A certain distance is covered at a certain speed. If half of this distance is covered in triple the time, the ratio of the two speeds is

Options

A. `5:1`
B. `2:1`
C. `6:1`
D. None of these

Answer

To solve the problem step-by-step, we will define the variables and use the relationships between distance, speed, and time.

### Step-by-Step Solution:

1. **Define the Variables:**
- Let the total distance be \( D \).
- Let the time taken to cover this distance be \( T \).
- The speed for the first part is given by the formula:
\[ S = \frac{D}{T} \]

2. **Determine Half the Distance and Time:**
- Half of the distance is \( \frac{D}{2} \).
- The time taken to cover this half distance is given as triple the original time, which is:
\[ 3T \]

3. **Calculate the New Speed:**
- The new speed \( S' \) for covering half the distance in triple the time can be calculated using the formula:
\[ S' = \frac{\text{Distance}}{\text{Time}} = \frac{\frac{D}{2}}{3T} \]
- Simplifying this gives:
\[ S' = \frac{D}{2 \cdot 3T} = \frac{D}{6T} \]

4. **Find the Ratio of the Two Speeds:**
- Now we have the original speed \( S = \frac{D}{T} \) and the new speed \( S' = \frac{D}{6T} \).
- The ratio of the two speeds \( \frac{S}{S'} \) is:
\[ \frac{S}{S'} = \frac{\frac{D}{T}}{\frac{D}{6T}} \]
- This simplifies to:
\[ \frac{S}{S'} = \frac{D}{T} \cdot \frac{6T}{D} = \frac{6T}{T} = 6 \]

5. **Express the Ratio:**
- Thus, the ratio of the two speeds \( S : S' \) is:
\[ S : S' = 6 : 1 \]

### Final Answer:
The ratio of the two speeds is \( 6 : 1 \).

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