A boat goes 4 km upstream and 4 km downstream in 1 hour. The same boat goes 5 km downstream and 3 km upstream in 55 minutes. What is the speed (in km/hr) of boat in still water?

Options

A. `6.5`
B. `7.75`
C. `9`
D. `10.5`

Answer

To solve the problem, we need to find the speed of the boat in still water (x) and the speed of the current (y). We can set up equations based on the information given in the problem.

### Step-by-Step Solution:

1. **Define Variables**:
- Let the speed of the boat in still water be \( x \) km/hr.
- Let the speed of the current be \( y \) km/hr.

2. **Set Up Equations from the First Scenario**:
- The boat goes 4 km upstream and 4 km downstream in 1 hour.
- The speed upstream is \( x - y \) and downstream is \( x + y \).
- The time taken to go upstream is \( \frac{4}{x - y} \) and downstream is \( \frac{4}{x + y} \).
- Therefore, we can write the equation:
\[ \frac{4}{x - y} + \frac{4}{x + y} = 1 \]
- Simplifying this, we multiply through by \( (x - y)(x + y) \):
\[ 4(x + y) + 4(x - y) = (x - y)(x + y) \]
\[ 4x + 4y + 4x - 4y = x^2 - y^2 \]
\[ 8x = x^2 - y^2 \quad \text{(Equation 1)} \]

3. **Set Up Equations from the Second Scenario**:
- The boat goes 5 km downstream and 3 km upstream in 55 minutes (which is \( \frac{55}{60} = \frac{11}{12} \) hours).
- The time taken to go downstream is \( \frac{5}{x + y} \) and upstream is \( \frac{3}{x - y} \).
- Therefore, we can write the equation:
\[ \frac{5}{x + y} + \frac{3}{x - y} = \frac{11}{12} \]
- Simplifying this, we multiply through by \( (x + y)(x - y) \):
\[ 5(x - y) + 3(x + y) = \frac{11}{12}(x^2 - y^2) \]
\[ 5x - 5y + 3x + 3y = \frac{11}{12}(x^2 - y^2) \]
\[ 8x - 2y = \frac{11}{12}(x^2 - y^2) \quad \text{(Equation 2)} \]

4. **Solve the Equations**:
- From Equation 1: \( x^2 - 8x - y^2 = 0 \)
- From Equation 2: Rearranging gives:
\[ 96x - 24y = 11(x^2 - y^2) \]
\[ 11x^2 - 96x + 11y^2 + 24y = 0 \]

5. **Substituting \( y^2 \) from Equation 1 into Equation 2**:
- Substitute \( y^2 = x^2 - 8x \) into the second equation and solve for \( x \).

6. **Final Calculation**:
- After solving the equations, we find \( x = 9 \) km/hr and \( y = 3 \) km/hr.

### Conclusion:
The speed of the boat in still water is \( \boxed{9} \) km/hr.