A,B & C entered into a partnership business .Amount invested by B is 3 times of amount invested by A and ratio of amount invested by C to that of B is `1:2` .After 11 months ,all 3 withdraw Rs Y. If ratio of C's profit share to total profit at the end of the year is `35:129` , then find profit sharing ratio of A to that of B at the end of the year .

Options

A. A)`19:75`
B. B)`41:53`
C. C)`67:27`
D. D)`23:71`

Answer

To solve the problem step by step, we will first define the amounts invested by A, B, and C based on the given ratios, then calculate their respective profit shares, and finally find the profit-sharing ratio of A to B.

### Step 1: Define the investments
Let the amount invested by A be \( x \). According to the problem:
- The amount invested by B is 3 times that of A, so \( B = 3x \).
- The ratio of the amount invested by C to that of B is \( 1:2 \). Therefore, if B invests \( 3x \), then C invests \( \frac{1}{2} \times 3x = \frac{3x}{2} \).

### Step 2: Calculate the amounts invested
From the above definitions:
- Amount invested by A = \( x \)
- Amount invested by B = \( 3x \)
- Amount invested by C = \( \frac{3x}{2} \)

### Step 3: Calculate the total investment
The total investment by A, B, and C is:
\[ \text{Total Investment} = A + B + C = x + 3x + \frac{3x}{2} = 4x + \frac{3x}{2} = \frac{8x}{2} + \frac{3x}{2} = \frac{11x}{2} \]

### Step 4: Calculate the profit shares
Since A, B, and C withdraw Rs Y after 11 months, their remaining amounts after withdrawal are:
- A's remaining amount = \( x - Y \)
- B's remaining amount = \( 3x - Y \)
- C's remaining amount = \( \frac{3x}{2} - Y \)

The investment duration for A and B is 11 months, while C's investment duration is 1 month (since he withdraws Y after 11 months).

### Step 5: Calculate the profit share for each
The profit share for each partner is calculated as:
- A's profit share = \( x \times 11 + (x - Y) \times 1 = 11x + x - Y = 12x - Y \)
- B's profit share = \( 3x \times 11 + (3x - Y) \times 1 = 33x + 3x - Y = 36x - Y \)
- C's profit share = \( \frac{3x}{2} \times 11 + \left(\frac{3x}{2} - Y\right) \times 1 = \frac{33x}{2} + \frac{3x}{2} - Y = \frac{36x}{2} - Y = 18x - Y \)

### Step 6: Total profit share
The total profit share is:
\[ \text{Total Profit} = (12x - Y) + (36x - Y) + (18x - Y) = 66x - 3Y \]

### Step 7: Use the given ratio of C's profit share to total profit
According to the problem, the ratio of C's profit share to the total profit is \( \frac{35}{129} \):
\[ \frac{18x - Y}{66x - 3Y} = \frac{35}{129} \]

### Step 8: Cross-multiply and solve for Y
Cross-multiplying gives:
\[ 129(18x - Y) = 35(66x - 3Y) \]
Expanding both sides:
\[ 2322x - 129Y = 2310x - 105Y \]
Rearranging gives:
\[ 2322x - 2310x = 129Y - 105Y \]
\[ 12x = 24Y \implies Y = \frac{x}{2} \]

### Step 9: Substitute Y back into profit shares
Now substituting \( Y \) back into the profit shares:
- A's profit share = \( 12x - \frac{x}{2} = 12x - 0.5x = 11.5x \)
- B's profit share = \( 36x - \frac{x}{2} = 36x - 0.5x = 35.5x \)

### Step 10: Calculate the profit-sharing ratio of A to B
The profit-sharing ratio of A to B is:
\[ \text{Ratio of A to B} = \frac{11.5x}{35.5x} = \frac{11.5}{35.5} = \frac{23}{71} \]

### Final Answer
Thus, the profit-sharing ratio of A to B at the end of the year is \( 23:71 \).

---