A, B and C start a business and the ratio of their investment is `8 : 5 : 3`. If at the end the ratio of their profits is `4 : 15 : 6`, then what is the ratio of time period of their investment?

Options

A. `1: 6 : 5`
B. `2 : 3: 4`
C. `1 : 6 : 4`
D. `2 : 6 : 3`

Answer

To solve the problem, we need to find the ratio of the time periods of investment for A, B, and C based on their investment and profit ratios.

### Step-by-Step Solution:

1. **Identify the Given Ratios**:
- Investment ratio of A, B, and C = 8 : 5 : 3
- Profit ratio of A, B, and C = 4 : 15 : 6

2. **Use the Formula for Profit**:
The profit earned by each partner in a business can be expressed as:
\[ \text{Profit} = \text{Investment} \times \text{Time} \]
Therefore, we can express the ratio of profits in terms of investments and time:
\[ \frac{P_A}{P_B} = \frac{I_A \times T_A}{I_B \times T_B} \]
This implies that:
\[ \frac{T_A}{T_B} = \frac{P_A \cdot I_B}{P_B \cdot I_A} \]

3. **Set Up the Ratios**:
We can derive the time ratio using the given investment and profit ratios:
\[ T_A : T_B : T_C = \frac{P_A}{I_A} : \frac{P_B}{I_B} : \frac{P_C}{I_C} \]

4. **Calculate Individual Time Ratios**:
- For A:
\[ T_A = \frac{P_A}{I_A} = \frac{4}{8} = \frac{1}{2} \]
- For B:
\[ T_B = \frac{P_B}{I_B} = \frac{15}{5} = 3 \]
- For C:
\[ T_C = \frac{P_C}{I_C} = \frac{6}{3} = 2 \]

5. **Combine the Ratios**:
Now we have:
\[ T_A : T_B : T_C = \frac{1}{2} : 3 : 2 \]

6. **Convert to Whole Numbers**:
To eliminate the fraction, multiply the entire ratio by 2:
\[ T_A : T_B : T_C = 1 : 6 : 4 \]

### Final Answer:
The ratio of the time periods of their investment is **1 : 6 : 4**.