The time take by a boat in upstream is double than the time taken by it in downstream, but distance covered by it in upstream is only 75% of distance covered by it in downstream. Find the ratio of speed of boat in still water to speed of current.

Options

A. ` 5 : 11`
B. ` 11 : 7`
C. `7 : 11`
D. `11 : 5`

Answer

To solve the problem, we need to find the ratio of the speed of the boat in still water (let's denote it as \( x \)) to the speed of the current (denote it as \( y \)).

### Step-by-Step Solution:

1. **Understanding the Distances**:
- Let the distance covered downstream be \( 4D \).
- The distance covered upstream is given as 75% of the downstream distance, which means:
\[ \text{Distance upstream} = 0.75 \times 4D = 3D \]

2. **Understanding the Speeds**:
- The speed of the boat in still water is \( x \).
- The speed of the current is \( y \).
- Therefore, the effective speed of the boat downstream (with the current) is \( x + y \).
- The effective speed of the boat upstream (against the current) is \( x - y \).

3. **Calculating the Times**:
- The time taken to travel downstream is given by:
\[ \text{Time downstream} = \frac{\text{Distance}}{\text{Speed}} = \frac{4D}{x + y} \]
- The time taken to travel upstream is:
\[ \text{Time upstream} = \frac{3D}{x - y} \]

4. **Setting Up the Equation**:
- According to the problem, the time taken upstream is double the time taken downstream:
\[ \frac{3D}{x - y} = 2 \times \frac{4D}{x + y} \]

5. **Simplifying the Equation**:
- Cancel \( D \) from both sides (assuming \( D \neq 0 \)):
\[ \frac{3}{x - y} = \frac{8}{x + y} \]
- Cross-multiplying gives:
\[ 3(x + y) = 8(x - y) \]

6. **Expanding and Rearranging**:
- Expanding both sides:
\[ 3x + 3y = 8x - 8y \]
- Rearranging the equation:
\[ 3y + 8y = 8x - 3x \]
\[ 11y = 5x \]

7. **Finding the Ratio**:
- Rearranging gives:
\[ \frac{x}{y} = \frac{11}{5} \]
- Therefore, the ratio of the speed of the boat in still water to the speed of the current is:
\[ x : y = 11 : 5 \]

### Final Answer:
The ratio of the speed of the boat in still water to the speed of the current is \( 11 : 5 \).