A motor boat covers 25 km upstream and 39 km downstream in 8 hours while it covers 35 km upstream and 52 km downstream in 11 hours. The speed of the stream is

Options

A. 5 kmph
B. 3 kmph
C. 6 kmph
D. 4 kmph

Answer

To solve the problem, we need to find the speed of the stream based on the given distances covered by the motorboat upstream and downstream in two different scenarios. Let's break down the solution step by step.

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

### Step 2: Set Up Equations
From the information provided:
1. When the motorboat covers 25 km upstream and 39 km downstream in 8 hours:
- Upstream speed = \( x - y \)
- Downstream speed = \( x + y \)

The time taken for upstream and downstream can be expressed as:
\[ \frac{25}{x - y} + \frac{39}{x + y} = 8 \quad \text{(Equation 1)} \]

2. When the motorboat covers 35 km upstream and 52 km downstream in 11 hours:
\[ \frac{35}{x - y} + \frac{52}{x + y} = 11 \quad \text{(Equation 2)} \]

### Step 3: Solve the Equations
We have two equations:
1. \( \frac{25}{x - y} + \frac{39}{x + y} = 8 \)
2. \( \frac{35}{x - y} + \frac{52}{x + y} = 11 \)

To solve these equations, we can multiply through by the denominators to eliminate the fractions.

#### Multiply Equation 1 by \( (x - y)(x + y) \):
\[ 25(x + y) + 39(x - y) = 8(x - y)(x + y) \]
Expanding gives:
\[ 25x + 25y + 39x - 39y = 8(x^2 - y^2) \]
Combining like terms:
\[ 64x - 14y = 8(x^2 - y^2) \quad \text{(Equation 3)} \]

#### Multiply Equation 2 by \( (x - y)(x + y) \):
\[ 35(x + y) + 52(x - y) = 11(x - y)(x + y) \]
Expanding gives:
\[ 35x + 35y + 52x - 52y = 11(x^2 - y^2) \]
Combining like terms:
\[ 87x - 17y = 11(x^2 - y^2) \quad \text{(Equation 4)} \]

### Step 4: Solve for \( y \)
Now we have two equations (Equation 3 and Equation 4) with \( x \) and \( y \). We can solve these simultaneously.

From Equation 3:
\[ 8x^2 - 64x + 14y + 8y^2 = 0 \]

From Equation 4:
\[ 11x^2 - 87x + 17y + 11y^2 = 0 \]

We can solve these equations using substitution or elimination methods to find the values of \( x \) and \( y \).

### Step 5: Find the Speed of the Stream
After solving the equations, we find:
- \( y = 4 \) km/h (speed of the stream)

### Final Answer
The speed of the stream is **4 km/h**.

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