A boat goes 24 km upstream and 28 km downstream in 6 hours. It goes 30km upstream and 21 km downstream in 6 hours and 30 minutes. The speed of the boat in still water is :

Options

A. 10 km/h
B. 4 km/h
C. 14 km/h
D. 6 km/h

Answer

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

### Step-by-Step Solution:

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

2. **Determine Speeds:**
- The speed of the boat upstream = \( x - y \)
- The speed of the boat downstream = \( x + y \)

3. **Set Up the First Equation:**
- From the first part of the question, the boat goes 24 km upstream and 28 km downstream in 6 hours.
- The time taken to go upstream = \( \frac{24}{x - y} \)
- The time taken to go downstream = \( \frac{28}{x + y} \)
- Therefore, we can write the equation:
\[ \frac{24}{x - y} + \frac{28}{x + y} = 6 \]

4. **Set Up the Second Equation:**
- From the second part of the question, the boat goes 30 km upstream and 21 km downstream in 6 hours and 30 minutes (which is 6.5 hours).
- The time taken to go upstream = \( \frac{30}{x - y} \)
- The time taken to go downstream = \( \frac{21}{x + y} \)
- Therefore, we can write the second equation:
\[ \frac{30}{x - y} + \frac{21}{x + y} = 6.5 \]

5. **Substituting Values:**
- To solve these equations, we can use a hit-and-trial method or substitution. Let's assume \( x + y = 14 \) and \( x - y = 6 \) based on the hint from the video.
- From \( x + y = 14 \) and \( x - y = 6 \), we can add these two equations:
\[ 2x = 20 \implies x = 10 \text{ km/h} \]
- Now, substituting \( x = 10 \) into \( x + y = 14 \):
\[ 10 + y = 14 \implies y = 4 \text{ km/h} \]

6. **Conclusion:**
- The speed of the boat in still water is \( x = 10 \) km/h.