A boat's man goes 48 km downstream in 8 hours and returns back in 12 hours. Find the speed of the boat in still water and the rate of the stream -----

Options

A. 5 km/hr, 1 km/hr
B. 10 km/hr, 2 km/hr
C. 6 km/hr, 1.5 km/hr
D. None of these

Answer

To solve the problem, we need to find the speed of the boat in still water and the rate of the stream based on the given information about downstream and upstream travel times.

### Step-by-Step Solution:

1. **Calculate Downstream Speed:**
- The boat travels 48 km downstream in 8 hours.
- Speed downstream = Distance / Time = 48 km / 8 hours = 6 km/h.

**Hint:** Remember that speed is calculated as distance divided by time.

2. **Calculate Upstream Speed:**
- The boat returns upstream covering the same distance of 48 km in 12 hours.
- Speed upstream = Distance / Time = 48 km / 12 hours = 4 km/h.

**Hint:** Use the same formula for speed to find the upstream speed.

3. **Let the Speed of the Boat in Still Water be 'b' and the Rate of the Stream be 's':**
- The downstream speed can be expressed as:
\[ b + s = 6 \] (1)
- The upstream speed can be expressed as:
\[ b - s = 4 \] (2)

**Hint:** Set up equations based on the relationship between the boat's speed, the stream's speed, and the calculated speeds.

4. **Solve the Equations:**
- Add equations (1) and (2):
\[ (b + s) + (b - s) = 6 + 4 \]
\[ 2b = 10 \implies b = 5 \text{ km/h} \]

**Hint:** Adding the two equations eliminates the stream speed, allowing you to solve for the boat speed.

5. **Find the Rate of the Stream:**
- Substitute \( b = 5 \) into equation (1):
\[ 5 + s = 6 \implies s = 6 - 5 = 1 \text{ km/h} \]

**Hint:** Use the value of 'b' to find 's' by substituting back into one of the original equations.

### Final Answers:
- Speed of the boat in still water (b) = 5 km/h
- Rate of the stream (s) = 1 km/h