Swati can row her boat at a speed of 5 km/hr in still water. If it takes her 1 hour more to row the boat 5.25 km upstream than to return downstream, find the speed of the stream.

Options

A. 5km/hr
B. 2 km/hr
C. 3 km/hr
D. 4 km/hr

Answer

To solve the problem step by step, we will follow these instructions:

### Step 1: Define Variables
Let the speed of the stream be \( x \) km/hr. The speed of Swati's boat in still water is given as 5 km/hr.

### Step 2: Determine Effective Speeds
- **Upstream Speed**: When Swati rows upstream, the effective speed is \( 5 - x \) km/hr.
- **Downstream Speed**: When she rows downstream, the effective speed is \( 5 + x \) km/hr.

### Step 3: Write the Time Equations
The distance Swati rows in both cases is 5.25 km.
- **Time taken to row upstream (t1)**:
\[ t_1 = \frac{5.25}{5 - x} \]
- **Time taken to row downstream (t2)**:
\[ t_2 = \frac{5.25}{5 + x} \]

### Step 4: Set Up the Equation
According to the problem, it takes her 1 hour more to row upstream than downstream:
\[ t_1 = t_2 + 1 \]
Substituting the expressions for \( t_1 \) and \( t_2 \):
\[ \frac{5.25}{5 - x} = \frac{5.25}{5 + x} + 1 \]

### Step 5: Clear the Fractions
To eliminate the fractions, we can multiply through by \( (5 - x)(5 + x) \):
\[ 5.25(5 + x) = 5.25(5 - x) + (5 - x)(5 + x) \]

### Step 6: Simplify the Equation
Expanding both sides:
\[ 5.25 \cdot 5 + 5.25x = 5.25 \cdot 5 - 5.25x + (25 - x^2) \]
This simplifies to:
\[ 5.25x + 5.25x = 25 - x^2 \]
\[ 10.5x = 25 - x^2 \]

### Step 7: Rearrange to Form a Quadratic Equation
Rearranging gives:
\[ x^2 + 10.5x - 25 = 0 \]

### Step 8: Solve the Quadratic Equation
Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
Here, \( a = 1, b = 10.5, c = -25 \):
\[ x = \frac{-10.5 \pm \sqrt{(10.5)^2 - 4 \cdot 1 \cdot (-25)}}{2 \cdot 1} \]
Calculating the discriminant:
\[ (10.5)^2 = 110.25 \]
\[ -4 \cdot 1 \cdot (-25) = 100 \]
\[ \sqrt{110.25 + 100} = \sqrt{210.25} = 14.5 \]
Now substituting back:
\[ x = \frac{-10.5 \pm 14.5}{2} \]

Calculating the two possible values:
1. \( x = \frac{4}{2} = 2 \) (valid)
2. \( x = \frac{-25}{2} \) (not valid since speed cannot be negative)

### Step 9: Conclusion
Thus, the speed of the stream is:
\[ \boxed{2 \text{ km/hr}} \]