Find respective ratio of time taken by the boat to travel 96 km upstream and 90 km in still water?
Statement 1: Speed of the boat in still water is 12 km/hr more than the speed of the stream
Statement 2: The boat can go 90 km downstream and 60 km upstream in 10 hr.

Options

A. Statement (1) alone is sufficient to answer the question but statement (2) alone is not sufficient to answer the question.
B. Statement (2) 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. Statements (1) and (2) taken together are not sufficient to answer the question.

Answer

To find the respective ratio of time taken by the boat to travel 96 km upstream and 90 km in still water, we will analyze the information provided in both statements step by step.

### Step 1: Define Variables
Let:
- \( V \) = speed of the stream (in km/hr)
- Speed of the boat in still water = \( V + 12 \) km/hr (from Statement 1)

### Step 2: Determine Speeds
From Statement 1, we know:
- Upstream speed = Speed of the boat - Speed of the stream = \( (V + 12) - V = 12 \) km/hr
- Downstream speed = Speed of the boat + Speed of the stream = \( (V + 12) + V = 2V + 12 \) km/hr

### Step 3: Analyze Statement 2
From Statement 2, we know:
- The boat can go 90 km downstream and 60 km upstream in 10 hours.
- Time taken to travel downstream = \( \frac{90}{2V + 12} \)
- Time taken to travel upstream = \( \frac{60}{12} \)

Setting up the equation:
\[ \frac{90}{2V + 12} + \frac{60}{12} = 10 \]

### Step 4: Simplify the Equation
Calculating the upstream time:
\[ \frac{60}{12} = 5 \text{ hours} \]
Thus, we have:
\[ \frac{90}{2V + 12} + 5 = 10 \]
Subtracting 5 from both sides:
\[ \frac{90}{2V + 12} = 5 \]

### Step 5: Solve for \( V \)
Cross-multiplying gives:
\[ 90 = 5(2V + 12) \]
Expanding:
\[ 90 = 10V + 60 \]
Subtracting 60 from both sides:
\[ 30 = 10V \]
Dividing by 10:
\[ V = 3 \text{ km/hr} \]

### Step 6: Calculate Speeds
Now substituting \( V \) back:
- Speed of the boat in still water = \( 3 + 12 = 15 \) km/hr
- Upstream speed = 12 km/hr
- Downstream speed = \( 2(3) + 12 = 18 \) km/hr

### Step 7: Calculate Time Taken
Now we can calculate the time taken for each journey:
- Time taken to travel 96 km upstream:
\[ \text{Time}_{upstream} = \frac{96}{12} = 8 \text{ hours} \]
- Time taken to travel 90 km in still water:
\[ \text{Time}_{still water} = \frac{90}{15} = 6 \text{ hours} \]

### Step 8: Find the Ratio
The ratio of time taken by the boat to travel 96 km upstream to the time taken to travel 90 km in still water is:
\[ \text{Ratio} = \frac{8}{6} = \frac{4}{3} \]

### Final Answer
The respective ratio of time taken by the boat to travel 96 km upstream and 90 km in still water is \( 4:3 \).

---