A boat covers 45 km distance in downstream and 16 km distance in upstream, takes total 9 hours. If the same boat covers 18 km distance in downstream in as much time, it covers 8 km distance in upstream. Then find the speed of boat in still water ?

Options

A. 5.5 km/hr
B. 7.5 km/hr
C. 6.5 km/hr
D. 9.5 km/hr

Answer

To solve the problem, we need to find the speed of the boat in still water (S) and the speed of the current (C).

### Step-by-Step Solution:

1. **Define Variables:**
- Let the speed of the boat in still water be \( S \) km/h.
- Let the speed of the current be \( C \) km/h.

2. **Set Up Equations from Given Information:**
- For downstream, the speed of the boat is \( S + C \).
- For upstream, the speed of the boat is \( S - C \).

3. **Use the First Condition:**
- The boat covers 45 km downstream and 16 km upstream in a total of 9 hours.
- The time taken to cover downstream is \( \frac{45}{S + C} \) hours.
- The time taken to cover upstream is \( \frac{16}{S - C} \) hours.
- Therefore, we can write the equation:
\[ \frac{45}{S + C} + \frac{16}{S - C} = 9 \]

4. **Use the Second Condition:**
- The boat covers 18 km downstream and 8 km upstream in the same amount of time.
- The time taken to cover downstream is \( \frac{18}{S + C} \) hours.
- The time taken to cover upstream is \( \frac{8}{S - C} \) hours.
- Therefore, we can write the equation:
\[ \frac{18}{S + C} = \frac{8}{S - C} \]

5. **Cross Multiply the Second Equation:**
- From \( \frac{18}{S + C} = \frac{8}{S - C} \), we get:
\[ 18(S - C) = 8(S + C) \]
- Expanding gives:
\[ 18S - 18C = 8S + 8C \]
- Rearranging gives:
\[ 10S = 26C \quad \Rightarrow \quad \frac{S}{C} = \frac{26}{10} = \frac{13}{5} \]

6. **Substitute \( S \) in Terms of \( C \):**
- Let \( S = \frac{13}{5}C \).

7. **Substitute \( S \) in the First Equation:**
- Substitute \( S \) in the first equation:
\[ \frac{45}{\frac{13}{5}C + C} + \frac{16}{\frac{13}{5}C - C} = 9 \]
- Simplifying gives:
\[ \frac{45}{\frac{18}{5}C} + \frac{16}{\frac{8}{5}C} = 9 \]
- This simplifies to:
\[ \frac{45 \cdot 5}{18C} + \frac{16 \cdot 5}{8C} = 9 \]
- Further simplifying gives:
\[ \frac{225}{18C} + \frac{80}{8C} = 9 \]
- Which simplifies to:
\[ \frac{225}{18C} + \frac{10}{C} = 9 \]

8. **Combine the Terms:**
- Combine the fractions:
\[ \frac{225 + 180}{18C} = 9 \]
- This leads to:
\[ \frac{405}{18C} = 9 \]
- Cross-multiplying gives:
\[ 405 = 162C \quad \Rightarrow \quad C = \frac{405}{162} = \frac{45}{18} = 2.5 \text{ km/h} \]

9. **Find \( S \):**
- Substitute \( C \) back into \( S = \frac{13}{5}C \):
\[ S = \frac{13}{5} \times 2.5 = 6.5 \text{ km/h} \]

### Final Answer:
The speed of the boat in still water is **6.5 km/h**.