Boats & Streams

Call the boat's speed in still water b and the stream's speed s.

• Downstream (with the current): b + s
• Upstream (against it): b − s

Reverse those two lines and you get the pair that solves most questions:

• b = (downstream + upstream)/2
• s = (downstream − upstream)/2

Four types:

1. Given b and s, find the two speeds — one addition, one subtraction.
2. Given the two speeds, find b and s — half the sum and half the difference.
3. Time for a one-way trip — pick the right speed first, then distance ÷ speed.
4. There and back in total time T. d/(b+s) + d/(b−s) = T, which simplifies to d = T(b² − s²)/(2b).

The single most common mistake is using the still-water speed for a journey that is actually on the river. Before dividing, write down which way the boat is going. A second guard: upstream time is always longer than downstream time for the same distance — if your answer says otherwise, the speeds are swapped.

Note that types 2 and 3 also cover "a man rows to a place and returns" — the same distance is being done at two different speeds, so the average speed is (b² − s²)/b, never b.

downstream = b + s · upstream = b − s · b = (down + up)/2 · s = (down − up)/2 · round trip: d = T(b² − s²)/(2b)

A man rows downstream at 15 km/h and upstream at 9 km/h. Find the speed of the boat in still water and of the stream. → Boat = (15 + 9)/2 = 12 km/h; stream = (15 − 9)/2 = 3 km/h. Sanity check: 12 + 3 = 15 downstream and 12 − 3 = 9 upstream, both matching the question. Note the stream is the half-difference, not the difference — 6 km/h would make the upstream speed 6, not 9.