Question

A, B and C started a business by investing \(^1_2\), \(^1_3\) and \(^1_6\)of the capital respectively. After \(^1_3\) of the total time, A withdrew his capital completely and after \(^1_4\) of the total time B withdrew his capital. C kept his capital for the full period. The ratio in which total profit is to be divided amongst the partners is

• A. 1:2:1
• B. 4:1:4
• C. 2:1:2
• D. 1:2:2

Answer 1

The Correct Option is C

To determine the ratio of the total profit that A, B, and C should receive, we need to consider both the amount of capital each invested and the time for which the capital was active in the business. Let's break down the problem and solve it step-by-step:

Step 1: Capital Contribution

• A invested \(\frac{1}{2}\) of the total capital.
• B invested \(\frac{1}{3}\) of the total capital.
• C invested \(\frac{1}{6}\) of the total capital.

Step 2: Time Duration of Investment

• A withdrew his capital after \(\frac{1}{3}\) of the total time, so A's capital was active for \(\frac{1}{3}\) of the time.
• B withdrew his capital after \(\frac{1}{4}\) of the total time, so B's capital was active for \(\frac{1}{4}\) of the time.
• C kept his investment for the entire time, so C's capital was active for the full period (1 unit of time).

Step 3: Calculate Weighted Investment

• The effective investment by A is \(\frac{1}{2} \times \frac{1}{3} = \frac{1}{6}\).
• The effective investment by B is \(\frac{1}{3} \times \frac{1}{4} = \frac{1}{12}\).
• The effective investment by C is \(\frac{1}{6} \times 1 = \frac{1}{6}\).

Step 4: Calculate the Ratio of Contributions

• First, find a common denominator to compare the contributions: \(\frac{1}{6} = \frac{2}{12}\) for A and C, \(\frac{1}{12} = \frac{1}{12}\) for B.
• A = \(\frac{2}{12}\), B = \(\frac{1}{12}\), C = \(\frac{2}{12}\)
• The ratio of contributions A:B:C is 2:1:2

Conclusion: The total profit should be divided in the ratio 2:1:2 among A, B, and C respectively.