Aayush contributes `1//4` of the capital for 15 months and Babloo received `2//3` of the profit. How ong Babloo's money was used ?

Options

A. 10 months
B. 9 months
C. 11 months
D. 7 months

Answer

To solve the problem, we need to determine how long Babloo's money was used based on the information given about the contributions and the profit-sharing.

### Step-by-Step Solution:

1. **Define the Capital Contributions**:
- Let the total capital be \( X \).
- Aayush contributes \( \frac{1}{4} \) of the capital, which is \( \frac{X}{4} \).
- Therefore, Babloo's contribution will be the remaining capital, which is:
\[ \text{Babloo's capital} = X - \frac{X}{4} = \frac{3X}{4} \]

2. **Determine the Time Contributions**:
- Aayush's money is used for 15 months.
- Let \( Y \) be the number of months Babloo's money was used.

3. **Calculate the Profit Shares**:
- According to the problem, Babloo receives \( \frac{2}{3} \) of the total profit.
- This implies that Aayush receives \( \frac{1}{3} \) of the total profit.

4. **Set Up the Equation Based on Profit Sharing**:
- The profit share is proportional to the product of capital and time. Therefore, we can set up the equation:
\[ \text{Aayush's share} : \text{Babloo's share} = \left(\frac{X}{4} \times 15\right) : \left(\frac{3X}{4} \times Y\right) \]
- Simplifying this gives:
\[ \frac{15}{4} : \frac{3Y}{4} \]
- Since Babloo receives \( \frac{2}{3} \) of the profit, we can equate:
\[ \frac{15}{4} : \frac{3Y}{4} = 1 : 2 \]

5. **Cross-Multiply to Solve for \( Y \)**:
- Cross-multiplying gives:
\[ 15 \cdot 2 = 3Y \]
- Thus:
\[ 30 = 3Y \implies Y = \frac{30}{3} = 10 \]

6. **Conclusion**:
- Babloo's money was used for **10 months**.