The ratio of the sum of the salaries of A and B to the difference of their salaries is 11 : 1, and the ratio of the sum of the salaries of B and C to the difference of their salaries is also 11 : 1. If A's salary is the highest, C's salary is the lowest, and the total of all three salaries is Rs 1,82,000, what is B's salary (in Rs)?

Introduction / Context:
This is an algebraic problem involving three salaries linked by conditions on the ratios of sums to differences. We are told how the sum of two salaries relates to their difference for pairs (A, B) and (B, C). We also know the order of the salaries (A highest, C lowest) and the total of all three. The goal is to determine B's salary. This type of question tests equation formation and manipulation using given ratios.

Given Data / Assumptions:
• Let salaries be A, B and C with A > B > C > 0. • (A + B) : (A − B) = 11 : 1. • (B + C) : (B − C) = 11 : 1. • Total salary A + B + C = Rs 1,82,000. • We are asked to find B's salary.

Concept / Approach:
From a ratio (sum) : (difference) = 11 : 1, we can write (sum)/(difference) = 11. For A and B, this gives (A + B)/(A − B) = 11. Solving this equation expresses B in terms of A. Similarly, from (B + C)/(B − C) = 11, we can express C in terms of B. Then we substitute these expressions into the total salary equation A + B + C = 1,82,000 and solve. This systematic substitution approach avoids guesswork and ensures consistency with all given conditions.

Step-by-Step Solution:
Step 1: From (A + B) : (A − B) = 11 : 1, write (A + B)/(A − B) = 11. Step 2: Cross multiply: A + B = 11(A − B). Step 3: Expand right side: A + B = 11A − 11B. Step 4: Bring terms to one side: A + B − 11A + 11B = 0, so -10A + 12B = 0. Step 5: Rearrange: 12B = 10A, so B = (10/12)A = (5/6)A. Step 6: Similarly, from (B + C)/(B − C) = 11, write B + C = 11(B − C). Step 7: Expand: B + C = 11B − 11C. Step 8: Bring terms together: B + C − 11B + 11C = 0 gives -10B + 12C = 0. Step 9: Rearrange: 12C = 10B, so C = (10/12)B = (5/6)B. Step 10: Substitute B in terms of A: B = (5/6)A. Then C = (5/6)B = (5/6) * (5/6)A = (25/36)A. Step 11: Use the total salary condition: A + B + C = 1,82,000. Step 12: Substitute B and C: A + (5/6)A + (25/36)A = 1,82,000. Step 13: Express in terms of a common denominator 36. A = (36/36)A, (5/6)A = (30/36)A, and (25/36)A = (25/36)A. Step 14: Sum the coefficients: (36 + 30 + 25)/36 * A = 1,82,000, so (91/36)A = 1,82,000. Step 15: Solve for A: A = 1,82,000 * (36/91). Step 16: Compute 1,82,000 * 36 / 91 = 72,000. Step 17: Now find B: B = (5/6)A = (5/6) * 72,000 = 60,000. Step 18: Find C: C = (25/36)A = (25/36) * 72,000 = 50,000.

Verification / Alternative Check:
Check the total: A + B + C = 72,000 + 60,000 + 50,000 = 1,82,000, which matches the given total. Now check the ratios. For A and B, (A + B)/(A − B) = (72,000 + 60,000)/(72,000 − 60,000) = 1,32,000 / 12,000 = 11. For B and C, (B + C)/(B − C) = (60,000 + 50,000)/(60,000 − 50,000) = 1,10,000 / 10,000 = 11. Both ratios are 11, confirming that the conditions are satisfied.

Why Other Options Are Wrong:
• 72,000 is A's salary, not B's. • 50,000 is C's salary, and 86,400 does not fit the derived relationships between A, B and C. • Only 60,000 is consistent with all the equations and the total salary.

Common Pitfalls:
A common mistake is to misinterpret the ratio of sum to difference and instead treat it as a simple difference-to-sum ratio. Another issue is algebraic sign errors when rearranging equations. When dealing with sums and differences, it helps to write each equation slowly and double check each move, especially when bringing terms from one side to another.

Final Answer:
B's salary is Rs 60,000.