A sum of Rs. 427 is divided among A, B and C such that three times the share of A, four times the share of B and seven times the share of C are all equal. What is the share of C in rupees?

Introduction / Context:
This is a ratio based division problem where a total sum is shared among three persons A, B and C under a special condition. The condition relates multiples of their shares rather than the shares directly. It is a typical question in the topic of ratio and proportion and partnership style divisions, often seen in competitive exams.

Given Data / Assumptions:

• Total sum to be divided is Rs. 427.
• Three times A's share, four times B's share and seven times C's share are all equal to one another.
• We must find the share of C.
• All shares are non negative real numbers.

Concept / Approach:
If 3 times A's share, 4 times B's share and 7 times C's share are all equal, then we can denote this common value by a single symbol, say k. Then A's share, B's share and C's share can each be expressed in terms of k. Once we do that, we can add the three shares to match the total of Rs. 427, solve for k, and then find C's share. This method uses basic algebra and ratio reasoning in a very direct way.

Step-by-Step Solution:
Step 1: Let the common value of 3A, 4B and 7C be k, that is 3A = 4B = 7C = k. Step 2: From 3A = k, we get A = k / 3. Step 3: From 4B = k, we get B = k / 4. Step 4: From 7C = k, we get C = k / 7. Step 5: The total sum is A + B + C = 427, so write k/3 + k/4 + k/7 = 427. Step 6: Find a common denominator for 3, 4 and 7, which is 84. Then k/3 = 28k/84, k/4 = 21k/84 and k/7 = 12k/84. Step 7: Add these fractions: A + B + C = (28k + 21k + 12k) / 84 = 61k / 84. Step 8: So 61k / 84 = 427. Therefore, k = 427 * 84 / 61. Step 9: Compute 427 / 61 = 7, so k = 7 * 84 = 588. Step 10: Now C = k / 7 = 588 / 7 = Rs. 84.

Verification / Alternative check:
Check the values of A and B as well. A = k / 3 = 588 / 3 = 196. B = k / 4 = 588 / 4 = 147. Now sum them: A + B + C = 196 + 147 + 84 = 427, which matches the total sum given. Also verify the condition: 3A = 3 * 196 = 588, 4B = 4 * 147 = 588 and 7C = 7 * 84 = 588. Since all three are equal to 588, the condition is satisfied and the solution is consistent.

Why Other Options Are Wrong:
If C were Rs. 64, 76, 98 or 72, then 7C would not equal 3A and 4B for any combination that adds up to Rs. 427 while respecting the common multiple concept. For example, if C were Rs. 64, then 7C would be 448, and constructing A and B such that 3A and 4B both equal 448 while still adding up correctly to 427 is impossible. Only C = Rs. 84 leads to a coherent set of values for A, B and C consistent with both the condition and the total sum.

Common Pitfalls:
Students sometimes misinterpret the condition and think that the ratio of A : B : C is 3 : 4 : 7, which is incorrect. The condition is about multiplied shares, not the shares themselves. Another error is in handling fractions or failing to find a correct common denominator, leading to algebraic mistakes. Careful manipulation of the fractions and explicit use of a single common value k makes these problems much easier and prevents confusion.

Final Answer:
The share of C is Rs. 84, corresponding to option C.