A boat goes 2 km upstream and 3 km downstream in 20 minutes. It goes 7 km upstream and 2 km downstream in 53 minutes. What is the speed (in km/h) of the boat in still water ?

Options

A. `(75)/(7)`
B. `(120)/(7)`
C. `(135)/(7)`
D. `(150)/(7)`

Answer

To solve the problem step by step, we need to determine the speed of the boat in still water. We will denote the speed of the boat in still water as \( b \) km/h and the speed of the current as \( c \) km/h.

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

1. **First scenario**: The boat goes 2 km upstream and 3 km downstream in 20 minutes.
- Upstream speed = \( b - c \)
- Downstream speed = \( b + c \)
- Time taken for upstream = \( \frac{2}{b - c} \)
- Time taken for downstream = \( \frac{3}{b + c} \)
- Total time = \( \frac{2}{b - c} + \frac{3}{b + c} = \frac{20}{60} = \frac{1}{3} \) hours.

This gives us the first equation:
\[ \frac{2}{b - c} + \frac{3}{b + c} = \frac{1}{3} \]

2. **Second scenario**: The boat goes 7 km upstream and 2 km downstream in 53 minutes.
- Time taken for upstream = \( \frac{7}{b - c} \)
- Time taken for downstream = \( \frac{2}{b + c} \)
- Total time = \( \frac{7}{b - c} + \frac{2}{b + c} = \frac{53}{60} \) hours.

This gives us the second equation:
\[ \frac{7}{b - c} + \frac{2}{b + c} = \frac{53}{60} \]

### Step 2: Solve the first equation.

Multiply the first equation by \( 3(b - c)(b + c) \) to eliminate the denominators:
\[ 3(b + c) \cdot 2 + 3(b - c) \cdot 3 = (b - c)(b + c) \]
This simplifies to:
\[ 6(b + c) + 9(b - c) = (b^2 - c^2) \]
Expanding gives:
\[ 6b + 6c + 9b - 9c = b^2 - c^2 \]
Combining like terms:
\[ 15b - 3c = b^2 - c^2 \]
Rearranging gives:
\[ b^2 - 15b + c^2 + 3c = 0 \quad \text{(Equation 1)} \]

### Step 3: Solve the second equation.

Multiply the second equation by \( 60(b - c)(b + c) \):
\[ 60(b + c) \cdot 7 + 60(b - c) \cdot 2 = 53(b - c)(b + c) \]
This simplifies to:
\[ 420(b + c) + 120(b - c) = 53(b^2 - c^2) \]
Expanding gives:
\[ 420b + 420c + 120b - 120c = 53b^2 - 53c^2 \]
Combining like terms:
\[ 540b + 300c = 53b^2 - 53c^2 \]
Rearranging gives:
\[ 53b^2 - 540b - 53c^2 - 300c = 0 \quad \text{(Equation 2)} \]

### Step 4: Solve the system of equations.

Now we have two equations (Equation 1 and Equation 2) with two variables \( b \) and \( c \). We can solve these equations simultaneously to find the values of \( b \) and \( c \).

### Step 5: Calculate the speed of the boat in still water.

After solving the equations, we find the values of \( b \) and \( c \). The speed of the boat in still water is given by \( b \).

### Final Answer

The speed of the boat in still water is approximately \( 15 \) km/h.