Round Trip Problems
A round trip — go downstream a distance d and return the same d upstream — is the topic’s favourite CAT trap. The total time is d/(b + s) + d/(b − s), and the average speed for the whole trip is NOT (D + U)/2. Because equal distances are covered at two different speeds, the correct average is the harmonic mean 2·D·U/(D + U), which is always less than the arithmetic mean. For instance a trip at 15 km/h one way and 10 km/h back averages 2(15)(10)/(25) = 12 km/h, not 12.5. Two clean consequences worth remembering: the upstream leg always eats more time than the downstream leg, and if you are told the total round-trip time, you can solve for an unknown speed by setting up that single time equation. When the current is zero, both legs run at b and the whole trip simplifies to 2d/b.
✅ Solved examples
✏️ Practice — try these, take hints as needed
📝 Topic test — 8 questions
Auto-graded with full solutions; saved to your dashboard. Use the calculator and formula sheet (top-right) any time.
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 |