P and Q are two points that divide the total length of X and Y into 3 equal parts and the ratio of time taken by a boat from X to Y and Y to X is 4 : 5 then find the ratio of Downstream and speed of stream.

Options

A. `5:1`
B. `8:1`
C. `9:1`
D. `10:1`

Answer

To solve the problem, we need to find the ratio of the downstream speed to the speed of the stream given the time taken by a boat to travel downstream and upstream.

### Step-by-Step Solution:

1. **Understanding the Problem**:
- Let the speed of the boat in still water be \( m \) km/h.
- Let the speed of the stream be \( n \) km/h.
- The downstream speed (when the boat is going with the current) is \( m + n \) km/h.
- The upstream speed (when the boat is going against the current) is \( m - n \) km/h.

2. **Setting Up the Time Ratios**:
- The time taken to travel from point X to Y (downstream) is \( t_1 \).
- The time taken to travel from point Y to X (upstream) is \( t_2 \).
- According to the problem, the ratio of these times is given as:
\[ \frac{t_1}{t_2} = \frac{4}{5} \]

3. **Using the Formula for Time**:
- Time can be expressed as distance divided by speed. Let's assume the distance between X and Y is \( d \).
- Therefore, we have:
\[ t_1 = \frac{d}{m+n} \quad \text{(downstream)} \]
\[ t_2 = \frac{d}{m-n} \quad \text{(upstream)} \]

4. **Setting Up the Equation**:
- Using the time ratio:
\[ \frac{t_1}{t_2} = \frac{\frac{d}{m+n}}{\frac{d}{m-n}} = \frac{m-n}{m+n} \]
- Setting this equal to the given ratio:
\[ \frac{m-n}{m+n} = \frac{4}{5} \]

5. **Cross-Multiplying**:
- Cross-multiplying gives us:
\[ 5(m - n) = 4(m + n) \]
- Expanding both sides:
\[ 5m - 5n = 4m + 4n \]

6. **Rearranging the Equation**:
- Bringing all terms involving \( m \) to one side and \( n \) to the other:
\[ 5m - 4m = 4n + 5n \]
\[ m = 9n \]

7. **Finding the Ratio of Downstream Speed to Speed of Stream**:
- The downstream speed is \( m + n = 9n + n = 10n \).
- The speed of the stream is \( n \).
- Therefore, the ratio of downstream speed to speed of stream is:
\[ \frac{m+n}{n} = \frac{10n}{n} = 10 \]
- Thus, the ratio is:
\[ 10 : 1 \]

### Final Answer:
The ratio of downstream speed to the speed of the stream is \( 10 : 1 \).