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 boat in still water?

Options

A. 20 km/h
B. 18 km/h
C. 15 km/h
D. 16 km/h

Answer

To solve the problem step by step, we will define the variables and set up equations based on the information provided.

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

### Step 2: Set Up the First Equation
The boat travels downstream for 120 km and then returns upstream for the same distance. The total time for the round trip is 15 hours. The downstream speed is \( x + y \) and the upstream speed is \( x - y \).

Using the formula for time, we can write:
\[ \text{Total time} = \text{Time downstream} + \text{Time upstream} \]
\[ 15 = \frac{120}{x + y} + \frac{120}{x - y} \]

### Step 3: Simplify the First Equation
Multiply through by \( (x + y)(x - y) \) to eliminate the denominators:
\[ 15(x + y)(x - y) = 120(x - y) + 120(x + y) \]
This simplifies to:
\[ 15(x^2 - y^2) = 240x \]
Rearranging gives us the first equation:
\[ 15x^2 - 15y^2 = 240x \quad \text{(Equation 1)} \]

### Step 4: Set Up the Second Equation
Now, if the speed of the river is doubled (i.e., \( 2y \)), the total time for the round trip becomes 24 hours. The new downstream speed is \( x + 2y \) and the upstream speed is \( x - 2y \).

Using the same time formula:
\[ 24 = \frac{120}{x + 2y} + \frac{120}{x - 2y} \]

### Step 5: Simplify the Second Equation
Multiply through by \( (x + 2y)(x - 2y) \):
\[ 24(x + 2y)(x - 2y) = 120(x - 2y) + 120(x + 2y) \]
This simplifies to:
\[ 24(x^2 - 4y^2) = 240x \]
Rearranging gives us the second equation:
\[ 24x^2 - 96y^2 = 240x \quad \text{(Equation 2)} \]

### Step 6: Solve the Equations
From Equation 1:
\[ 15x^2 - 240x - 15y^2 = 0 \]
From Equation 2:
\[ 24x^2 - 240x - 96y^2 = 0 \]

### Step 7: Eliminate One Variable
We can express \( y^2 \) in terms of \( x^2 \) using these equations. From Equation 1:
\[ 15y^2 = 15x^2 - 240x \implies y^2 = x^2 - 16x \quad \text{(1)} \]

Substituting \( y^2 \) from (1) into Equation 2:
\[ 24x^2 - 240x - 96(x^2 - 16x) = 0 \]
This simplifies to:
\[ 24x^2 - 240x - 96x^2 + 1536x = 0 \]
Combining like terms gives:
\[ -72x^2 + 1296x = 0 \]
Factoring out \( 24x \):
\[ 24x(-3x + 54) = 0 \]
Thus, \( x = 0 \) or \( x = 18 \).

### Step 8: Find the Speed of the Boat
Since \( x = 0 \) is not a valid solution, we have:
\[ x = 18 \text{ km/h} \]

### Final Answer
The speed of the boat in still water is **18 km/h**.

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