Boats and Streams: Upstream, Downstream and Round Trips

A boat on a river has its own speed through the water, called its speed in still water (b), and the river carries it along at the speed of the stream (s). Going with the current, downstream, the two add. Going against it, upstream, the stream takes speed away.

Adding and subtracting these two equations gives the formulas that most questions need:

• Boat speed in still water: b = (D + U) ÷ 2
• Speed of the stream: s = (D − U) ÷ 2

These are not extra facts to memorise. They come from solving two simple equations, so if you forget them in the exam you can rebuild them in a few seconds.

When b and s are given, find the speed for the leg the question asks about and divide.

The stream speed in the same example is (14 − 6) ÷ 2 = 4 km/h. A quick check: 10 + 4 = 14 and 10 − 4 = 6, matching the given speeds.

If distances and times are given instead of speeds, convert each leg to a speed first.

A round trip covers the same distance d twice, once at each speed. The total time is d/(b + s) + d/(b − s). Do not average the speeds: the boat spends more time on the slow leg, so the slow speed counts for more.

To find the distance from a given total time, write the total as d multiplied by the sum of the reciprocals of the two speeds.

For the same distance, time is inversely proportional to speed. If going upstream takes k times as long as going downstream, then (b + s) : (b − s) = k : 1, and solving gives b : s = (k + 1) : (k − 1).

The same structure appears in wind-assisted flight and in escalator questions, where a walker's speed combines with the escalator's.

• ✘ Using b where the question needs D or U.
• ✘ Taking half the sum of D and U as the stream speed. Half the sum is the boat; half the difference is the stream.
• ✘ Averaging upstream and downstream speeds for a round trip.
• ✘ Forgetting that the upstream speed must be positive, so b must be greater than s.

Relative speed works the same way for trains crossing each other. Round trips share their logic with the average speed formula. More formulas are in the Quant cheat code.

How do I remember which speed is upstream?

Upstream means against the current, so the stream slows the boat and its speed is b minus s. Downstream means with the current, so the stream helps and the speed is b plus s. The upstream speed is always the smaller one.

Is the average speed of a round trip equal to the still-water speed?

No. The boat spends longer on the slow upstream leg, so the round-trip average is (b + s)(b − s)/b, which is less than b. Only when there is no current are the two the same.

All problems and solutions on this page are original. No outside facts are relied on; every numeric answer is recomputed by software before publishing.

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