A boat can go 30 km downstream and 24 km upstream in 2 hours 27 minutes. Also, it can go 20 km downstream and 8 km upstream in 74 minutes. What is the speed of the boat in still water in km/h ?

Options

A. 18
B. 20
C. 24
D. 22

Answer

To solve the problem of finding the speed of the boat in still water, we can break it down into a series of steps.

### Step-by-Step Solution:

1. **Convert Time to Hours**:
- The first scenario gives us 2 hours and 27 minutes. We convert this to hours:
\[ 2 \text{ hours} + \frac{27 \text{ minutes}}{60} = 2 + 0.45 = 2.45 \text{ hours} \]
- The second scenario gives us 74 minutes. We convert this to hours:
\[ \frac{74 \text{ minutes}}{60} = 1.2333 \text{ hours} \text{ (approximately)} \]

2. **Set Up Equations**:
- Let \( A \) be the speed of the boat in still water (km/h) and \( B \) be the speed of the current (km/h).
- From the first scenario (30 km downstream and 24 km upstream):
\[ \frac{30}{A + B} + \frac{24}{A - B} = 2.45 \]
- From the second scenario (20 km downstream and 8 km upstream):
\[ \frac{20}{A + B} + \frac{8}{A - B} = 1.2333 \]

3. **Clear the Denominators**:
- Multiply through by the denominators to eliminate fractions:
- For the first equation:
\[ 30(A - B) + 24(A + B) = 2.45(A + B)(A - B) \]
- For the second equation:
\[ 20(A - B) + 8(A + B) = 1.2333(A + B)(A - B) \]

4. **Expand and Rearrange**:
- Expand both equations and rearrange them to isolate terms involving \( A \) and \( B \).

5. **Solve the System of Equations**:
- You will end up with two equations in terms of \( A \) and \( B \). Solve these equations simultaneously to find the values of \( A \) and \( B \).

6. **Calculate the Speed of the Boat**:
- Once you have the values of \( A \) and \( B \), the speed of the boat in still water is simply \( A \).