Rs. 60,500 is divided among A, B and C such that A receives two ninths of the amount received by B and C together, and B receives three sevenths of the amount received by A and C together. What is the share of C in rupees?

Introduction / Context:
This is a ratio and algebra based division problem where the total sum is divided among three persons A, B and C according to conditions involving combinations of their shares. Instead of a direct ratio, the relationships are given in terms of fractions of sums of two shares. We must convert these conditions into equations, solve for the three shares using algebra and then identify C's share.

Given Data / Assumptions:

• Total sum to be divided among A, B and C is Rs. 60,500.
• A receives two ninths of the sum of B and C, that is A = (2/9) * (B + C).
• B receives three sevenths of the sum of A and C, that is B = (3/7) * (A + C).
• All shares are non negative and their sum is exactly Rs. 60,500.
• We need to find the value of C.

Concept / Approach:
We translate the verbal conditions into algebraic equations in A, B and C. The first condition gives a direct expression for A in terms of B and C. The second gives B in terms of A and C. Along with the total sum equation A + B + C = 60500, we have three equations in three unknowns. Solving this system yields unique values for A, B and C. This type of problem is an example of simultaneous linear equations in the context of sharing money.

Step-by-Step Solution:
Step 1: Write the first condition: A = (2/9) * (B + C). Step 2: Write the second condition: B = (3/7) * (A + C). Step 3: Write the sum condition: A + B + C = 60500. Step 4: From A = (2/9)(B + C), express A in terms of B and C as A = 2(B + C) / 9. Step 5: From B = (3/7)(A + C), express B in terms of A and C as B = 3(A + C) / 7. Step 6: Substitute these expressions into the total sum equation. Using algebraic methods or symbolic solving (for example, with standard linear elimination), we obtain A = 11000, B = 18150 and C = 31350. Step 7: Therefore, C's share is Rs. 31350.

Verification / Alternative check:
Verify both conditions with the obtained values. First, B + C = 18150 + 31350 = 49500. Then (2/9)(B + C) = (2/9) * 49500 = 11000, which matches A. Second, A + C = 11000 + 31350 = 42350. Then (3/7)(A + C) = (3/7) * 42350 = 18150, which matches B. Finally, check the total: 11000 + 18150 + 31350 = 60500. All conditions are satisfied, confirming that the solution is correct.

Why Other Options Are Wrong:
If C were Rs. 29850, 30120, 37250 or 28650, it would not be possible to find values of A and B that satisfy both the fractional conditions and the total sum exactly. For example, trying to adjust A and B for C = 29850 would break at least one of the equations A = (2/9)(B + C) or B = (3/7)(A + C). Only C = Rs. 31350 leads to consistent values for A and B and preserves the total sum of Rs. 60500.

Common Pitfalls:
Mistakes often arise when translating the conditions into equations or when handling fractions. Some learners may misread the statements and write expressions like A = (2/9)B + C instead of A = (2/9)(B + C). Others may get lost in the algebra and forget to check that their final answers satisfy all original conditions. Carefully writing each equation and verifying with substitution at the end is the safest approach.

Final Answer:
The share of C is Rs. 31350, which corresponds to option C.