A boat travels for 7 hours. If it travels 4 hours downstream and 3 hours upstream than it covers the distance of 116 km. But if it travels 3 hours downstream and 4 hours upstream, it covers a distance of 108 km. Find the speed of a boat.

Answer

To solve the problem step by step, we will use the information provided about the boat's travel downstream and upstream to set up equations.

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

### Step 2: Set Up the First Equation
In the first scenario, the boat travels 4 hours downstream and 3 hours upstream, covering a total distance of 116 km.

- Speed downstream = \( x + y \)
- Speed upstream = \( x - y \)

The distance covered downstream in 4 hours is:
\[ \text{Distance downstream} = 4(x + y) \]

The distance covered upstream in 3 hours is:
\[ \text{Distance upstream} = 3(x - y) \]

According to the problem:
\[ 4(x + y) + 3(x - y) = 116 \]

Expanding this equation:
\[ 4x + 4y + 3x - 3y = 116 \]
Combining like terms:
\[ 7x + y = 116 \] (Equation 1)

### Step 3: Set Up the Second Equation
In the second scenario, the boat travels 3 hours downstream and 4 hours upstream, covering a total distance of 108 km.

The distance covered downstream in 3 hours is:
\[ \text{Distance downstream} = 3(x + y) \]

The distance covered upstream in 4 hours is:
\[ \text{Distance upstream} = 4(x - y) \]

According to the problem:
\[ 3(x + y) + 4(x - y) = 108 \]

Expanding this equation:
\[ 3x + 3y + 4x - 4y = 108 \]
Combining like terms:
\[ 7x - y = 108 \] (Equation 2)

### Step 4: Solve the System of Equations
Now we have a system of two equations:
1. \( 7x + y = 116 \) (Equation 1)
2. \( 7x - y = 108 \) (Equation 2)

To eliminate \( y \), we can add both equations:
\[ (7x + y) + (7x - y) = 116 + 108 \]
This simplifies to:
\[ 14x = 224 \]

### Step 5: Solve for \( x \)
Now, divide both sides by 14:
\[ x = \frac{224}{14} = 16 \]

### Step 6: Conclusion
The speed of the boat in still water is:
\[ \boxed{16 \text{ km/h}} \]