A,B and C invested Rs . 5000 , Rs , 10000 and Rs . 20000 in a partnership business respectively .
After 'x' months A doubled his investment and after 'y' months B increased his investment by 50 % .if at the end of the year they distributed the profit earned in the ratio `10:15:24` , then find `(x+y)`.

Options

A. A)18
B. B)12
C. C)15
D. D)10

Answer

To solve the problem, we need to calculate the total investment of A, B, and C over the time they invested, and then find the values of \(x\) and \(y\) based on the profit-sharing ratio.

### Step-by-Step Solution:

1. **Initial Investments**:
- A invests Rs. 5000
- B invests Rs. 10000
- C invests Rs. 20000

2. **Investment Duration**:
- A invests for \(x\) months at Rs. 5000, then doubles his investment to Rs. 10000 for the remaining \(12 - x\) months.
- B invests Rs. 10000 for \(y\) months, then increases his investment by 50% (to Rs. 15000) for the remaining \(12 - y\) months.
- C invests Rs. 20000 for the entire 12 months.

3. **Calculate the Effective Capital Contribution**:
- A's contribution:
\[ \text{A's contribution} = 5000 \times x + 10000 \times (12 - x) = 5000x + 120000 - 10000x = 120000 - 5000x \]
- B's contribution:
\[ \text{B's contribution} = 10000 \times y + 15000 \times (12 - y) = 10000y + 180000 - 15000y = 180000 - 5000y \]
- C's contribution:
\[ \text{C's contribution} = 20000 \times 12 = 240000 \]

4. **Profit Sharing Ratio**:
- The profit is shared in the ratio \(10:15:24\).
- Let’s denote the contributions as \(A:C = 10:24\) and \(B:C = 15:24\).

5. **Setting Up the Equations**:
- From A and C:
\[ \frac{120000 - 5000x}{240000} = \frac{10}{24} \]
Cross-multiplying gives:
\[ 24(120000 - 5000x) = 10 \times 240000 \]
Simplifying:
\[ 2880000 - 120000x = 2400000 \]
\[ 120000x = 480000 \implies x = 4 \]

- From B and C:
\[ \frac{180000 - 5000y}{240000} = \frac{15}{24} \]
Cross-multiplying gives:
\[ 24(180000 - 5000y) = 15 \times 240000 \]
Simplifying:
\[ 4320000 - 120000y = 3600000 \]
\[ 120000y = 720000 \implies y = 6 \]

6. **Finding \(x + y\)**:
\[ x + y = 4 + 6 = 10 \]

### Final Answer:
Thus, the value of \(x + y\) is **10**.