Boats and Streams

Speed of boat in still water = 15 kmph. Speed of current = 75% of 15 = 11.25 kmph. Downstream speed = 15 + 11.25 = 26.25 kmph. Upstream speed = 15 - 11.25 = 3.75 kmph. Let distance be d. d/26.25 + d/3.75 = 16. Solving: d(1/26.25 + 1/3.75) = 16, which gives d = 52.5 km. Total distance = 2d = 105 km.

Let boat speed in still water be 2x and stream speed be x. Downstream speed = 2x + x = 3x. Upstream speed = 2x - x = x. Total time = 72/(3x) + 72/x = 24/x + 72/x = 96/x = 24 hours. Solving gives x = 4 km/h. Boat speed in still water = 2x = 8 km/h. Verification: downstream speed = 12 km/h (72/12 = 6 hours), upstream speed = 4 km/h (72/4 = 18 hours), total = 24 hours.

Let downstream time be T. Upstream time is 1.4T (40% more). Speed ratio = distance/time ratio = 1/1.4T : 1/T. Boat speed = (downstream + upstream)/2, current = (downstream - upstream)/2. This gives boat speed 500% more than current.

Let current speed = c km/hr. Boat speed in still water = c + 300% of c = 4c km/hr. Downstream speed = (4c + c) = 5c km/hr. Upstream speed = (4c - c) = 3c km/hr. Time = distance/speed, so 60/5c + 60/3c = 8 hours. This gives 12/c + 20/c = 8, so 32/c = 8, giving c = 4 km/hr. Upstream speed = 3c = 12 km/hr.

Let distance be D km. Downstream: speed = 20 + s, time = 16/60 hr. Upstream: speed = 20 - s, time = 24/60 hr. D/(20+s) = 16/60 and D/(20-s) = 24/60. Dividing: (20-s)/(20+s) = 16/24 = 2/3. Cross-multiply: 60 - 3s = 40 + 2s, so 5s = 20 and s = 4 km/hr. Option B (5 km/hr) is a distractor from incorrect calculation.

Let current speed = c km/hr. Then boat speed = 3c km/hr. Downstream speed = 3c + c = 4c, upstream speed = 3c - c = 2c. Total time = 72/(4c) + 72/(2c) = 18/c + 36/c = 54/c = 10.8 hours. Solving: c = 5 km/hr. Downstream speed = 4c = 20 km/hr. Option A (15) would mean current = 3.75, giving time 72/18.75 + 72/7.5 = 10.56 hours, incorrect.

Speed ratios are 2:3:4. For a fixed distance, time is inversely proportional to speed. If speeds are in ratio 2:3:4, then times are in ratio 1/2 : 1/3 : 1/4. Converting to integers with LCM 12: 6:4:3. Option A incorrectly uses the same ratio as speed. Options B and C don't represent the inverse relationship correctly.

दिया है: नाव की speed in still water = 36 km/hr, upstream distance = 56 km, upstream time = 1 hour 45 minutes = 7/4 hours। Upstream speed = 56 / (7/4) = 32 km/hr। Therefore stream speed = 36 - 32 = 4 km/hr। Downstream speed = 36 + 4 = 40 km/hr। Downstream time = 56 / 40 = 1.4 hours = 1 hour 24 minutes। अगर stream speed a है तो upstream speed = boat_speed - a और downstream = boat_speed + a।

Let boat speed in still water be b and stream speed be s. Downstream speed = b + s, upstream speed = b - s. Time is inversely proportional to speed for same distance. Given downstream time = 12 hours, upstream time = 24 hours, so upstream takes twice as long, meaning upstream speed is half of downstream speed. Therefore: (b - s) = 1/2(b + s). Solving: 2b - 2s = b + s, so b = 3s, giving ratio b:s = 3:1. We can verify: if b = 3k and s = k, downstream speed = 4k, upstream speed = 2k - indeed upstream is half of downstream.

Let boat speed = 4x, stream speed = x. Downstream speed = 5x, upstream speed = 3x. In 2 hours downstream: 10x km. In 2 hours upstream: 6x km. Difference = 4x = 5 km, so x = 1.25. Boat speed in still water = 4x = 5 kmph. The question asks for boat speed, not the difference itself.

Let b = boat speed in still water, s = stream speed. Downstream speed = b + s, upstream speed = b - s. First trip: 24/(b-s) + 36/(b+s) = 6. Second trip: 36/(b-s) + 24/(b+s) = 6.5. Solving these gives b = 10 km/h, s = 2 km/h. Time for 1080 km in still water = 1080 / 10 = 108 hours. Option D is correct.

Let stream speed = x km/hr. Downstream: 90/(15+x), Upstream: 90/(15-x). Total: 90/(15+x) + 90/(15-x) = 13.5. Substituting x=5: 90/20 + 90/10 = 4.5 + 9 = 13.5 hours. This matches exactly.

Let boat speed in still water = b km/hr, stream speed = 4 km/hr. Upstream speed = b-4, downstream speed = b+4. Time for 40 km upstream = 40/(b-4). Time for 60 km downstream = 60/(b+4). Given ratio 4:5, so 40/(b-4) ÷ 60/(b+4) = 4/5. This gives 40(b+4) × 5 = 60(b-4) × 4, so 200(b+4) = 240(b-4), giving 200b + 800 = 240b - 960, so 40b = 1760, b = 44 km/hr.

Let the boat's speed in still water be x km/hr. Upstream speed is (x-2) km/hr and downstream speed is (x+2) km/hr. Time upstream + time downstream = 10/(x-2) + 10/(x+2) = 55/60 hours. Solving: 20x/(x²-4) = 11/12, which gives 11x² - 240x - 44 = 0. The valid solution is x = 22 km/hr.

Downstream speed = 30 km/hr, upstream speed = 30/2 = 15 km/hr. Speed in still water = (downstream + upstream)/2 = (30 + 15)/2 = 22.5 km/hr. This equals 45/2 km/hr (Option C). The mixed fraction format '45 2' represents 45/2.