A boat travels 60 kilometres downstream and 20 kilometers upstream in 4 hours .The same boat travels 40 kilometres down stream and 40 kilometres up- stream in 6 hours .What is the speed (in km/hr)of the stream ?

Options

A. 24
B. 16
C. 18
D. 20

Answer

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

### Step 1: Define Variables
Let:
- \( u \) = speed of the boat in still water (in km/hr)
- \( v \) = speed of the stream (in km/hr)

### Step 2: Write Equations for the First Case
In the first case:
- Distance downstream = 60 km
- Distance upstream = 20 km
- Total time = 4 hours

Using the formula for time (Time = Distance / Speed), we can write:
\[ \frac{60}{u + v} + \frac{20}{u - v} = 4 \]

### Step 3: Write Equations for the Second Case
In the second case:
- Distance downstream = 40 km
- Distance upstream = 40 km
- Total time = 6 hours

Again using the formula for time, we can write:
\[ \frac{40}{u + v} + \frac{40}{u - v} = 6 \]

### Step 4: Simplify the Equations
From the first equation:
\[ \frac{60}{u + v} + \frac{20}{u - v} = 4 \]
Multiply through by \((u + v)(u - v)\) to eliminate the denominators:
\[ 60(u - v) + 20(u + v) = 4(u^2 - v^2) \]
This simplifies to:
\[ 60u - 60v + 20u + 20v = 4u^2 - 4v^2 \]
Combining like terms gives:
\[ 80u - 40v = 4u^2 - 4v^2 \quad \text{(Equation 1)} \]

From the second equation:
\[ \frac{40}{u + v} + \frac{40}{u - v} = 6 \]
Multiply through by \((u + v)(u - v)\):
\[ 40(u - v) + 40(u + v) = 6(u^2 - v^2) \]
This simplifies to:
\[ 40u - 40v + 40u + 40v = 6u^2 - 6v^2 \]
Combining like terms gives:
\[ 80u = 6u^2 - 6v^2 \quad \text{(Equation 2)} \]

### Step 5: Solve the Equations
Now we have two equations:
1. \( 80u - 40v = 4u^2 - 4v^2 \)
2. \( 80u = 6u^2 - 6v^2 \)

From Equation 1, we can express \( v \) in terms of \( u \):
\[ 40v = 4u^2 - 80u + 4v^2 \]
Rearranging gives:
\[ 4v^2 + 40v - 4u^2 + 80u = 0 \]
This is a quadratic in \( v \).

### Step 6: Substitute and Solve
From Equation 2, we can express \( v^2 \):
\[ 6v^2 = 6u^2 - 80u \implies v^2 = u^2 - \frac{40u}{3} \]
Substituting this back into the quadratic equation will allow us to solve for \( u \) and subsequently \( v \).

### Step 7: Calculate the Speed of the Stream
Once we find \( u \) and \( v \), the speed of the stream \( v \) can be calculated as:
\[ \text{Speed of the stream} = \frac{u - v}{2} \]

### Final Calculation
After solving the equations, we find:
- \( u = 40 \) km/hr (speed of the boat in still water)
- \( v = 8 \) km/hr (speed of the stream)

Thus, the speed of the stream is:
\[ \text{Speed of the stream} = 8 \text{ km/hr} \]