A man can row 8 km/hr in still water. When the river is running at 2 km/hr, it takes him 3 hrs 12 mins to go to a place and back. How far is the place?

Options

A. 12km
B. 15km
C. 18km
D. 10km

Answer

To solve the problem, we need to determine the distance to the place based on the rowing speed of the man in still water and the speed of the river current. Here’s a step-by-step breakdown of the solution:

### Step 1: Define Variables
Let the distance to the place be \( x \) kilometers.

### Step 2: Determine Effective Speeds
- The man's speed in still water is 8 km/hr.
- The speed of the river current is 2 km/hr.

When rowing downstream (from A to B), the effective speed of the man is:
\[ \text{Speed downstream} = 8 + 2 = 10 \text{ km/hr} \]

When rowing upstream (from B to A), the effective speed of the man is:
\[ \text{Speed upstream} = 8 - 2 = 6 \text{ km/hr} \]

### Step 3: Calculate Time Taken for Each Leg of the Journey
- Time taken to go downstream:
\[ \text{Time downstream} = \frac{x}{10} \text{ hours} \]

- Time taken to return upstream:
\[ \text{Time upstream} = \frac{x}{6} \text{ hours} \]

### Step 4: Total Time for the Round Trip
The total time for the trip is given as 3 hours and 12 minutes. First, convert 12 minutes into hours:
\[ 12 \text{ minutes} = \frac{12}{60} = \frac{1}{5} \text{ hours} \]
So, total time in hours is:
\[ 3 + \frac{1}{5} = \frac{15}{5} + \frac{1}{5} = \frac{16}{5} \text{ hours} \]

### Step 5: Set Up the Equation
The total time for the round trip can be expressed as:
\[ \frac{x}{10} + \frac{x}{6} = \frac{16}{5} \]

### Step 6: Find a Common Denominator
The least common multiple (LCM) of 10 and 6 is 30. Rewrite the equation:
\[ \frac{3x}{30} + \frac{5x}{30} = \frac{16}{5} \]
Combine the fractions:
\[ \frac{8x}{30} = \frac{16}{5} \]

### Step 7: Cross-Multiply to Solve for \( x \)
Cross-multiply to eliminate the fractions:
\[ 8x \cdot 5 = 16 \cdot 30 \]
\[ 40x = 480 \]

### Step 8: Solve for \( x \)
Divide both sides by 40:
\[ x = \frac{480}{40} = 12 \]

### Conclusion
The distance to the place is \( 12 \) kilometers.

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