A man swins in a river with and against water at the rate of `15 kmh ^(-1) and 5 kmh^(-1)` . Find the man's speed in still water and the speed of the river.

Answer

To find the man's speed in still water and the speed of the river, we can use the following approach:

### Step 1: Define Variables
Let:
- \( v_m \) = man's speed in still water (in km/h)
- \( v_r \) = speed of the river (in km/h)

### Step 2: Set Up Equations
When the man swims with the current, his effective speed is the sum of his speed in still water and the speed of the river:
\[ v_m + v_r = 15 \, \text{km/h} \quad \text{(1)} \]

When the man swims against the current, his effective speed is the difference between his speed in still water and the speed of the river:
\[ v_m - v_r = 5 \, \text{km/h} \quad \text{(2)} \]

### Step 3: Solve the Equations
We can solve the two equations simultaneously.

1. From equation (1):
\[ v_m + v_r = 15 \]

2. From equation (2):
\[ v_m - v_r = 5 \]

Now, we can add both equations:
\[ (v_m + v_r) + (v_m - v_r) = 15 + 5 \]
This simplifies to:
\[ 2v_m = 20 \]
Dividing both sides by 2 gives:
\[ v_m = 10 \, \text{km/h} \]

Now, substitute \( v_m \) back into equation (1) to find \( v_r \):
\[ 10 + v_r = 15 \]
Subtracting 10 from both sides gives:
\[ v_r = 5 \, \text{km/h} \]

### Step 4: Conclusion
The man's speed in still water is \( 10 \, \text{km/h} \) and the speed of the river is \( 5 \, \text{km/h} \).

### Summary of Results
- Man's speed in still water, \( v_m = 10 \, \text{km/h} \)
- Speed of the river, \( v_r = 5 \, \text{km/h} \)