S, T and U start a business and their capitals are in the ratio of `3: 4 : 6`. At the end" they receive the profit in the ratio of `1 : 2 : 3`. What will be the respective ratio of time period for which they contribute their capitals?

Options

A. `3 : 2 : 2`
B. `2 : 3 : 3`
C. `2 : 2 : 3 `
D. `4 : 5 : 3`

Answer

To solve the problem, we need to find the respective ratio of the time periods for which S, T, and U contributed their capitals based on the given ratios of their capitals and profits.

### Step-by-Step Solution:

1. **Identify the Ratios**:
- The capital ratio of S, T, and U is given as \( 3:4:6 \).
- The profit ratio of S, T, and U is given as \( 1:2:3 \).

2. **Assign Variables**:
- Let the capitals of S, T, and U be \( 3x, 4x, \) and \( 6x \) respectively, where \( x \) is a common multiplier.
- Let the profits of S, T, and U be \( 1y, 2y, \) and \( 3y \) respectively, where \( y \) is a common multiplier.

3. **Use the Relationship Between Capital, Time, and Profit**:
- The relationship states that \( \text{Capital} \times \text{Time} = \text{Profit} \).
- Therefore, for each person, we can write:
- For S: \( 3x \times t_S = 1y \)
- For T: \( 4x \times t_T = 2y \)
- For U: \( 6x \times t_U = 3y \)

4. **Express Time in Terms of Profit and Capital**:
- Rearranging the equations gives us:
- \( t_S = \frac{1y}{3x} \)
- \( t_T = \frac{2y}{4x} = \frac{y}{2x} \)
- \( t_U = \frac{3y}{6x} = \frac{y}{2x} \)

5. **Find the Ratio of Time Periods**:
- Now we can express the time periods in a ratio:
- \( t_S : t_T : t_U = \frac{1y}{3x} : \frac{y}{2x} : \frac{y}{2x} \)
- To simplify, we can eliminate \( y \) and \( x \) from the ratios:
- This gives us \( t_S : t_T : t_U = \frac{1}{3} : \frac{1}{2} : \frac{1}{2} \)

6. **Finding a Common Denominator**:
- The common denominator for \( 3, 2, 2 \) is \( 6 \).
- So, we convert the ratios:
- \( t_S = \frac{1}{3} = \frac{2}{6} \)
- \( t_T = \frac{1}{2} = \frac{3}{6} \)
- \( t_U = \frac{1}{2} = \frac{3}{6} \)

7. **Final Ratio**:
- Therefore, the final ratio of time periods is:
- \( t_S : t_T : t_U = 2 : 3 : 3 \)

### Conclusion:
The respective ratio of the time periods for which S, T, and U contributed their capitals is \( 2 : 3 : 3 \).