The time taken by a man to row downstream is `(3)/(5)` the of the time taken by him to row upstream. If the product of the speeds of the man and the current is (both taken in kmph) 36, then find the speed of the man and the current .

Options

A. 12 kmph, 3 kmph
B. 10 kmph, 2 kmph
C. 9 kmph, 2 kmph
D. 10 kmph, 3 kmph

Answer

To solve the problem step by step, we will define the variables, set up the equations based on the given information, and solve for the speeds of the man and the current.

### Step-by-Step Solution:

1. **Define Variables**:
- Let the speed of the man be \( U \) km/h.
- Let the speed of the current be \( V \) km/h.

2. **Establish Relationships**:
- The speed of the man rowing upstream (against the current) is \( U - V \).
- The speed of the man rowing downstream (with the current) is \( U + V \).

3. **Set Up Time Relationships**:
- Let the distance rowed be \( D \) km.
- The time taken to row upstream is given by:
\[ T_{up} = \frac{D}{U - V} \]
- The time taken to row downstream is given by:
\[ T_{down} = \frac{D}{U + V} \]
- According to the problem, the time taken to row downstream is \(\frac{3}{5}\) of the time taken to row upstream:
\[ T_{down} = \frac{3}{5} T_{up} \]

4. **Substitute the Time Expressions**:
\[ \frac{D}{U + V} = \frac{3}{5} \cdot \frac{D}{U - V} \]
- Since \( D \) is common on both sides, we can cancel it out (assuming \( D \neq 0 \)):
\[ \frac{1}{U + V} = \frac{3}{5} \cdot \frac{1}{U - V} \]

5. **Cross Multiply**:
\[ 5 = 3 \cdot \frac{U + V}{U - V} \]
- Rearranging gives:
\[ 5(U - V) = 3(U + V) \]

6. **Expand and Rearrange**:
\[ 5U - 5V = 3U + 3V \]
\[ 5U - 3U = 3V + 5V \]
\[ 2U = 8V \]
- Simplifying gives:
\[ U = 4V \]

7. **Use the Product of Speeds**:
- We know that the product of the speeds of the man and the current is given as:
\[ U \cdot V = 36 \]
- Substitute \( U = 4V \) into the equation:
\[ 4V \cdot V = 36 \]
\[ 4V^2 = 36 \]
\[ V^2 = 9 \]
\[ V = 3 \text{ km/h} \]

8. **Find the Speed of the Man**:
- Now substitute \( V \) back to find \( U \):
\[ U = 4V = 4 \cdot 3 = 12 \text{ km/h} \]

### Final Answer:
- The speed of the man is \( 12 \) km/h.
- The speed of the current is \( 3 \) km/h.