Boats and Streams

Speed downstream = 8 + v, upstream = 8 - v where v is river speed. Time = 60/(8+v) + 60/(8-v) = 20 hours. Solving: 60(8-v + 8+v)/((8+v)(8-v)) = 20. 60×16/(64-v²) = 20. 960 = 20(64-v²). 48 = 64 - v². v² = 16, v = 4 km/h.

Let boat speed = b, stream speed = s. Downstream: b + s = 16. Upstream: b - s = 11. Adding equations: 2b = 27, so b = 13.5 kmph. Then s = 16 - 13.5 = 2.5 kmph. Alternatively, using formula: stream speed = (downstream - upstream)/2 = (16-11)/2 = 2.5 kmph. So option D (2.5 kmph) is correct.

Downstream speed = boat speed + stream speed = 13 + 4 = 17 km/h. Time = distance/speed = 68/17 = 4 hours. This is straightforward - the stream aids the boat, making it faster than in still water.

Speed of boat in still water = (downstream speed + upstream speed)/2 = (13 + 7)/2 = 10 km/hr. The speed of the stream would be (downstream - upstream)/2 = (13 - 7)/2 = 3 km/hr. This is the standard formula for problems involving boats and streams.

Against stream: speed = 4 km / (20/60) hr = 12 kmph. With stream: speed = 4 km / (15/60) hr = 16 kmph. Let boat speed = b, current = c. Then b - c = 12 and b + c = 16. Solving: b = 14 kmph, c = 2 kmph. Rate of current is 2 km/hr. Option A (3 km/hr) and C (2.5 km/hr) don't satisfy both conditions.

Let upstream speed = u, downstream = d, boat speed in still water = b, current speed = c. Then u = b-c, d = b+c. From given data: 30/u + 44/d = 10 and 40/u + 55/d = 13. Solving: u = 5, d = 11. Then b = (u+d)/2 = (5+11)/2 = 8 km/hr. Option B is correct.

Let boat speed = b kmph. Speed upstream = b - 3, downstream = b + 3. Total time = 42/(b-3) + 42/(b+3) = 306/60 = 5.1 hours. Solving: 42(1/(b-3) + 1/(b+3)) = 5.1. After cross-multiplication: 42(b+3 + b-3)/((b-3)(b+3)) = 5.1. This gives 84b/(b²-9) = 5.1, or 84b = 5.1b² - 45.9. Rearranging: 5.1b² - 84b - 45.9 = 0. Using quadratic formula: b = (84 ± √(7056 + 936.36))/10.2 = (84 ± 89.4)/10.2. Taking positive root: b = 17 kmph. Option B is correct. Options A, C, D, E are computational errors.

Let boat speed in still water be x km/hr. Upstream speed = x-5, downstream speed = x+5. Time = distance/speed, so 10/(x-5) + 10/(x+5) = 50/60 hours. Solving gives x = 25 km/hr.

Using time = distance/speed: Upstream speed = 22-c, downstream = 22+c. Total time = 10/(22-c) + 10/(22+c) = 55/60 hours. Solving gives c=2 km/hr. Negative current speed is physically impossible.

Downstream speed (8 km/hr) is boat + current, upstream speed (5 km/hr) is boat - current. Still water speed = (8 + 5) / 2 = 6.5 km/hr. Current speed is (8 - 5) / 2 = 1.5 km/hr. This formula works for all boat-stream problems.

Downstream speed = 1 km/7.5 min = 8 km/hr. Upstream speed = 5 km/hr. Let b = boat speed, c = current speed. b+c=8, b-c=5. Adding: 2b=13, so b=6.5 km/hr.

Speed along stream (downstream) = 18 km/hr, speed in still water = 12 km/hr. Stream speed = 18 - 12 = 6 km/hr. Speed against stream = 12 - 6 = 6 km/hr.

Downstream speed = 100/10 = 10 km/hr (boat + stream). Upstream speed = 75/15 = 5 km/hr (boat - stream). Stream speed = (10 - 5)/2 = 2.5 km/hr. This is derived from the standard formula: if downstream speed is D and upstream speed is U, then stream speed = (D - U)/2.

Let boat speed in still water = v km/hr. Downstream speed = v+3, upstream speed = v-3. Time = distance/speed, so 12/(v+3) + 12/(v-3) = 3. Solving: 12(v-3) + 12(v+3) = 3(v+3)(v-3) gives 12v - 36 + 12v + 36 = 3(v² - 9), so 24v = 3v² - 27, giving v² - 8v - 9 = 0, so (v-9)(v+1) = 0. v=9 km/hr (positive). Downstream: 12/12=1 hr, upstream: 12/6=2 hr, total 3 hr. ✓

Upstream speed = Distance/Time = 15/(80/60 hrs) = 15/(4/3) = 11.25 km/hr. Using Upstream speed = Boat speed - Stream speed: 11.25 = Boat speed - 5, so Boat speed = 16.25 km/hr.