A boat travels 65 km downstream in 6 hrs 30 minutes, when it is travelling in a stream (`S_1`). If this boat travels in another stream (`S_2`) whose speed is 10% more, than that of `S_1` it covers 58 km upstream in 10 hrs. What is the speed of `S_2`? (in km/hrs). (Assume the boat’s speed in still water in both streams remains the same)

Options

A. 2.2
B. 1.5
C. 1.8
D. 2.4

Answer

To solve the problem step by step, we will first define the variables and then use the information provided to set up equations.

### Step 1: Define Variables
Let the speed of stream `S1` be \( x \) km/hr.
Then, the speed of stream `S2`, which is 10% more than `S1`, will be:
\[ S2 = 1.1x = \frac{11x}{10} \text{ km/hr} \]

### Step 2: Calculate Downstream Speed for `S1`
The boat travels 65 km downstream in 6 hours and 30 minutes.
Convert 6 hours and 30 minutes into hours:
\[ 6 \text{ hours } 30 \text{ minutes} = 6.5 \text{ hours} \]
The speed downstream is given by:
\[ \text{Speed} = \frac{\text{Distance}}{\text{Time}} = \frac{65 \text{ km}}{6.5 \text{ hrs}} = 10 \text{ km/hr} \]
Since downstream speed is the sum of the boat's speed in still water and the stream's speed:
\[ \text{Speed of boat} + S1 = 10 \]
Let the speed of the boat in still water be \( b \):
\[ b + x = 10 \quad \text{(Equation 1)} \]

### Step 3: Calculate Upstream Speed for `S2`
The boat travels 58 km upstream in 10 hours.
The speed upstream is:
\[ \text{Speed} = \frac{58 \text{ km}}{10 \text{ hrs}} = 5.8 \text{ km/hr} \]
Since upstream speed is the difference between the boat's speed and the stream's speed:
\[ b - S2 = 5.8 \]
Substituting \( S2 = \frac{11x}{10} \):
\[ b - \frac{11x}{10} = 5.8 \quad \text{(Equation 2)} \]

### Step 4: Solve the Equations
Now we have two equations:
1. \( b + x = 10 \)
2. \( b - \frac{11x}{10} = 5.8 \)

From Equation 1, we can express \( b \):
\[ b = 10 - x \]

Substituting \( b \) in Equation 2:
\[ 10 - x - \frac{11x}{10} = 5.8 \]
Multiply everything by 10 to eliminate the fraction:
\[ 100 - 10x - 11x = 58 \]
Combine like terms:
\[ 100 - 21x = 58 \]
Rearranging gives:
\[ 21x = 100 - 58 \]
\[ 21x = 42 \]
\[ x = 2 \text{ km/hr} \]

### Step 5: Calculate Speed of `S2`
Now, substitute \( x \) back to find the speed of `S2`:
\[ S2 = \frac{11x}{10} = \frac{11 \times 2}{10} = \frac{22}{10} = 2.2 \text{ km/hr} \]

### Final Answer
The speed of stream `S2` is \( \boxed{2.2} \) km/hr.

---