Given below in each question there are two statements (I) and (II). You must determine, which statement is enough to give the answer of question. Also, there are five alternatives given, you have to choose one alternative as your answer of the questions:
What is the speed of boat in still water when the upstream speed of boat is equal to the speed of stream?
I. Time required to cover certain distance upstream is 24 sec.
II. Time required to cover certain distance downstream is 8 sec.

Options

A. A)only statement I
B. B)Only statement II
C. C)Both I and II together
D. D)Both together are
not sufficient

Answer

To solve the problem, we need to determine the speed of the boat in still water when the upstream speed of the boat is equal to the speed of the stream. We will analyze both statements provided to see if either one or both together can give us the required information.

### Step-by-Step Solution:

1. **Understanding the Problem:**
- Let the speed of the boat in still water be \( V \).
- Let the speed of the stream (current) be \( S \).
- According to the problem, the upstream speed of the boat is equal to the speed of the stream, which means \( V - S = S \) or \( V = 2S \).

2. **Analyzing Statement I:**
- Statement I states that the time required to cover a certain distance upstream is 24 seconds.
- The upstream speed is \( V - S = S \) (from the problem statement).
- Therefore, the distance \( d \) can be expressed as:
\[ d = \text{upstream speed} \times \text{time} = S \times 24 \]
- This gives us \( d = 24S \).
- However, we cannot determine \( V \) (the speed of the boat in still water) from this statement alone because we have two variables \( V \) and \( S \) and only one equation.

3. **Analyzing Statement II:**
- Statement II states that the time required to cover a certain distance downstream is 8 seconds.
- The downstream speed is \( V + S \).
- Therefore, the distance \( d \) can be expressed as:
\[ d = \text{downstream speed} \times \text{time} = (V + S) \times 8 \]
- This gives us \( d = 8(V + S) \).
- Again, we have two variables \( V \) and \( S \) and only one equation, so we cannot determine \( V \) from this statement alone.

4. **Combining Both Statements:**
- From Statement I, we have \( d = 24S \).
- From Statement II, we have \( d = 8(V + S) \).
- Setting these two equations equal gives us:
\[ 24S = 8(V + S) \]
- Simplifying this:
\[ 24S = 8V + 8S \]
\[ 16S = 8V \]
\[ V = 2S \]
- This confirms that we can find \( V \) using both statements together.

### Conclusion:
- **Statement I alone is not sufficient.**
- **Statement II alone is not sufficient.**
- **Both statements together are sufficient to determine the speed of the boat in still water.**

### Final Answer:
The correct option is **D: Both statements together are not sufficient.**