A boat can cover 160 km downstream in half of the time in which it can cover the same distance upstream. If in three hours boat can cover 96 km downstream, then find the speed of stream?

Options

A. 4 km/hr
B. 16 km/hr
C. 12 km/hr
D. 8 km/hr

Answer

To solve the problem step by step, we need to find the speed of the stream based on the information given about the boat's movement downstream and upstream.

### Step 1: Define Variables
Let:
- \( x \) = speed of the boat in still water (in km/h)
- \( y \) = speed of the stream (in km/h)

### Step 2: Determine Downstream Speed
From the problem, we know that the boat can cover 96 km downstream in 3 hours. We can calculate the downstream speed:
\[ \text{Speed downstream} = \frac{\text{Distance}}{\text{Time}} = \frac{96 \text{ km}}{3 \text{ hours}} = 32 \text{ km/h} \]
This gives us the equation:
\[ x + y = 32 \quad \text{(1)} \]

### Step 3: Determine Time for Downstream and Upstream
The problem states that the boat can cover 160 km downstream in half the time it takes to cover the same distance upstream.

Let \( t \) be the time taken to cover 160 km upstream. Then, the time taken to cover 160 km downstream is \( \frac{t}{2} \).

Using the formula for time:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \]
For downstream:
\[ \frac{160}{x + y} = \frac{t}{2} \quad \text{(2)} \]
For upstream:
\[ \frac{160}{x - y} = t \quad \text{(3)} \]

### Step 4: Substitute and Solve for \( t \)
From equation (2):
\[ t = \frac{320}{x + y} \quad \text{(4)} \]
From equation (3):
\[ t = \frac{160}{x - y} \quad \text{(5)} \]

### Step 5: Set Equations Equal
Setting equations (4) and (5) equal gives:
\[ \frac{320}{x + y} = \frac{160}{x - y} \]
Cross-multiplying:
\[ 320(x - y) = 160(x + y) \]
Expanding both sides:
\[ 320x - 320y = 160x + 160y \]
Rearranging gives:
\[ 320x - 160x = 320y + 160y \]
\[ 160x = 480y \]
Dividing both sides by 80:
\[ 2x = 6y \quad \text{or} \quad x = 3y \quad \text{(6)} \]

### Step 6: Substitute \( x \) in Equation (1)
Substituting \( x = 3y \) into equation (1):
\[ 3y + y = 32 \]
\[ 4y = 32 \]
\[ y = 8 \text{ km/h} \]

### Step 7: Conclusion
The speed of the stream is \( y = 8 \text{ km/h} \).