A man covers 39 km upstream an 116 km downstream in 7 hrs. He also covers 65 km upstream and 87 km downstream in 8 hrs. Find the speed of boat in still water.

Options

A. 21 km/hr
B. 27 km/hr
C. 18 km/hr
D. 29 km/hr

Answer

To solve the problem, we need to determine the speed of the boat in still water (denoted as \( x \)) and the speed of the stream (denoted as \( y \)). We will set up equations based on the information provided in the question.

### Step 1: Set Up the Equations
From the problem, we have two scenarios:

1. A man covers 39 km upstream and 116 km downstream in 7 hours.
2. He also covers 65 km upstream and 87 km downstream in 8 hours.

Using the formula for speed, we can express the time taken for upstream and downstream travel as follows:

- Upstream speed = \( x - y \)
- Downstream speed = \( x + y \)

Using this, we can set up our first equation from the first scenario:

\[ \frac{39}{x - y} + \frac{116}{x + y} = 7 \tag{1} \]

For the second scenario, we set up the second equation:

\[ \frac{65}{x - y} + \frac{87}{x + y} = 8 \tag{2} \]

### Step 2: Solve the Equations
To solve these equations, we will first express \( \frac{1}{x - y} \) and \( \frac{1}{x + y} \) in terms of a common variable.

Let:
\[ a = \frac{1}{x - y} \quad \text{and} \quad b = \frac{1}{x + y} \]

Then we can rewrite our equations as:

1. \( 39a + 116b = 7 \)
2. \( 65a + 87b = 8 \)

### Step 3: Solve for \( a \) and \( b \)
We can solve these two equations simultaneously.

From equation (1):
\[ 39a + 116b = 7 \tag{1} \]

From equation (2):
\[ 65a + 87b = 8 \tag{2} \]

To eliminate one variable, we can multiply equation (1) by 5 and equation (2) by 3:

\[ 195a + 580b = 35 \tag{3} \]
\[ 195a + 261b = 24 \tag{4} \]

Now, subtract equation (4) from equation (3):

\[ (580b - 261b) = 35 - 24 \]
\[ 319b = 11 \implies b = \frac{11}{319} \approx 0.0345 \]

Now substitute \( b \) back into one of the equations to find \( a \). Using equation (1):

\[ 39a + 116 \left(\frac{11}{319}\right) = 7 \]
\[ 39a + \frac{1276}{319} = 7 \]
\[ 39a = 7 - \frac{1276}{319} \]
\[ 39a = \frac{2233 - 1276}{319} = \frac{957}{319} \]
\[ a = \frac{957}{39 \times 319} \]

### Step 4: Find \( x \) and \( y \)
Now we can find \( x \) and \( y \) using the relationships:

\[ x - y = \frac{1}{a} \quad \text{and} \quad x + y = \frac{1}{b} \]

Substituting the values of \( a \) and \( b \):

1. \( x - y = \frac{39 \times 319}{957} \)
2. \( x + y = \frac{319}{11} \)

Now we can solve these two equations to find \( x \) and \( y \).

### Step 5: Solve for \( x \)
Adding the two equations gives:

\[ 2x = \left(\frac{39 \times 319}{957} + \frac{319}{11}\right) \]

Calculating this will give us \( x \), the speed of the boat in still water.

### Final Calculation
After performing the calculations, we find that:

\[ x = 21 \text{ km/h} \]

### Conclusion
The speed of the boat in still water is **21 km/h**.