Mallah can row 40 km upstream and 55 km downstream in 13 h and 30 km upstraem and 44 km downstream in 10 hours what is the speed of Mallah in still water?

Options

A. 6 km/h
B. 12 km/h
C. 3 km/h
D. 8 km/h

Answer

To solve the problem, we need to determine the speed of Mallah in still water based on the information provided about his rowing upstream and downstream.

### Step-by-Step Solution:

1. **Define Variables:**
- Let \( x \) be the speed of Mallah in still water (in km/h).
- Let \( y \) be the speed of the stream (in km/h).

2. **Set Up Equations from Given Information:**
- From the first scenario:
- Distance upstream = 40 km
- Distance downstream = 55 km
- Total time = 13 hours
- Upstream speed = \( x - y \)
- Downstream speed = \( x + y \)
- Time taken = Distance / Speed
- Therefore, the equation becomes:
\[ \frac{40}{x - y} + \frac{55}{x + y} = 13 \]

- From the second scenario:
- Distance upstream = 30 km
- Distance downstream = 44 km
- Total time = 10 hours
- The equation becomes:
\[ \frac{30}{x - y} + \frac{44}{x + y} = 10 \]

3. **Multiply Each Equation to Eliminate Fractions:**
- For the first equation:
\[ 40(x + y) + 55(x - y) = 13(x - y)(x + y) \]
Simplifying gives:
\[ 40x + 40y + 55x - 55y = 13(x^2 - y^2) \]
\[ 95x - 15y = 13(x^2 - y^2) \]

- For the second equation:
\[ 30(x + y) + 44(x - y) = 10(x - y)(x + y) \]
Simplifying gives:
\[ 30x + 30y + 44x - 44y = 10(x^2 - y^2) \]
\[ 74x - 14y = 10(x^2 - y^2) \]

4. **Rearranging the Equations:**
- We now have two equations:
1. \( 13x^2 - 95x + 15y = 13y^2 \)
2. \( 10x^2 - 74x + 14y = 10y^2 \)

5. **Solve the System of Equations:**
- We can solve these equations simultaneously to find the values of \( x \) and \( y \).
- After substituting and solving, we find:
- \( x + y = 11 \) (downstream speed)
- \( x - y = 5 \) (upstream speed)

6. **Add and Solve for \( x \):**
- Adding the two equations:
\[ (x + y) + (x - y) = 11 + 5 \]
\[ 2x = 16 \implies x = 8 \text{ km/h} \]

7. **Conclusion:**
- The speed of Mallah in still water is \( 8 \) km/h.