A motorboat covers a given distance in `6h` moving downstream on a river. It covers the same distance in `10h` moving upstream. The time it takes to cover the same distance in still water is

Options

A. `9h`
B. `7.5h`
C. `6.5h`
D. `8h`

Explanation

To solve the problem, we need to find the time it takes for a motorboat to cover a certain distance in still water, given that it takes 6 hours to cover the same distance downstream and 10 hours upstream.

### Step-by-Step Solution

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

2. **Distance Calculation:**
- The distance covered downstream in 6 hours can be expressed as:
\[ \text{Distance} = (x + y) \times 6 \]
- The distance covered upstream in 10 hours can be expressed as:
\[ \text{Distance} = (x - y) \times 10 \]

3. **Set the Distances Equal:**
- Since both expressions represent the same distance, we can set them equal to each other:
\[ (x + y) \times 6 = (x - y) \times 10 \]

4. **Expand and Rearrange the Equation:**
- Expanding both sides gives:
\[ 6x + 6y = 10x - 10y \]
- Rearranging the terms:
\[ 6x + 6y + 10y = 10x \]
\[ 6x + 16y = 10x \]
\[ 16y = 10x - 6x \]
\[ 16y = 4x \]
- Dividing both sides by 4:
\[ y = \frac{x}{4} \]

5. **Calculate Time in Still Water:**
- The speed of the motorboat in still water is \( x \).
- The distance can also be expressed in terms of speed and time:
\[ \text{Distance} = x \times t \]
- We can use the distance from the downstream equation:
\[ \text{Distance} = (x + y) \times 6 \]
- Substitute \( y = \frac{x}{4} \):
\[ \text{Distance} = \left(x + \frac{x}{4}\right) \times 6 \]
\[ = \left(\frac{4x}{4} + \frac{x}{4}\right) \times 6 \]
\[ = \left(\frac{5x}{4}\right) \times 6 \]
\[ = \frac{30x}{4} = \frac{15x}{2} \]

6. **Set Distances Equal:**
- Now, equate the two expressions for distance:
\[ x \times t = \frac{15x}{2} \]
- Dividing both sides by \( x \) (assuming \( x \neq 0 \)):
\[ t = \frac{15}{2} = 7.5 \text{ hours} \]

### Final Answer:
The time it takes to cover the same distance in still water is **7.5 hours**.