A man can row at 10 km/h in still water. If he takes total 5 h to go to a place 24 km away and return, then the speed of the water current is

Options

A. 2km /h
B. 3 km/h
C. `1/2` km/h
D. 1km /h

Answer

To solve the problem step by step, we will use the information given in the question and apply the concepts of speed, distance, and time.

### Step-by-Step Solution:

1. **Identify the Variables**:
- Let the speed of the man in still water be \( x = 10 \) km/h.
- Let the speed of the water current be \( y \) km/h.

2. **Understand the Journey**:
- The man travels to a place 24 km away and then returns, making the total distance traveled \( 24 + 24 = 48 \) km.
- The total time taken for the round trip is given as 5 hours.

3. **Calculate the Speeds**:
- When rowing upstream (against the current), the effective speed is \( x - y \) km/h.
- When rowing downstream (with the current), the effective speed is \( x + y \) km/h.

4. **Set Up the Time Equation**:
- The time taken to go upstream (to the place) is given by:
\[ \text{Time upstream} = \frac{24}{x - y} \]
- The time taken to return downstream is given by:
\[ \text{Time downstream} = \frac{24}{x + y} \]
- The total time for the journey is:
\[ \frac{24}{x - y} + \frac{24}{x + y} = 5 \]

5. **Substitute the Value of \( x \)**:
- Substitute \( x = 10 \) into the equation:
\[ \frac{24}{10 - y} + \frac{24}{10 + y} = 5 \]

6. **Multiply Through by the Denominators**:
- To eliminate the fractions, multiply through by \( (10 - y)(10 + y) \):
\[ 24(10 + y) + 24(10 - y) = 5(10 - y)(10 + y) \]

7. **Simplify the Equation**:
- Expanding both sides:
\[ 240 + 24y + 240 - 24y = 5(100 - y^2) \]
- This simplifies to:
\[ 480 = 500 - 5y^2 \]

8. **Rearranging the Equation**:
- Rearranging gives:
\[ 5y^2 = 500 - 480 \]
- Thus:
\[ 5y^2 = 20 \]
- Dividing by 5:
\[ y^2 = 4 \]

9. **Finding the Value of \( y \)**:
- Taking the square root:
\[ y = \sqrt{4} = 2 \]
- Therefore, the speed of the water current is \( y = 2 \) km/h.

### Final Answer:
The speed of the water current is **2 km/h**.