A motorboat travelling at the some speed, can cover 25 km upstream and 39 km downstream in 8 h. At the same speed, it can travel 35 km upstream and 52 km downstream in 11 h. The speed of the stream is

Options

A. 2 km/h
B. 3 km/h
C. 4 km/h
D. 5 km/h

Answer

To solve the problem, we need to find the speed of the stream. Let's denote:

- Speed of the motorboat = \( x \) km/h
- Speed of the stream = \( y \) km/h

### Step 1: Set up the equations based on the given information

From the problem, we have two scenarios:

1. **First Scenario**: The motorboat travels 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:
\[ \frac{25}{x - y} + \frac{39}{x + y} = 8 \quad \text{(1)} \]

2. **Second Scenario**: The motorboat travels 35 km upstream and 52 km downstream in 11 hours.
- The time taken for this scenario can be expressed as:
\[ \frac{35}{x - y} + \frac{52}{x + y} = 11 \quad \text{(2)} \]

### Step 2: Solve the equations

We have two equations (1) and (2):

1. \(\frac{25}{x - y} + \frac{39}{x + y} = 8\)
2. \(\frac{35}{x - y} + \frac{52}{x + y} = 11\)

Let's denote \( a = x - y \) and \( b = x + y \). Then we can rewrite the equations as:

1. \(\frac{25}{a} + \frac{39}{b} = 8\)
2. \(\frac{35}{a} + \frac{52}{b} = 11\)

### Step 3: Solve for \( a \) and \( b \)

From the first equation, we can express \( b \) in terms of \( a \):
\[ \frac{39}{b} = 8 - \frac{25}{a} \implies b = \frac{39a}{8a - 25} \]

Substituting \( b \) in the second equation:
\[ \frac{35}{a} + \frac{52(8a - 25)}{39a} = 11 \]

Multiplying through by \( 39a \) to eliminate the denominators:
\[ 35 \cdot 39 + 52(8a - 25) = 11 \cdot 39a \]
\[ 1365 + 416a - 1300 = 429a \]
\[ 65 = 429a - 416a \]
\[ 65 = 13a \implies a = 5 \]

Now substituting \( a = 5 \) back to find \( b \):
\[ b = \frac{39 \cdot 5}{8 \cdot 5 - 25} = \frac{195}{40 - 25} = \frac{195}{15} = 13 \]

### Step 4: Find \( x \) and \( y \)

Now we have:
- \( a = x - y = 5 \)
- \( b = x + y = 13 \)

Adding these two equations:
\[ (x - y) + (x + y) = 5 + 13 \implies 2x = 18 \implies x = 9 \]

Subtracting the first from the second:
\[ (x + y) - (x - y) = 13 - 5 \implies 2y = 8 \implies y = 4 \]

### Conclusion

The speed of the stream \( y \) is \( 4 \) km/h.