Still Water Speed
The two recovery formulas are the heart of CAT-level boat problems: b = (D + U)/2 and s = (D − U)/2, where D and U are the downstream and upstream speeds. So the still-water speed is the average of the two effective speeds, and the stream speed is half their difference — memorise both. When the data is given as times for the same distance, a slicker route exists: since time is inversely proportional to speed, b : s = (t↑ + t↓) : (t↑ − t↓). For example, if upstream takes twice as long as downstream, t↑ : t↓ = 2 : 1, so b : s = 3 : 1. This ratio trick lets you answer many questions without ever finding the actual distance. Always sanity-check that your b comes out larger than s; a stream faster than the boat is physically impossible and signals an arithmetic slip.
✅ Solved examples
✏️ Practice — try these, take hints as needed
📝 Topic test — 8 questions
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Formula Reference Sheet
Effective speeds & recovery
| Downstream speed | D = b + s |
|---|---|
| Upstream speed | U = b − s |
| Still-water (boat) speed | b = (D + U) / 2 |
| Stream (current) speed | s = (D − U) / 2 |
| Time = Distance ÷ speed | t = d / (b ± s) |
Round trips & ratios
| Round-trip average speed | 2·D·U / (D + U) (harmonic mean) |
|---|---|
| b : s from times (same distance) | b : s = (t↑ + t↓) : (t↑ − t↓) |
| b : s from distances (equal time) | b : s = (d↓ + d↑) : (d↓ − d↑) |
| Total time for distance d each way | d/(b+s) + d/(b−s) |
| Still-water vs stream time ratio | t_still : t_against = (b−s) : b |