X and Y stared a business with capitals `₹20000` and `₹ 25000`. After feq months Z joined them with a capital of `₹30000`. If the share of Z in the annual profit of `₹50000` is `₹14000`, then after how many months from the beginning did Z join ?

Options

A. 7
B. 6
C. 3
D. 5

Explanation

To solve the problem step by step, we need to analyze the contributions of X, Y, and Z to the business and how their investments relate to the profit distribution.

### Step-by-Step Solution:

1. **Identify the Capitals Invested:**
- X's capital = ₹20,000
- Y's capital = ₹25,000
- Z's capital = ₹30,000

2. **Determine the Total Profit:**
- Total annual profit = ₹50,000
- Z's share of the profit = ₹14,000

3. **Calculate the Ratio of Shares:**
- Let the time that Z worked in months be \( x \).
- X and Y worked for 12 months.
- The effective capital contribution for each partner can be calculated as:
- X's contribution = ₹20,000 * 12 months = ₹240,000
- Y's contribution = ₹25,000 * 12 months = ₹300,000
- Z's contribution = ₹30,000 * \( x \) months = ₹30,000x

4. **Set Up the Equation Based on Profit Sharing:**
- The total effective capital contributions = X's contribution + Y's contribution + Z's contribution
- Total contribution = ₹240,000 + ₹300,000 + ₹30,000x = ₹540,000 + ₹30,000x
- The ratio of the contributions is used to determine the share of profit:
- Z's share of profit = (Z's contribution / Total contribution) * Total profit
- Thus, we have:
\[ 14,000 = \left(\frac{30,000x}{540,000 + 30,000x}\right) \times 50,000 \]

5. **Simplify the Equation:**
- Cross-multiplying gives:
\[ 14,000(540,000 + 30,000x) = 50,000 \times 30,000x \]
- Expanding both sides:
\[ 7,560,000,000 + 420,000x = 1,500,000x \]

6. **Rearranging the Equation:**
- Bringing all terms involving \( x \) to one side:
\[ 7,560,000,000 = 1,500,000x - 420,000x \]
- This simplifies to:
\[ 7,560,000,000 = 1,080,000x \]

7. **Solving for \( x \):**
- Dividing both sides by 1,080,000:
\[ x = \frac{7,560,000,000}{1,080,000} = 7 \]
- Therefore, Z worked for 7 months.

8. **Determine When Z Joined:**
- Since Z worked for 7 months and X and Y worked for the entire 12 months:
- Z joined after \( 12 - 7 = 5 \) months from the start.

### Final Answer:
Z joined the business after **5 months** from the beginning.