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. (a)9.5 h
B. (b)7.5 h
C. (c)6.5 h
D. (d)8 h

Answer

To solve the problem, we need to determine the time it takes for a motorboat to cover the same distance in still water, given the time taken to cover the distance downstream and upstream.

### Step-by-Step Solution:

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

2. **Determine Downstream and Upstream Speeds:**
- When moving downstream, the effective speed of the boat is \( V + x \).
- When moving upstream, the effective speed of the boat is \( V - x \).

3. **Set Up Distance Equations:**
- The distance covered downstream in 6 hours can be expressed as:
\[ \text{Distance} = \text{Speed} \times \text{Time} = (V + x) \times 6 \]
- The distance covered upstream in 10 hours can be expressed as:
\[ \text{Distance} = (V - x) \times 10 \]

4. **Equate the Distances:**
- Since the distances are the same, we can set the two expressions equal to each other:
\[ 6(V + x) = 10(V - x) \]

5. **Expand and Rearrange the Equation:**
- Expanding both sides gives:
\[ 6V + 6x = 10V - 10x \]
- Rearranging the equation to isolate terms involving \( V \) and \( x \):
\[ 6V + 6x + 10x = 10V \]
\[ 6V + 16x = 10V \]
- Simplifying further:
\[ 4V = 16x \]
\[ V = 4x \]

6. **Calculate the Distance:**
- Substitute \( V = 4x \) back into the distance equation for downstream:
\[ \text{Distance} = 6(V + x) = 6(4x + x) = 6(5x) = 30x \]

7. **Calculate Time in Still Water:**
- The speed in still water is \( V = 4x \).
- The time taken to cover the same distance in still water is given by:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} = \frac{30x}{4x} = \frac{30}{4} = 7.5 \text{ hours} \]

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

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