A person goes 60 km (upstream and downstream) in 10 hours. Time taken to cover 3 km downstream is equal to time taken to cover 2 km upstream. Find the speed of current and speed of boat in still water.

Answer

To solve the problem step by step, we will define the variables, set up equations based on the information given, and solve for the speed of the boat in still water and the speed of the current.

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

### Step 2: Set Up the Equations
1. The speed of the boat downstream (with the current) is \( x + y \) km/h.
2. The speed of the boat upstream (against the current) is \( x - y \) km/h.

### Step 3: Use the Time Information
According to the problem, the time taken to cover 3 km downstream is equal to the time taken to cover 2 km upstream.

Using the formula for time \( \text{Time} = \frac{\text{Distance}}{\text{Speed}} \):
- Time taken to cover 3 km downstream:
\[ \text{Time}_{\text{downstream}} = \frac{3}{x + y} \]
- Time taken to cover 2 km upstream:
\[ \text{Time}_{\text{upstream}} = \frac{2}{x - y} \]

Setting these two times equal gives us the first equation:
\[ \frac{3}{x + y} = \frac{2}{x - y} \]

### Step 4: Cross-Multiply to Solve for \( x \) and \( y \)
Cross-multiplying gives:
\[ 3(x - y) = 2(x + y) \]
Expanding both sides:
\[ 3x - 3y = 2x + 2y \]
Rearranging the equation:
\[ 3x - 2x = 2y + 3y \]
This simplifies to:
\[ x = 5y \quad \text{(Equation 1)} \]

### Step 5: Use the Total Time Information
The total distance covered upstream and downstream is 60 km, and the total time taken is 10 hours. Therefore, we can write:
\[ \frac{60}{x + y} + \frac{60}{x - y} = 10 \]

### Step 6: Simplify the Total Time Equation
Multiplying through by \( (x + y)(x - y) \) to eliminate the denominators:
\[ 60(x - y) + 60(x + y) = 10(x^2 - y^2) \]
This simplifies to:
\[ 60x - 60y + 60x + 60y = 10(x^2 - y^2) \]
Combining like terms:
\[ 120x = 10(x^2 - y^2) \]
Dividing through by 10:
\[ 12x = x^2 - y^2 \quad \text{(Equation 2)} \]

### Step 7: Substitute Equation 1 into Equation 2
From Equation 1, substitute \( y = \frac{x}{5} \) into Equation 2:
\[ 12x = x^2 - \left(\frac{x}{5}\right)^2 \]
This becomes:
\[ 12x = x^2 - \frac{x^2}{25} \]
Combining the terms:
\[ 12x = x^2 \left(1 - \frac{1}{25}\right) \]
This simplifies to:
\[ 12x = x^2 \left(\frac{24}{25}\right) \]
Rearranging gives:
\[ x^2 - 12 \cdot \frac{25}{24} x = 0 \]
Factoring out \( x \):
\[ x \left(x - 12 \cdot \frac{25}{24}\right) = 0 \]
Thus, \( x = 0 \) or \( x = 12.5 \).

### Step 8: Find \( y \)
Using \( x = 12.5 \) in Equation 1:
\[ y = \frac{12.5}{5} = 2.5 \]

### Conclusion
The speed of the boat in still water is \( 12.5 \) km/h, and the speed of the current is \( 2.5 \) km/h.