A sum of Rs 427 is to be divided among A, B and C so 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 question tests proportional thinking in a slightly indirect way. Instead of giving a direct ratio, it expresses relationships using equal multiples of the unknown shares. Such problems help train the ability to convert verbal relationships into algebraic expressions involving a common constant.

Given Data / Assumptions:

• Total amount to be divided = Rs 427.
• Three times the share of A, four times the share of B and seven times the share of C are all equal.
• We must find the value of C share.

Concept / Approach:
If three times A share, four times B share and seven times C share are equal, then there is a common constant k such that 3A = 4B = 7C = k. From this, we can express A, B and C in terms of k, then use the total sum to solve for k and hence find C. This is equivalent to a ratio setup but expressed in terms of equal products.

Step-by-Step Solution:
Step 1: Let 3A = 4B = 7C = k. Step 2: Then A = k / 3, B = k / 4 and C = k / 7. Step 3: Total amount is A + B + C = 427. Step 4: Substitute the expressions: k / 3 + k / 4 + k / 7 = 427. Step 5: Find the common denominator 84 and rewrite: (28k + 21k + 12k) / 84 = 427. Step 6: This simplifies to 61k / 84 = 427. Step 7: Therefore, k = 427 * 84 / 61 = 427 * (84 / 61) = 427 * (84 / 61). Step 8: Since 61 * 7 = 427, we get k = 7 * 84 = 588. Step 9: Now C = k / 7 = 588 / 7 = 84.

Verification / Alternative check:
Compute A and B to verify. A = k / 3 = 588 / 3 = 196, B = k / 4 = 588 / 4 = 147 and C = 84. Total amount is 196 + 147 + 84 = 427, which matches the given sum. Also, 3A = 588, 4B = 588 and 7C = 588, so the condition about equal multiples holds exactly.

Why Other Options Are Wrong:
Values such as 140, 196, 240 or 112 do not satisfy both the total sum and the equal multiples condition when substituted into the equations. For example, if C were 196, then 7C would be much larger than 3A or 4B for any division totalling 427, so the core constraint would break.

Common Pitfalls:
Learners sometimes assume the ratio of A : B : C is 3 : 4 : 7 directly, which is incorrect because the given statement is about 3A, 4B and 7C being equal, not A, B and C themselves. Forgetting to use a common constant k or mismanaging the fraction arithmetic when summing k / 3, k / 4 and k / 7 can also cause errors.

Final Answer:
The share of C is Rs 84.