Boats and Streams
Downstream the river helps, upstream it fights. Boat speed is the average of the two; stream speed is half the gap.
Why this topic matters
Rowing with the river's flow, the water carries you: you go faster. Against it, the water pushes back: you go slower.
Boats and streams is relative speed with the river as the second mover. Two formulas cover every question.
Downstream and upstream
Boat speed in still water b, stream speed s. Downstream (with the flow) = b + s. Upstream (against) = b - s.
Going back: b = (down + up) / 2, s = (down - up) / 2.
- Down = 12 km/h, Up = 6 km/h.Distance / time for each.
- b = (12 + 6)/2 = 9. s = (12 - 6)/2 = 3.
- Boat 9 km/h, stream 3 km/h.
- Down 6 km/h, up 4 km/h.
- d/6 + d/4 = 5/4 hours.75 minutes = 5/4 h.
- (2d + 3d)/12 = 5/4, so 5d = 15.Common denominator 12.
- d = 3 km.
Formula summary
Shortcuts and tricks
Speeds from distances
Use it when: '20 km down in 2 h, 12 km up in 2 h'.
- Down 10, up 6.
- Boat 8, stream 2.
- (10 + 6)/2.
Before you move on, you should be able to...
- write downstream and upstream speeds
- get boat and stream speeds from them
- solve round-trip time questions
Practice questions (4)
Try each one before opening the solution. Or practise them one by one so your score is saved.
easy Boat 15 km/h in still water, stream 3 km/h. Time for 36 km down and back?
- 36/18 + 36/12.
Answer: 5 h
medium A boat takes twice as long upstream as downstream for the same distance. Boat 9 km/h. Stream?
- b + s = 2(b - s) → b = 3s.
Answer: 3 km/h
hard A man rows 48 km and back in 14 h. He can row 4 km downstream in the time he rows 3 km upstream. Stream speed?
- Down:up = 4:3 = 4k:3k.
- 12/k + 16/k = 14 → k = 2.
- Down 8, up 6, stream 1.
Answer: 1 km/h
hard A boat goes 30 km upstream and 44 km downstream in 10 h, and 40 km up and 55 km down in 13 h. Stream speed?
- Up 5, down 11 fits both (6 + 4 = 10, 8 + 5 = 13).
- Stream = (11 - 5)/2.
Answer: 3 km/h
Class plan (2 hours, for trainers)
A tested order for teaching this topic in one 2-hour session. Present mode follows the same order.
| 10 min | Warm-up. |
| 30 min | Down and up with the river picture. |
| 25 min | Finding b and s. |
| 20 min | Round trips. |
| 30 min | Practice set. |
| 5 min | Recap. |