A man rowed against a stream flowing `1 (1)/(2)` km/hr to a certain point and then turned back, stopping 2 km, short of the place where he originally started. If the whole time occupied in rowing be 2 hr 10 min and his uniform speed in still water be `4(1)/(2)` km/hr. Find the distance the man went up the stream.

Options

A. 7 km
B. 3.5km
C. 4 km
D. 5 km

Answer

To solve the problem step by step, we need to break down the information given and use the relevant formulas.

### Step 1: Identify the given information
- Speed of the stream (S) = 1.5 km/hr
- Speed of the man in still water (M) = 4.5 km/hr
- Total time taken = 2 hours 10 minutes = \(2 + \frac{10}{60} = \frac{13}{6}\) hours
- The man stopped 2 km short of his starting point on the way back.

### Step 2: Calculate the effective speeds
- **Upstream speed (U)** = Speed of man - Speed of stream = \(M - S = 4.5 - 1.5 = 3\) km/hr
- **Downstream speed (D)** = Speed of man + Speed of stream = \(M + S = 4.5 + 1.5 = 6\) km/hr

### Step 3: Define the distance traveled upstream
Let the distance traveled upstream be \(x\) km. Therefore, the distance traveled downstream (on the way back) will be \(x - 2\) km (since he stopped 2 km short).

### Step 4: Write the time equations
Using the formula for time, which is \( \text{Time} = \frac{\text{Distance}}{\text{Speed}} \):

- Time taken to go upstream = \( \frac{x}{3} \) hours
- Time taken to come downstream = \( \frac{x - 2}{6} \) hours

### Step 5: Set up the equation for total time
The total time taken for the journey is given as \( \frac{13}{6} \) hours. Therefore, we can set up the equation:

\[ \frac{x}{3} + \frac{x - 2}{6} = \frac{13}{6} \]

### Step 6: Solve the equation
To solve the equation, we first find a common denominator, which is 6:

\[ \frac{2x}{6} + \frac{x - 2}{6} = \frac{13}{6} \]

Combining the fractions on the left side:

\[ \frac{2x + x - 2}{6} = \frac{13}{6} \]

This simplifies to:

\[ \frac{3x - 2}{6} = \frac{13}{6} \]

Now, we can eliminate the denominators by multiplying both sides by 6:

\[ 3x - 2 = 13 \]

Adding 2 to both sides gives:

\[ 3x = 15 \]

Dividing both sides by 3 results in:

\[ x = 5 \]

### Conclusion
The distance the man went upstream is **5 km**.