Boats and Streams: Formulas, Shortcuts and Solved Examples — MCQs with Answers (Set 1)

For a round trip, the distance covered downstream equals the distance covered upstream. Average speed for such a trip is not the simple average of the two speeds. It is also not just the boat’s still-water speed. It is the harmonic mean of the two speeds: 2(u+v)(u-v) / [(u+v)+(u-v)].

Let a = downstream speed and b = upstream speed. Downstream speed = u + v and upstream speed = u – v, where u is still-water speed and v is stream speed. Subtracting the upstream equation from the downstream one cancels u: (u+v) – (u-v) = 2v. So a – b = 2v, which gives stream speed v = (a – b) / 2.

Let a = downstream speed and b = upstream speed. Downstream speed = u + v and upstream speed = u – v, where u is still-water speed and v is stream speed. Adding these two equations cancels v: (u+v) + (u-v) = 2u. So a + b = 2u, which gives still-water speed u = (a + b) / 2.

Let u = boat speed in still water and v = stream speed. Moving upstream, the current flows against the boat. So the current’s speed is subtracted from the boat’s own speed. Upstream speed = u – v.

Let u = boat speed in still water and v = stream speed. Moving downstream, the current flows the same way as the boat. So the current’s speed adds to the boat’s own speed. Downstream speed = u + v.

The stream flows in one fixed direction. Moving in the opposite direction is called moving upstream. In this direction, the current pushes back against the boat.

The stream flows in one fixed direction. Moving in that same direction is called moving downstream. In this direction, the current helps push the boat along.

Let upstream speed = x km/hr and downstream speed = y km/hr. First trip: 41/x + 57/y = 12 hours. Second trip: 18/x + 51/y = 7 hours. Solving these two equations together gives x = 5 km/hr and y = 15 km/hr. Downstream = B + S and upstream = B – S, so still-water speed = (x + y)/2. Speed in still water = (5 + 15)/2 = 20/2 = 10 km/hr. Check: 41/5 + 57/15 = 12 hours, matching the first trip.

Let upstream speed = x km/hr and downstream speed = y km/hr. First trip: 27/x + 60/y = 21 hours. Second trip: 16/x + 16/y = 10 hours. Solving these two equations together gives x = 2 km/hr and y = 8 km/hr. Downstream = B + S and upstream = B – S, so still-water speed = (x + y)/2. Speed in still water = (2 + 8)/2 = 10/2 = 5 km/hr. Check: 27/2 + 60/8 = 21 hours, matching the first trip.

Let boat speed in still water = B = 16 km/hr and stream speed = S = 4 km/hr. Downstream speed = B + S = 16 + 4 = 20 km/hr. Upstream speed = B – S = 16 – 4 = 12 km/hr. For a round trip with equal distances, average speed is the harmonic mean of the two speeds, not their simple average. Average speed = (2 x downstream speed x upstream speed) / (downstream speed + upstream speed). Average speed = (2 x 20 x 12) / (20 + 12) = 480 / 32 = 15 km/hr. Check: the plain average of 20 and 12 is 16 km/hr, not 15 km/hr, confirming the harmonic-mean formula is needed instead.

Let boat speed in still water = B = 12 km/hr and stream speed = S = 6 km/hr. Downstream speed = B + S = 12 + 6 = 18 km/hr. Upstream speed = B – S = 12 – 6 = 6 km/hr. For a round trip with equal distances, average speed is the harmonic mean of the two speeds, not their simple average. Average speed = (2 x downstream speed x upstream speed) / (downstream speed + upstream speed). Average speed = (2 x 18 x 6) / (18 + 6) = 216 / 24 = 9 km/hr. Check: the plain average of 18 and 6 is 12 km/hr, not 9 km/hr, confirming the harmonic-mean formula is needed instead.

Let boat speed in still water = B = 18 km/hr and stream speed = S = 12 km/hr. Downstream speed = B + S = 18 + 12 = 30 km/hr. Upstream speed = B – S = 18 – 12 = 6 km/hr. For a round trip with equal distances, average speed is the harmonic mean of the two speeds, not their simple average. Average speed = (2 x downstream speed x upstream speed) / (downstream speed + upstream speed). Average speed = (2 x 30 x 6) / (30 + 6) = 360 / 36 = 10 km/hr. Check: the plain average of 30 and 6 is 18 km/hr, not 10 km/hr, confirming the harmonic-mean formula is needed instead.

Let boat speed in still water = B = 16 km/hr and stream speed = S = 12 km/hr. Downstream speed = B + S = 16 + 12 = 28 km/hr. Upstream speed = B – S = 16 – 12 = 4 km/hr. For a round trip with equal distances, average speed is the harmonic mean of the two speeds, not their simple average. Average speed = (2 x downstream speed x upstream speed) / (downstream speed + upstream speed). Average speed = (2 x 28 x 4) / (28 + 4) = 224 / 32 = 7 km/hr. Check: the plain average of 28 and 4 is 16 km/hr, not 7 km/hr, confirming the harmonic-mean formula is needed instead.

Let boat speed in still water = B = 9 km/hr and stream speed = S = 6 km/hr. Downstream speed = B + S = 9 + 6 = 15 km/hr. Upstream speed = B – S = 9 – 6 = 3 km/hr. For a round trip with equal distances, average speed is the harmonic mean of the two speeds, not their simple average. Average speed = (2 x downstream speed x upstream speed) / (downstream speed + upstream speed). Average speed = (2 x 15 x 3) / (15 + 3) = 90 / 18 = 5 km/hr. Check: the plain average of 15 and 3 is 9 km/hr, not 5 km/hr, confirming the harmonic-mean formula is needed instead.

Let boat speed in still water = B = 20 km/hr and stream speed = S = 10 km/hr. Downstream speed = B + S = 20 + 10 = 30 km/hr. Upstream speed = B – S = 20 – 10 = 10 km/hr. For a round trip with equal distances, average speed is the harmonic mean of the two speeds, not their simple average. Average speed = (2 x downstream speed x upstream speed) / (downstream speed + upstream speed). Average speed = (2 x 30 x 10) / (30 + 10) = 600 / 40 = 15 km/hr. Check: the plain average of 30 and 10 is 20 km/hr, not 15 km/hr, confirming the harmonic-mean formula is needed instead.

Let boat speed in still water = B = 8 km/hr and stream speed = S = 4 km/hr. Downstream speed = B + S = 8 + 4 = 12 km/hr. Upstream speed = B – S = 8 – 4 = 4 km/hr. For a round trip with equal distances, average speed is the harmonic mean of the two speeds, not their simple average. Average speed = (2 x downstream speed x upstream speed) / (downstream speed + upstream speed). Average speed = (2 x 12 x 4) / (12 + 4) = 96 / 16 = 6 km/hr. Check: the plain average of 12 and 4 is 8 km/hr, not 6 km/hr, confirming the harmonic-mean formula is needed instead.

Let boat speed in still water = B = 9 km/hr and stream speed = S = 3 km/hr. Downstream speed = B + S = 9 + 3 = 12 km/hr. Upstream speed = B – S = 9 – 3 = 6 km/hr. For a round trip with equal distances, average speed is the harmonic mean of the two speeds, not their simple average. Average speed = (2 x downstream speed x upstream speed) / (downstream speed + upstream speed). Average speed = (2 x 12 x 6) / (12 + 6) = 144 / 18 = 8 km/hr. Check: the plain average of 12 and 6 is 9 km/hr, not 8 km/hr, confirming the harmonic-mean formula is needed instead.

Let boat speed in still water = B = 18 km/hr and stream speed = S = 6 km/hr. Downstream speed = B + S = 18 + 6 = 24 km/hr. Upstream speed = B – S = 18 – 6 = 12 km/hr. For a round trip with equal distances, average speed is the harmonic mean of the two speeds, not their simple average. Average speed = (2 x downstream speed x upstream speed) / (downstream speed + upstream speed). Average speed = (2 x 24 x 12) / (24 + 12) = 576 / 36 = 16 km/hr. Check: the plain average of 24 and 12 is 18 km/hr, not 16 km/hr, confirming the harmonic-mean formula is needed instead.

Let boat speed in still water = B = 9 km/hr and stream speed = S = 1 km/hr. Downstream speed = B + S = 9 + 1 = 10 km/hr. Upstream speed = B – S = 9 – 1 = 8 km/hr. Let d be the one-way distance in km. Time downstream = d/10 and time upstream = d/8. Total time: d/10 + d/8 = 1 hour 21 minutes. Solving this equation gives d = 6 km. Check: time downstream = 6/10 and time upstream = 6/8; these add up to 1 hour 21 minutes.

Let boat speed in still water = B = 7 km/hr and stream speed = S = 3 km/hr. Downstream speed = B + S = 7 + 3 = 10 km/hr. Upstream speed = B – S = 7 – 3 = 4 km/hr. Let d be the one-way distance in km. Time downstream = d/10 and time upstream = d/4. Total time: d/10 + d/4 = 7 hours 42 minutes. Solving this equation gives d = 22 km. Check: time downstream = 22/10 and time upstream = 22/4; these add up to 7 hours 42 minutes.

Let boat speed in still water = B = 8 km/hr and stream speed = S = 6 km/hr. Downstream speed = B + S = 8 + 6 = 14 km/hr. Upstream speed = B – S = 8 – 6 = 2 km/hr. Let d be the one-way distance in km. Time downstream = d/14 and time upstream = d/2. Total time: d/14 + d/2 = 16 hours. Solving this equation gives d = 28 km. Check: time downstream = 28/14 and time upstream = 28/2; these add up to 16 hours.

Let boat speed in still water = B = 8 km/hr and stream speed = S = 4 km/hr. Downstream speed = B + S = 8 + 4 = 12 km/hr. Upstream speed = B – S = 8 – 4 = 4 km/hr. Let d be the one-way distance in km. Time downstream = d/12 and time upstream = d/4. Total time: d/12 + d/4 = 7 hours 20 minutes. Solving this equation gives d = 22 km. Check: time downstream = 22/12 and time upstream = 22/4; these add up to 7 hours 20 minutes.

Let boat speed in still water = B = 11 km/hr and stream speed = S = 7 km/hr. Downstream speed = B + S = 11 + 7 = 18 km/hr. Upstream speed = B – S = 11 – 7 = 4 km/hr. Let d be the one-way distance in km. Time downstream = d/18 and time upstream = d/4. Total time: d/18 + d/4 = 9 hours 10 minutes. Solving this equation gives d = 30 km. Check: time downstream = 30/18 and time upstream = 30/4; these add up to 9 hours 10 minutes.

Let boat speed in still water = B = 11 km/hr and stream speed = S = 1 km/hr. Downstream speed = B + S = 11 + 1 = 12 km/hr. Upstream speed = B – S = 11 – 1 = 10 km/hr. Let d be the one-way distance in km. Time downstream = d/12 and time upstream = d/10. Total time: d/12 + d/10 = 2 hours 45 minutes. Solving this equation gives d = 15 km. Check: time downstream = 15/12 and time upstream = 15/10; these add up to 2 hours 45 minutes.

Let boat speed in still water = B = 12 km/hr and stream speed = S = 8 km/hr. Downstream speed = B + S = 12 + 8 = 20 km/hr. Upstream speed = B – S = 12 – 8 = 4 km/hr. Let d be the one-way distance in km. Time downstream = d/20 and time upstream = d/4. Total time: d/20 + d/4 = 6 hours. Solving this equation gives d = 20 km. Check: time downstream = 20/20 and time upstream = 20/4; these add up to 6 hours.