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

Options

A. `4 : 1`
B. `1 : 4`
C. `2 : 1`
D. `1 : 2`

Answer

To solve the problem, we need to find the ratio of two speeds based on the given conditions. Let's break it down step by step.

### Step 1: Define the Variables
Let:
- \( D \) = total distance
- \( T \) = time taken to cover the distance at the original speed
- \( S \) = original speed

### Step 2: Calculate the Original Speed
The formula for speed is given by:
\[ S = \frac{D}{T} \]

### Step 3: Analyze the New Conditions
According to the problem, half of the distance \( \left(\frac{D}{2}\right) \) is covered in double the time \( (2T) \).

### Step 4: Calculate the New Speed
The new speed \( S_1 \) can be calculated as follows:
\[ S_1 = \frac{\text{Distance}}{\text{Time}} = \frac{\frac{D}{2}}{2T} = \frac{D}{2 \cdot 2T} = \frac{D}{4T} \]

### Step 5: Find the Ratio of the Two Speeds
Now, we need to find the ratio of the original speed \( S \) to the new speed \( S_1 \):
\[ \text{Ratio} = \frac{S}{S_1} = \frac{\frac{D}{T}}{\frac{D}{4T}} \]

### Step 6: Simplify the Ratio
To simplify the ratio:
\[ \text{Ratio} = \frac{D}{T} \cdot \frac{4T}{D} = \frac{4DT}{DT} = 4 \]

Thus, the ratio of the two speeds is:
\[ \text{Ratio} = 4:1 \]

### Final Answer
The ratio of the two speeds is \( 4:1 \).

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