A man can row 60 km in downstream and 35 km in upstream in 9 hours. Also, he can row 49 km in upstream and 75 km in downstream in 12 hours. Find the rate of the current.

Options

A. 6 kmph
B. 3 kmph
C. 7.5 kmph
D. 4 kmph

Answer

To solve the problem step by step, we will first define the variables and then create equations based on the information given in the question.

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

### Step 2: Set Up the Equations
From the problem, we have two scenarios:

1. **First Scenario**:
- Distance downstream: 60 km
- Distance upstream: 35 km
- Total time: 9 hours

The speed downstream is \( x + y \) and the speed upstream is \( x - y \).

The equation for this scenario can be written as:
\[ \frac{60}{x+y} + \frac{35}{x-y} = 9 \]

2. **Second Scenario**:
- Distance downstream: 75 km
- Distance upstream: 49 km
- Total time: 12 hours

The equation for this scenario can be written as:
\[ \frac{75}{x+y} + \frac{49}{x-y} = 12 \]

### Step 3: Solve the Equations
Now we have two equations:
1. \( \frac{60}{x+y} + \frac{35}{x-y} = 9 \) (Equation 1)
2. \( \frac{75}{x+y} + \frac{49}{x-y} = 12 \) (Equation 2)

To solve these equations, we can multiply both sides of each equation by the denominators to eliminate the fractions.

**From Equation 1**:
Multiply through by \((x+y)(x-y)\):
\[ 60(x-y) + 35(x+y) = 9(x+y)(x-y) \]

**From Equation 2**:
Multiply through by \((x+y)(x-y)\):
\[ 75(x-y) + 49(x+y) = 12(x+y)(x-y) \]

### Step 4: Simplify the Equations
Now we will simplify both equations.

For Equation 1:
\[ 60x - 60y + 35x + 35y = 9(x^2 - y^2) \]
Combine like terms:
\[ 95x - 25y = 9(x^2 - y^2) \]

For Equation 2:
\[ 75x - 75y + 49x + 49y = 12(x^2 - y^2) \]
Combine like terms:
\[ 124x - 26y = 12(x^2 - y^2) \]

### Step 5: Solve for \(x\) and \(y\)
Now we have two equations in \(x\) and \(y\). We can solve these equations simultaneously.

From the first equation:
\[ 9x^2 - 95x + 25y + 9y^2 = 0 \]

From the second equation:
\[ 12x^2 - 124x + 26y + 12y^2 = 0 \]

### Step 6: Find the Value of \(y\)
After solving these equations, we find that:
- \( x = 11 \) km/h (speed of man in still water)
- \( y = 4 \) km/h (speed of current)

### Final Answer
The rate of the current is \( y = 4 \) km/h.