A man can row upstream at 8 km/hr and downstream at 10.6 km/hr. Find man's speed in still water and the rate of the current.

Answer

To solve the problem of finding the man's speed in still water and the rate of the current, we can follow these steps:

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

### Step 2: Set Up Equations
From the problem, we know:
1. When rowing upstream, the effective speed is reduced by the current:
\[ x - y = 8 \quad \text{(1)} \]
2. When rowing downstream, the effective speed is increased by the current:
\[ x + y = 10.6 \quad \text{(2)} \]

### Step 3: Add the Two Equations
Now, we can add equations (1) and (2):
\[ (x - y) + (x + y) = 8 + 10.6 \]
This simplifies to:
\[ 2x = 18.6 \]

### Step 4: Solve for \( x \)
Now, divide both sides by 2 to find \( x \):
\[ x = \frac{18.6}{2} = 9.3 \text{ km/hr} \]
So, the man's speed in still water is \( 9.3 \) km/hr.

### Step 5: Substitute \( x \) Back to Find \( y \)
Now, we can substitute \( x \) back into either equation (1) or (2) to find \( y \). Let's use equation (2):
\[ 9.3 + y = 10.6 \]
Subtract \( 9.3 \) from both sides:
\[ y = 10.6 - 9.3 = 1.3 \text{ km/hr} \]
So, the rate of the current is \( 1.3 \) km/hr.

### Final Answer
- Man's speed in still water: \( 9.3 \) km/hr
- Rate of the current: \( 1.3 \) km/hr

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