Aman, Kamal and Ratan start a business. The ratio of their investments is `12 : 4 : 3`. If at the end the ratio of their profits is `3 : 16:9`, what is the ratio of time periods of their investments ?

Options

A. `9 : 64 : 81`
B. ` 4 : 36 : 9`
C. ` 1 : 16 : 12`
D. `3 : 64 : 9`

Answer

To find the ratio of the time periods of the investments made by Aman, Kamal, and Ratan, we can use the relationship between capital, time, and profit. The formula we will use is:

\[ \text{Capital} \times \text{Time} = \text{Profit} \]

From this, we can derive that:

\[ \text{Time} = \frac{\text{Profit}}{\text{Capital}} \]

### Step-by-Step Solution:

1. **Identify the Ratios**:
- The ratio of investments (capital) of Aman, Kamal, and Ratan is given as:
\[ \text{Aman : Kamal : Ratan} = 12 : 4 : 3 \]
- The ratio of profits is given as:
\[ \text{Aman : Kamal : Ratan} = 3 : 16 : 9 \]

2. **Set Up the Time Ratio**:
- According to the formula, the ratio of time periods can be calculated as:
\[ \text{Time} \propto \frac{\text{Profit}}{\text{Capital}} \]
- Therefore, we can express the time periods for Aman, Kamal, and Ratan as:
\[ \text{Time}_A = \frac{3}{12}, \quad \text{Time}_K = \frac{16}{4}, \quad \text{Time}_R = \frac{9}{3} \]

3. **Calculate Each Time Period**:
- For Aman:
\[ \text{Time}_A = \frac{3}{12} = \frac{1}{4} \]
- For Kamal:
\[ \text{Time}_K = \frac{16}{4} = 4 \]
- For Ratan:
\[ \text{Time}_R = \frac{9}{3} = 3 \]

4. **Form the Ratio of Time Periods**:
- Now we have:
\[ \text{Time}_A : \text{Time}_K : \text{Time}_R = \frac{1}{4} : 4 : 3 \]
- To eliminate the fractions, we can multiply each term by 4 (the least common multiple of the denominators):
\[ 1 : 16 : 12 \]

5. **Final Ratio**:
- The ratio of the time periods of their investments is:
\[ \text{Time Ratio} = 1 : 16 : 12 \]

### Final Answer:
The ratio of the time periods of their investments is \( 1 : 16 : 12 \).