The ratio of the speed of Aman, Kamal and Manan is 4 : 5 : 6 respectively. What is the ratio of the time taken by Aman, Kamal and Manan respectively to cover the same distance?

Options

A. `10 : 12 : 15`
B. `6 : 5: 4`
C. `4 : 5 : 6`
D. `15 : 12 : 10`

Answer

To solve the problem, we need to find the ratio of the time taken by Aman, Kamal, and Manan to cover the same distance, given their speeds.

### Step-by-Step Solution:

1. **Identify the speeds of Aman, Kamal, and Manan**:
The speeds are given in the ratio of 4:5:6.
Let's denote the speeds as:
- Speed of Aman = 4x
- Speed of Kamal = 5x
- Speed of Manan = 6x
(where x is a common multiplier)

2. **Understand the relationship between speed and time**:
The relationship between speed, time, and distance is given by the formula:
\[ \text{Distance} = \text{Speed} \times \text{Time} \]
When the distance is constant, the time taken is inversely proportional to the speed. This means:
\[ \text{Time} \propto \frac{1}{\text{Speed}} \]

3. **Set up the ratio of time taken**:
Since the time taken is inversely proportional to the speed, we can express the time taken by each person as follows:
- Time taken by Aman = \( \frac{1}{\text{Speed of Aman}} = \frac{1}{4x} \)
- Time taken by Kamal = \( \frac{1}{\text{Speed of Kamal}} = \frac{1}{5x} \)
- Time taken by Manan = \( \frac{1}{\text{Speed of Manan}} = \frac{1}{6x} \)

4. **Form the ratio of time taken**:
The ratio of the times taken by Aman, Kamal, and Manan can be written as:
\[ \text{Time ratio} = \frac{1}{4x} : \frac{1}{5x} : \frac{1}{6x} \]
To simplify this, we can take the reciprocal of each term:
\[ \text{Time ratio} = 5x : 4x : 6x \]
Since \( x \) is a common factor, we can cancel it out:
\[ \text{Time ratio} = 5 : 4 : 6 \]

5. **Final ratio**:
The final ratio of the time taken by Aman, Kamal, and Manan to cover the same distance is:
\[ \text{Time ratio} = 5 : 4 : 6 \]