A motor boat went downstream for 120 km and immediately returned. It took the boat 15 hours to complete the round trip. If the speed of the river were twice as high, the trip downstream and back would take 24 hours.
What is the speed of the stream?

Options

A. 3.5 km/h
B. 4 km/h
C. 6 km/h
D. 8 km/h

Answer

To solve the problem, we need to determine the speed of the stream (y) given the conditions of the motorboat's journey downstream and upstream.

### Step-by-Step Solution:

1. **Define Variables:**
- Let the speed of the motorboat in still water be \( x \) km/h.
- Let the speed of the stream be \( y \) km/h.

2. **Set Up the First Equation:**
- The boat travels downstream for 120 km and then returns upstream for 120 km. The total time for the round trip is 15 hours.
- The time taken to go downstream is \( \frac{120}{x + y} \) and the time taken to return upstream is \( \frac{120}{x - y} \).
- Therefore, we can write the equation:
\[ \frac{120}{x + y} + \frac{120}{x - y} = 15 \]
- Multiplying through by \( (x + y)(x - y) \) to eliminate the denominators gives:
\[ 120(x - y) + 120(x + y) = 15(x^2 - y^2) \]
- Simplifying this results in:
\[ 240x = 15(x^2 - y^2) \]
- Dividing through by 15, we obtain:
\[ 16x = x^2 - y^2 \quad \text{(Equation 1)} \]

3. **Set Up the Second Equation:**
- If the speed of the river doubles, the new speed of the stream is \( 2y \).
- The total time for the round trip now becomes 24 hours.
- The time taken to go downstream is \( \frac{120}{x + 2y} \) and the time taken to return upstream is \( \frac{120}{x - 2y} \).
- Thus, we can write:
\[ \frac{120}{x + 2y} + \frac{120}{x - 2y} = 24 \]
- Multiplying through by \( (x + 2y)(x - 2y) \) gives:
\[ 120(x - 2y) + 120(x + 2y) = 24(x^2 - 4y^2) \]
- Simplifying this results in:
\[ 240x = 24(x^2 - 4y^2) \]
- Dividing through by 24, we obtain:
\[ 10x = x^2 - 4y^2 \quad \text{(Equation 2)} \]

4. **Solve the Equations:**
- Now we have two equations:
- \( x^2 - y^2 = 16x \) (Equation 1)
- \( x^2 - 4y^2 = 10x \) (Equation 2)
- From Equation 1, we can express \( y^2 \):
\[ y^2 = x^2 - 16x \]
- Substitute \( y^2 \) into Equation 2:
\[ x^2 - 4(x^2 - 16x) = 10x \]
- Simplifying gives:
\[ x^2 - 4x^2 + 64x = 10x \]
\[ -3x^2 + 54x = 0 \]
\[ 3x(x - 18) = 0 \]
- Thus, \( x = 0 \) or \( x = 18 \). Since speed cannot be zero, we have \( x = 18 \) km/h.

5. **Find the Speed of the Stream:**
- Substitute \( x = 18 \) back into the expression for \( y^2 \):
\[ y^2 = 18^2 - 16 \times 18 = 324 - 288 = 36 \]
- Therefore, \( y = 6 \) km/h.

### Final Answer:
The speed of the stream is \( \boxed{6} \) km/h.