Malla can row 40 km upstream and 55 km downstream in 13 h and 30 km upstream and 44 km downstream in 10 hours. What is the speed of Malla 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 step by step, we will define the variables and set up equations based on the information given in the question.

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

### Step 2: Set Up Equations
From the question, we can derive two equations based on the time taken for upstream and downstream journeys.

1. **First Journey**:
Malla rows 40 km upstream and 55 km downstream in 13 hours.
- Time taken to row upstream = \( \frac{40}{x - y} \)
- Time taken to row downstream = \( \frac{55}{x + y} \)
- Therefore, the equation becomes:
\[ \frac{40}{x - y} + \frac{55}{x + y} = 13 \]

2. **Second Journey**:
Malla rows 30 km upstream and 44 km downstream in 10 hours.
- Time taken to row upstream = \( \frac{30}{x - y} \)
- Time taken to row downstream = \( \frac{44}{x + y} \)
- Therefore, the equation becomes:
\[ \frac{30}{x - y} + \frac{44}{x + y} = 10 \]

### Step 3: Solve the First Equation
We can rewrite the first equation:
\[ \frac{40}{x - y} + \frac{55}{x + y} = 13 \]
This can be rearranged to find a common denominator:
\[ \frac{40(x + y) + 55(x - y)}{(x - y)(x + y)} = 13 \]
Expanding the numerator:
\[ 40x + 40y + 55x - 55y = 13(x^2 - y^2) \]
Combining like terms gives:
\[ 95x - 15y = 13(x^2 - y^2) \]

### Step 4: Solve the Second Equation
Now, we can rewrite the second equation:
\[ \frac{30}{x - y} + \frac{44}{x + y} = 10 \]
Rearranging similarly:
\[ \frac{30(x + y) + 44(x - y)}{(x - y)(x + y)} = 10 \]
Expanding the numerator:
\[ 30x + 30y + 44x - 44y = 10(x^2 - y^2) \]
Combining like terms gives:
\[ 74x - 14y = 10(x^2 - y^2) \]

### Step 5: Solve the System of Equations
We now have two equations:
1. \( 95x - 15y = 13(x^2 - y^2) \)
2. \( 74x - 14y = 10(x^2 - y^2) \)

We can solve these equations simultaneously. To simplify, we can assume values for \( x + y \) and \( x - y \) and check for consistency.

### Step 6: Assume Values
Assume \( x + y = 11 \) km/h. Then we can find \( x - y \) using the second equation:
\[ \frac{30}{x - y} + \frac{44}{11} = 10 \]
This simplifies to:
\[ \frac{30}{x - y} + 4 = 10 \]
Thus,
\[ \frac{30}{x - y} = 6 \implies x - y = 5 \text{ km/h} \]

### Step 7: Solve for \( x \) and \( y \)
Now we have:
1. \( x + y = 11 \)
2. \( x - y = 5 \)

Adding these two equations:
\[ 2x = 16 \implies x = 8 \text{ km/h} \]

### Step 8: Conclusion
The speed of Malla in still water is \( \boxed{8} \) km/h.