Amit and Brijesh started a business with initial investments in the ratio of `12:11` and their annual profits were in the ratio of `4:1` . If Amit invested the money for 11 months, then for what time Brijesh invested the money ?

Options

A. 9 months
B. 3 months
C. 5 months
D. 10 months

Answer

To solve the problem step by step, we will use the information provided about the investments and profits of Amit and Brijesh.

### Step-by-Step Solution:

1. **Identify the Investments:**
- Let Amit's investment be represented as \( 12x \) (since the ratio of their investments is \( 12:11 \)).
- Therefore, Brijesh's investment will be \( 11x \).

2. **Identify the Time Periods:**
- Amit invested his money for \( 11 \) months.
- Let Brijesh's investment period be \( y \) months.

3. **Calculate the Investment Contribution:**
- The contribution to the business from Amit can be calculated as:
\[ \text{Amit's Contribution} = \text{Investment} \times \text{Time} = 12x \times 11 = 132x \]
- The contribution from Brijesh can be calculated as:
\[ \text{Brijesh's Contribution} = \text{Investment} \times \text{Time} = 11x \times y = 11xy \]

4. **Set Up the Ratio of Contributions:**
- The ratio of their contributions is equal to the ratio of their profits, which is given as \( 4:1 \).
- Therefore, we can set up the equation:
\[ \frac{132x}{11xy} = \frac{4}{1} \]

5. **Cross-Multiply to Solve for \( y \):**
- Cross-multiplying gives:
\[ 132x \cdot 1 = 4 \cdot 11xy \]
- Simplifying this, we have:
\[ 132x = 44xy \]

6. **Cancel \( x \) from Both Sides:**
- Since \( x \) is common in both terms (assuming \( x \neq 0 \)), we can cancel it out:
\[ 132 = 44y \]

7. **Solve for \( y \):**
- Dividing both sides by \( 44 \):
\[ y = \frac{132}{44} = 3 \]

8. **Conclusion:**
- Therefore, Brijesh invested the money for \( 3 \) months.

### Final Answer:
Brijesh invested the money for **3 months**.