The ratio of the speed of A, B and C is 2 : 3 : 6 respectively. What is the ratio of the time taken by A, B and C respectively to cover the same distance?

Options

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

Answer

To solve the problem, we need to find the ratio of the time taken by A, B, and C to cover the same distance, given the ratio of their speeds.

### Step-by-Step Solution:

1. **Understand the Relationship Between Speed and Time:**
The relationship between speed, distance, and time is given by the formula:
\[ \text{Speed} = \frac{\text{Distance}}{\text{Time}} \]
This implies that:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \]
Hence, time is inversely proportional to speed when the distance is constant.

2. **Identify the Given Ratios:**
We are given the ratio of speeds of A, B, and C as:
\[ \text{Speed of A} : \text{Speed of B} : \text{Speed of C} = 2 : 3 : 6 \]

3. **Express the Speeds:**
Let's denote the speeds of A, B, and C as:
\[ \text{Speed of A} = 2x, \quad \text{Speed of B} = 3x, \quad \text{Speed of C} = 6x \]
where \( x \) is a common multiplier.

4. **Calculate the Time Taken:**
Since the distance is the same for all three, we can express the time taken by each as follows:
\[ \text{Time of A} = \frac{\text{Distance}}{2x}, \quad \text{Time of B} = \frac{\text{Distance}}{3x}, \quad \text{Time of C} = \frac{\text{Distance}}{6x} \]

5. **Find the Ratios of Time:**
To find the ratio of the times taken by A, B, and C, we can write:
\[ \text{Time of A} : \text{Time of B} : \text{Time of C} = \frac{1}{2x} : \frac{1}{3x} : \frac{1}{6x} \]

6. **Eliminate the Common Factor:**
We can eliminate \( x \) from the ratios:
\[ = \frac{1}{2} : \frac{1}{3} : \frac{1}{6} \]

7. **Convert to a Common Denominator:**
To simplify, we can find a common denominator for 2, 3, and 6, which is 6:
\[ = \frac{3}{6} : \frac{2}{6} : \frac{1}{6} \]

8. **Final Ratio:**
Thus, the ratio of the times taken by A, B, and C is:
\[ = 3 : 2 : 1 \]

### Final Answer:
The ratio of the time taken by A, B, and C to cover the same distance is \( 3 : 2 : 1 \).