Boats & Streams Notes

Last reviewed 22 Sept 2026 · 5 min read

Downstream and upstream

Let $b$ be the speed of the boat in still water and $s$ the speed of the stream, with $b > s$ .

FormulaThe four relations

Downstream d = b + s, Upstream u = b - s

b = \frac{d + u}{2}, s = \frac{d - u}{2}

The last two are the whole chapter: any question that gives the two journey speeds gives the boat and the stream at once.

A log, raft or floating object has no speed of its own, so it travels at the speed of the stream, $s$ .

Useful consequences

FormulaShortcuts worth knowing

If upstream takes $n$ times as long as downstream over the same distance, then $u : d = 1 : n$ and

\frac{b}{s} = \frac{n + 1}{n - 1}

Round trip over a distance $D$ each way:

Total time = \frac{D}{b + s} + \frac{D}{b - s} = \frac{2 D b}{b ^{2} - s ^{2}}, Average speed = \frac{b ^{2} - s ^{2}}{b}

A boat that takes the same time to go $x$ km downstream and $y$ km upstream has $\frac{x}{b + s} = \frac{y}{b - s}$ .

The same formulas describe an aeroplane flying with or against a wind, and a swimmer in a current — only the names change.

Worked examples

Worked ExampleExample 1 — both ways

A boat does 12 km/h in still water and the stream flows at 3 km/h. Find the time for 45 km downstream and 45 km upstream.

Solution. Downstream $= 15$ km/h → $\frac{45}{15} =$ 3 hours. Upstream $= 9$ km/h → $\frac{45}{9} =$ 5 hours.

Worked ExampleExample 2 — finding both speeds

A boat covers 30 km downstream in 2 hours and the same 30 km upstream in 3 hours. Find the speed of the boat and of the stream.

Solution. $d = 15$ , $u = 10$ km/h. $b = \frac{15 + 10}{2} =$ 12.5 km/h, $s = \frac{15 - 10}{2} =$ 2.5 km/h.

Worked ExampleExample 3 — n times as long

A boat takes twice as long to go upstream as downstream over the same distance. Find the ratio of the boat's speed to the stream's.

Solution. $\frac{b}{s} = \frac{n + 1}{n - 1} = \frac{3}{1}$ → 3 : 1.

Worked ExampleExample 4 — round trip

A boat does 10 km/h in still water and the stream 2 km/h. It goes 48 km downstream and returns. Find the total time and the average speed.

Solution. $\frac{48}{12} + \frac{48}{8} = 4 + 6 =$ 10 hours for 96 km → average $=$ 9.6 km/h, which is also $\frac{b ^{2} - s ^{2}}{b} = \frac{100 - 4}{10}$ .

Worked ExampleExample 5 — a return journey

A man rows 18 km downstream in 2 hours and returns in 3 hours. Find both speeds.

Solution. $d = 9$ , $u = 6$ → $b =$ 7.5 km/h, $s =$ 1.5 km/h.

Worked ExampleExample 6 — equal times

A boat goes 20 km downstream in the same time as 12 km upstream. The stream runs at 2 km/h. Find the boat's speed.

Solution. $\frac{20}{b + 2} = \frac{12}{b - 2} \Rightarrow 20 b - 40 = 12 b + 24 \Rightarrow 8 b = 64$ → 8 km/h.

Worked ExampleExample 7 — a floating log

A boat does 9 km/h in still water; the stream 3 km/h. How long will a log take to drift 12 km downstream?

Solution. A log moves at the stream's speed: $\frac{12}{3} =$ 4 hours. The boat's speed is irrelevant.

Worked ExampleExample 8 — total time given

A boat whose still-water speed is 8 km/h takes 4 hours 30 minutes to row to a place 16 km away and back. Find the speed of the stream.

Solution. $\frac{16}{8 + s} + \frac{16}{8 - s} = 4.5 \Rightarrow \frac{16 \times 16}{64 - s ^{2}} = 4.5 \Rightarrow 256 = 288 - 4.5 s^{2}$ $\Rightarrow s^{2} = \frac{32}{4.5} = \frac{64}{9}$ → $s =$ 8/3 km/h (about 2.67 km/h).

Worked ExampleExample 9 — a plane and the wind

An aircraft flies 900 km with a 50 km/h tail wind in 3 hours. How long will the return flight take against the same wind?

Solution. Ground speed out $= 300$ km/h, so the still-air speed is $250$ km/h. Return ground speed $= 200$ km/h → $\frac{900}{200} =$ 4.5 hours.

Worked ExampleExample 10 — distance from a difference

A boat's still-water speed is 15 km/h and the stream's 5 km/h. If the upstream trip takes 2 hours longer than the downstream trip over the same distance, find the distance.

Solution. $\frac{D}{10} - \frac{D}{20} = 2 \Rightarrow \frac{D}{20} = 2$ → 40 km.

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Boats & Streams

Downstream and upstream

Useful consequences

Worked examples

This chapter is in the syllabus of

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