A, B and C enter into a partnership and their shares are in the ratio `1/2 : 1/3 : 1/4`. After 2 months, A withdraws half of his capital and after 10 months, a profit af Rs. 378 is divided among them. What is B's share ?

Options

A. Rs. 129
B. Rs. 144
C. Rs. 156
D. Rs. 168

Answer

To solve the problem step by step, let's break it down:

### Step 1: Determine the ratio of A, B, and C's investments.
The shares of A, B, and C are given in the ratio \( \frac{1}{2} : \frac{1}{3} : \frac{1}{4} \).

To simplify this ratio, we can find a common denominator. The least common multiple (LCM) of the denominators (2, 3, and 4) is 12.

Now, we convert each fraction to have a denominator of 12:
- A's share: \( \frac{1}{2} = \frac{6}{12} \)
- B's share: \( \frac{1}{3} = \frac{4}{12} \)
- C's share: \( \frac{1}{4} = \frac{3}{12} \)

Thus, the ratio of A : B : C becomes \( 6 : 4 : 3 \).

### Step 2: Calculate the effective capital contribution of each partner.
Let’s assume the total capital contributed by A, B, and C is represented by these ratios:
- A contributes \( 6x \)
- B contributes \( 4x \)
- C contributes \( 3x \)

### Step 3: Account for A's withdrawal after 2 months.
A withdraws half of his capital after 2 months. Therefore, after 2 months:
- A's capital becomes \( \frac{6x}{2} = 3x \) (remaining half).

### Step 4: Calculate the time each partner’s capital was invested.
- A's capital (6x for 2 months, then 3x for 10 months):
- For the first 2 months: \( 6x \times 2 = 12x \)
- For the next 10 months: \( 3x \times 10 = 30x \)
- Total for A: \( 12x + 30x = 42x \)

- B's capital (4x for 12 months):
- Total for B: \( 4x \times 12 = 48x \)

- C's capital (3x for 12 months):
- Total for C: \( 3x \times 12 = 36x \)

### Step 5: Calculate the total capital contribution.
Now, we sum up the contributions:
- Total contribution = \( 42x + 48x + 36x = 126x \)

### Step 6: Determine the profit-sharing ratio.
The profit-sharing ratio based on their contributions is:
- A's share: \( 42x \)
- B's share: \( 48x \)
- C's share: \( 36x \)

### Step 7: Calculate the total profit and the value of each unit.
The total profit is Rs. 378. The total parts in the ratio are:
- Total parts = \( 42 + 48 + 36 = 126 \)

Now, we find the value of one part:
\[ \text{Value of one part} = \frac{378}{126} = 3 \]

### Step 8: Calculate B's share.
B's share in terms of x is \( 48x \). Thus:
\[ B's \, share = 48 \times 3 = 144 \]

### Final Answer:
B's share of the profit is Rs. 144.

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