's, Laspeyres', and Paasche's indices. Pay close attention to the algebraic manipulation, especially when solving the quadratic equation. Both solutions should be presented if mathematically valid.