The speed of a boat in still water is 15 km/hr. It can go 30 km upstream and return downstream to the original point in 4 hours 30 minutes. Find the speed of the stream.

Answer

To solve the problem, we will follow these steps:

### Step 1: Define Variables
Let the speed of the boat in still water be \( b = 15 \) km/hr and the speed of the stream be \( x \) km/hr.

### Step 2: Determine Upstream and Downstream Speeds
- The speed of the boat going upstream (against the current) is given by:
\[ \text{Upstream speed} = b - x = 15 - x \text{ km/hr} \]
- The speed of the boat going downstream (with the current) is given by:
\[ \text{Downstream speed} = b + x = 15 + x \text{ km/hr} \]

### Step 3: Calculate Time Taken for Each Journey
- The time taken to travel 30 km upstream is:
\[ t_1 = \frac{30}{15 - x} \text{ hours} \]
- The time taken to travel 30 km downstream is:
\[ t_2 = \frac{30}{15 + x} \text{ hours} \]

### Step 4: Set Up the Equation for Total Time
The total time for the round trip is given as 4 hours and 30 minutes, which can be converted to hours:
\[ 4 \text{ hours } 30 \text{ minutes} = 4.5 \text{ hours} = \frac{9}{2} \text{ hours} \]
Thus, we can write the equation:
\[ t_1 + t_2 = \frac{9}{2} \]
Substituting the expressions for \( t_1 \) and \( t_2 \):
\[ \frac{30}{15 - x} + \frac{30}{15 + x} = \frac{9}{2} \]

### Step 5: Simplify the Equation
To simplify, we can multiply through by \( (15 - x)(15 + x) \) to eliminate the denominators:
\[ 30(15 + x) + 30(15 - x) = \frac{9}{2}(15 - x)(15 + x) \]
This simplifies to:
\[ 30 \cdot 15 + 30x + 30 \cdot 15 - 30x = \frac{9}{2}(225 - x^2) \]
\[ 900 = \frac{9}{2}(225 - x^2) \]

### Step 6: Multiply Both Sides by 2
To eliminate the fraction, multiply both sides by 2:
\[ 1800 = 9(225 - x^2) \]

### Step 7: Divide by 9
Now, divide both sides by 9:
\[ 200 = 225 - x^2 \]

### Step 8: Rearrange to Solve for \( x^2 \)
Rearranging gives:
\[ x^2 = 225 - 200 = 25 \]

### Step 9: Take the Square Root
Taking the square root of both sides:
\[ x = \sqrt{25} = 5 \]
Since speed cannot be negative, we have:
\[ x = 5 \text{ km/hr} \]

### Conclusion
The speed of the stream is \( 5 \) km/hr.

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