A boat can go 30 km downstream and 24 km upstream in 2 hours 27 minutes. Also, it can go 10 km downstream and 4 km upstream in 37 minutes. What is the speed of the boat upstream (in km/h)?

Options

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

Answer

To solve the problem, we will use the information given about the boat's travel downstream and upstream to find the speed of the boat upstream.

### Step-by-Step Solution:

1. **Convert Time to Hours:**
- The total time for the first scenario is 2 hours and 27 minutes.
- Convert 27 minutes to hours: \( \frac{27}{60} = 0.45 \) hours.
- Therefore, total time = \( 2 + 0.45 = 2.45 \) hours.

2. **Set Up the First Equation:**
- Let \( a \) be the speed of the boat in still water (in km/h) and \( b \) be the speed of the current (in km/h).
- The speed downstream = \( a + b \) and the speed upstream = \( a - b \).
- For the first scenario:
- Distance downstream = 30 km, Distance upstream = 24 km.
- Time taken downstream = \( \frac{30}{a + b} \) and time taken upstream = \( \frac{24}{a - b} \).
- Therefore, we can write the equation:
\[ \frac{30}{a + b} + \frac{24}{a - b} = 2.45 \quad \text{(Equation 1)} \]

3. **Convert Time for the Second Scenario:**
- The total time for the second scenario is 37 minutes.
- Convert 37 minutes to hours: \( \frac{37}{60} \approx 0.6167 \) hours.

4. **Set Up the Second Equation:**
- For the second scenario:
- Distance downstream = 10 km, Distance upstream = 4 km.
- Time taken downstream = \( \frac{10}{a + b} \) and time taken upstream = \( \frac{4}{a - b} \).
- Therefore, we can write the equation:
\[ \frac{10}{a + b} + \frac{4}{a - b} = 0.6167 \quad \text{(Equation 2)} \]

5. **Multiply Equation 1 by a Common Denominator:**
- Multiply through by \( (a + b)(a - b) \):
\[ 30(a - b) + 24(a + b) = 2.45(a + b)(a - b) \]
- Simplifying gives:
\[ 30a - 30b + 24a + 24b = 2.45(a^2 - b^2) \]
- Combine like terms:
\[ 54a - 6b = 2.45(a^2 - b^2) \quad \text{(Equation 3)} \]

6. **Multiply Equation 2 by a Common Denominator:**
- Multiply through by \( (a + b)(a - b) \):
\[ 10(a - b) + 4(a + b) = 0.6167(a + b)(a - b) \]
- Simplifying gives:
\[ 10a - 10b + 4a + 4b = 0.6167(a^2 - b^2) \]
- Combine like terms:
\[ 14a - 6b = 0.6167(a^2 - b^2) \quad \text{(Equation 4)} \]

7. **Solve the System of Equations:**
- Now we have two equations (Equation 3 and Equation 4) that can be solved simultaneously.
- Substituting values or using elimination methods will yield the values of \( a \) and \( b \).

8. **Find the Speed Upstream:**
- Once we have \( a \) and \( b \), the speed of the boat upstream is calculated as \( a - b \).

### Final Calculation:
After solving the equations, we find:
- \( a - b = 20 \) km/h (speed of the boat upstream).