A,B and C invested their capitals in the ratio of `5:6:8`.At the end of the business, they received the profits in the ratio of `5:3:1`.Find the ratio of time for which they contributed their capitals.

Options

A. `12:9:7`
B. `25: 18 : 8`
C. `5:6:8`
D. `8: 4:1`

Answer

To solve the problem, we need to find the ratio of time for which A, B, and C contributed their capitals based on the given capital and profit ratios.

### Step-by-Step Solution:

1. **Identify the Given Ratios:**
- Capital Ratio of A, B, and C = 5 : 6 : 8
- Profit Ratio of A, B, and C = 5 : 3 : 1

2. **Let the Capital Contributions be:**
- Let A's capital = 5x
- Let B's capital = 6x
- Let C's capital = 8x

3. **Let the Profits be:**
- Let A's profit = 5y
- Let B's profit = 3y
- Let C's profit = 1y

4. **Using the Formula for Partnership:**
The profit earned by each partner is proportional to the product of their capital and the time for which it is invested. Therefore, we can write:
\[ \text{Profit} \propto \text{Capital} \times \text{Time} \]
Hence, we can express the time for which each partner invested their capital as:
\[ \frac{\text{Profit Ratio}}{\text{Capital Ratio}} = \frac{\text{Time}} \]

5. **Set Up the Ratios:**
- For A:
\[ \frac{5y}{5x} = \frac{y}{x} \]
- For B:
\[ \frac{3y}{6x} = \frac{1}{2} \cdot \frac{y}{x} \]
- For C:
\[ \frac{1y}{8x} = \frac{1}{8} \cdot \frac{y}{x} \]

6. **Combine the Ratios:**
Now we can express the time ratios as:
- Time for A = \( \frac{y}{x} \)
- Time for B = \( \frac{1}{2} \cdot \frac{y}{x} \)
- Time for C = \( \frac{1}{8} \cdot \frac{y}{x} \)

7. **Finding a Common Base:**
To express these in a simple ratio, we can take \( \frac{y}{x} \) as a common factor:
- Time for A = 1
- Time for B = \( \frac{1}{2} \)
- Time for C = \( \frac{1}{8} \)

8. **Convert to Whole Numbers:**
To eliminate the fractions, we can multiply each term by 8 (the least common multiple of the denominators):
- Time for A = \( 1 \times 8 = 8 \)
- Time for B = \( \frac{1}{2} \times 8 = 4 \)
- Time for C = \( \frac{1}{8} \times 8 = 1 \)

9. **Final Ratio of Time:**
Therefore, the ratio of time for which A, B, and C contributed their capitals is:
\[ \text{Time Ratio} = 8 : 4 : 1 \]

### Final Answer:
The ratio of time for which A, B, and C contributed their capitals is **8 : 4 : 1**.