The speed of the stream is approximately 64.6 km/hr.
Let's denote the speed of the boat in still water as B km/hr, the speed of the stream as S km/hr, and the time it takes to travel the same distance upstream and downstream as T1 and T2 hours, respectively.
We are given that the boat can travel 75 km downstream in 2 hours, so we have:
75 = B * 2 = S * 2
From this, we can find the speed of the boat in still water and the speed of the stream:
S = 75 / 2 = 37.5 km/hr
B = 75 / 2 = 37.5 km/hr
Now, let's consider the given information about the boat's speeds when the stream is doubled and the boat's speed is doubled:
Downstream speed: 2B + 2S = 2 * 37.5 + 2 * 37.5 = 75 + 75 = 150 km/hr
Upstream speed: B - S = 37.5 - 37.5 = 0 km/hr (assuming the boat can't move upstream against the stream)
When the boat's speed is doubled and the stream's speed is tripled, we have:
Downstream speed: 2B + 2S = 2 * 75 + 2 * 37.5 = 150 + 75 = 225 km/hr
Upstream speed: B - S = 37.5 - 37.5 = 0 km/hr (assuming the boat can't move upstream against the stream)
We are given that the boat can travel 275 km downstream in 6 hours. With the new speeds, we can find the time it takes to travel 275 km downstream:
T2 = 275 / 225 = 1.238 hours
Now, we can use the information that the time taken to travel the same distance upstream is 1.8 times the time taken to travel the same distance downstream to find the speed of the stream:
T1 = 1.8 * T2 = 1.8 * 1.238 = 2.14 hours
Using this information, we can find the distance the boat can travel upstream in 2 hours:
D = B * T1 = 37.5 * 2.14 = 82.18 km
Since the boat can travel 75 km downstream in 2 hours, we can find the time it takes to travel 82.18 km downstream:
T3 = 82.18 / 75 = 1.1067 hours
Now, we can find the speed of the stream:
S = 75 / T3 = 75 / 1.1067 ≈ 64.6 km/hr
So, the speed of the stream is approximately 64.6 km/hr.