Let's call the speed of the stream "S" km/hr.

When the boat is going downstream (i.e., in the same direction as the stream), its effective speed is the sum of its speed in still water and the speed of the stream:

Speed downstream = Speed in still water + Speed of stream = (15 + S) km/hr

When the boat is going upstream (i.e., against the direction of the stream), its effective speed is the difference between its speed in still water and the speed of the stream:

Speed upstream = Speed in still water - Speed of stream = (15 - S) km/hr

Now, we know that the boat travels 30 km downstream and then returns the same distance upstream, so the total distance covered is 2 * 30 km = 60 km.

We are also given that the total time taken for the round trip is 4 hours and 30 minutes, which is equivalent to 4.5 hours.

We can use the formula: Time = Distance / Speed

For the downstream journey, the time taken is 30 km / (15 + S) km/hr. For the upstream journey, the time taken is 30 km / (15 - S) km/hr.

So, the total time taken for the round trip is:

(30 / (15 + S)) + (30 / (15 - S)) = 4.5 hours

Now, we can solve this equation for S:

(30 / (15 + S)) + (30 / (15 - S)) = 4.5

Multiply both sides by (15 + S)(15 - S) to get rid of the denominators:

30(15 - S) + 30(15 + S) = 4.5(15 + S)(15 - S)

Now, simplify the equation:

30(15 - S) + 30(15 + S) = 4.5(225 - S^2)

Expand and simplify further:

450 - 30S + 450 + 30S = 1012.5 - 4.5S^2

Combine like terms:

900 = 1012.5 - 4.5S^2

Move 1012.5 to the left side:

4.5S^2 = 1012.5 - 900

4.5S^2 = 112.5

Now, divide by 4.5 to isolate S^2:

S^2 = 112.5 / 4.5 S^2 = 25

Take the square root of both sides to find S:

S = √25 S = 5 km/hr

So, the speed of the stream is 5 km/hr.