The following questions are accompanied by two statements (I) and (II). You have to determine which statements(s) is/are sufficient/ necessary to answer the questions.
What will be the speed of boat in still water?
I. Ratio of speed of boat in downstream to that of in upstream is 16:9.
II. Boat covers 80 km in downstream in 2.5 hours,

Options

A. Statement (I) alone is sufficient to answer the question but statement (II) alone is not sufficient to answer the question
B. Statement (II) alone is sufficient to answer the question but statement (I) alone is not sufficient to answer the question.
C. Both the statements taken together are necessary to answer the question, but neither of the statements alone is sufficient to answer the question.
D. Either statement (I) or statement (II) by itself is sufficient to answer the question.

Answer

To determine the speed of the boat in still water, we will analyze the two statements provided and see how they contribute to finding the answer.

### Step 1: Understand the Question
We need to find the speed of the boat in still water. The speed of the boat in still water can be calculated using the formula:
\[ \text{Speed of boat in still water} = \frac{\text{Speed downstream} + \text{Speed upstream}}{2} \]

### Step 2: Analyze Statement I
**Statement I:** The ratio of the speed of the boat in downstream to that of upstream is 16:9.

Let the speed of the boat in still water be \( K \).
- Let the speed of the current be \( C \).
- The speed downstream (SD) can be expressed as \( K + C \).
- The speed upstream (SU) can be expressed as \( K - C \).

From the ratio given:
\[ \frac{K + C}{K - C} = \frac{16}{9} \]
Cross-multiplying gives:
\[ 9(K + C) = 16(K - C) \]
Expanding this:
\[ 9K + 9C = 16K - 16C \]
Rearranging gives:
\[ 7K = 25C \quad \Rightarrow \quad C = \frac{7K}{25} \]

Now substituting \( C \) back into the expressions for downstream and upstream speeds:
\[ SD = K + \frac{7K}{25} = \frac{32K}{25} \]
\[ SU = K - \frac{7K}{25} = \frac{18K}{25} \]

We have the speeds in terms of \( K \), but we cannot find the exact value of \( K \) from this statement alone. Therefore, **Statement I alone is not sufficient.**

### Step 3: Analyze Statement II
**Statement II:** The boat covers 80 km downstream in 2.5 hours.

From this statement, we can find the downstream speed:
\[ \text{Speed downstream} = \frac{\text{Distance}}{\text{Time}} = \frac{80 \text{ km}}{2.5 \text{ hours}} = 32 \text{ km/h} \]

### Step 4: Combine Statements I and II
Now we have:
- From Statement I, we know the ratio of speeds.
- From Statement II, we know the downstream speed is 32 km/h.

Using the downstream speed:
\[ SD = K + C = 32 \text{ km/h} \]

Now, using the expression we derived from Statement I:
\[ SD = \frac{32K}{25} = 32 \]
From this, we can solve for \( K \):
\[ 32K = 32 \times 25 \quad \Rightarrow \quad K = 25 \text{ km/h} \]

### Conclusion
Thus, the speed of the boat in still water is \( 25 \text{ km/h} \).

### Final Answer
Both statements together are necessary to answer the question, but neither statement alone is sufficient.