A boat covers 48 km upstream and 72 km downstream in 12 hours, while it covers 72 km upstream and 48 km downstream in 13 hours. The speed of stream is :

Options

A. 2 km/h
B. 2.2 km/h
C. 2.5 km/h
D. 4 km/h

Answer

To solve the problem, we need to determine the speed of the stream based on the information given about the boat's travel times upstream and downstream. Let's break it down step by step.

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

### Step 2: Set Up Equations
From the problem, we have two scenarios:

1. **First Scenario**: The boat covers 48 km upstream and 72 km downstream in 12 hours.
- Upstream speed = \( x - y \)
- Downstream speed = \( x + y \)

The equation for the first scenario can be written as:
\[ \frac{48}{x - y} + \frac{72}{x + y} = 12 \]

2. **Second Scenario**: The boat covers 72 km upstream and 48 km downstream in 13 hours.
- The equation for the second scenario can be written as:
\[ \frac{72}{x - y} + \frac{48}{x + y} = 13 \]

### Step 3: Simplify the Equations
Now we will simplify these equations.

**Equation 1**:
\[ \frac{48}{x - y} + \frac{72}{x + y} = 12 \]
Multiplying through by \( (x - y)(x + y) \):
\[ 48(x + y) + 72(x - y) = 12(x^2 - y^2) \]
Expanding:
\[ 48x + 48y + 72x - 72y = 12x^2 - 12y^2 \]
Combining like terms:
\[ 120x - 24y = 12x^2 - 12y^2 \quad \text{(Equation 1)} \]

**Equation 2**:
\[ \frac{72}{x - y} + \frac{48}{x + y} = 13 \]
Multiplying through by \( (x - y)(x + y) \):
\[ 72(x + y) + 48(x - y) = 13(x^2 - y^2) \]
Expanding:
\[ 72x + 72y + 48x - 48y = 13x^2 - 13y^2 \]
Combining like terms:
\[ 120x + 24y = 13x^2 - 13y^2 \quad \text{(Equation 2)} \]

### Step 4: Solve the System of Equations
Now we have a system of two equations:
1. \( 12x^2 - 12y^2 - 120x + 24y = 0 \)
2. \( 13x^2 - 13y^2 - 120x - 24y = 0 \)

To solve this, we can use substitution or elimination.

### Step 5: Assume Values
To simplify, we can assume values for \( x + y \) and \( x - y \). Let's assume:
- \( x + y = 12 \)
- \( x - y = 8 \)

### Step 6: Solve for \( x \) and \( y \)
Now we can solve these two equations:
1. \( x + y = 12 \)
2. \( x - y = 8 \)

Adding these two equations:
\[ 2x = 20 \implies x = 10 \]

Subtracting the second from the first:
\[ 2y = 4 \implies y = 2 \]

### Conclusion
Thus, the speed of the stream \( y \) is 2 km/h.

### Final Answer
The speed of the stream is **2 km/h**.

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