Ratio of speed of a boat in upstream to in downstream is 11:14. If boat travel 22 km in upstream and 28 km downstream in 2 hour, then find the speed of stream is what percent of speed of boat in downstream.

Options

A. `10(5)/(7)%`
B. `12 %`
C. `13(7)/(11)%`
D. `13 %`

Answer

To solve the problem step-by-step, we will follow these steps:

### Step 1: Understand the Given Ratios
We are given the ratio of the speed of the boat in upstream to downstream as 11:14. Let's denote:
- Speed of the boat in still water = \( x \) km/h
- Speed of the stream = \( y \) km/h

From the given ratio:
- Speed in upstream = \( x - y \)
- Speed in downstream = \( x + y \)

### Step 2: Set Up the Ratio
From the ratio provided:
\[ \frac{x - y}{x + y} = \frac{11}{14} \]

### Step 3: Cross-Multiply to Solve for x and y
Cross-multiplying gives us:
\[ 14(x - y) = 11(x + y) \]
Expanding both sides:
\[ 14x - 14y = 11x + 11y \]
Rearranging the equation:
\[ 14x - 11x = 14y + 11y \]
\[ 3x = 25y \]
This simplifies to:
\[ \frac{x}{y} = \frac{25}{3} \]

### Step 4: Calculate the Time Taken for Upstream and Downstream
We know the boat travels 22 km upstream and 28 km downstream in 2 hours. Let’s denote:
- Time taken upstream = \( t_1 \)
- Time taken downstream = \( t_2 \)

From the distance formula:
\[ t_1 = \frac{22}{x - y} \]
\[ t_2 = \frac{28}{x + y} \]
Given that \( t_1 + t_2 = 2 \) hours:
\[ \frac{22}{x - y} + \frac{28}{x + y} = 2 \]

### Step 5: Substitute x and y
Using \( x = \frac{25}{3}y \), we can substitute \( x \) in the equation:
\[ x - y = \frac{25}{3}y - y = \frac{22}{3}y \]
\[ x + y = \frac{25}{3}y + y = \frac{28}{3}y \]

Now substituting these into the time equation:
\[ \frac{22}{\frac{22}{3}y} + \frac{28}{\frac{28}{3}y} = 2 \]
This simplifies to:
\[ \frac{22 \cdot 3}{22y} + \frac{28 \cdot 3}{28y} = 2 \]
\[ \frac{3}{y} + \frac{3}{y} = 2 \]
\[ \frac{6}{y} = 2 \]
Thus, solving for \( y \):
\[ y = 3 \text{ km/h} \]

### Step 6: Find the Speed of the Boat in Downstream
Now substituting \( y \) back to find \( x \):
\[ x = \frac{25}{3} \cdot 3 = 25 \text{ km/h} \]
Thus, the speed of the boat in downstream:
\[ x + y = 25 + 3 = 28 \text{ km/h} \]

### Step 7: Find the Speed of Stream as a Percentage of Speed of Boat in Downstream
Now, we need to find what percent the speed of the stream \( y \) is of the speed of the boat in downstream:
\[ \text{Percentage} = \left(\frac{y}{x + y}\right) \times 100 = \left(\frac{3}{28}\right) \times 100 \]
Calculating this gives:
\[ \text{Percentage} = \frac{300}{28} \approx 10.71\% \]

### Final Answer
The speed of the stream is approximately \( 10.71\% \) of the speed of the boat in downstream.