About Boats and Streams
A boat on a river has two speeds: its own speed in still water, and the speed of the current. Going downstream the two add together, and going upstream the current works against the boat and they subtract. Nearly every question in the topic is solved by writing those two speeds down and then using the ordinary distance formula on each leg.
What you need to understand
- Downstream speed is the speed of the boat plus the speed of the stream, and upstream speed is the boat minus the stream.
- The speed of the boat in still water is half the sum of the downstream and upstream speeds.
- The speed of the stream is half the difference between the downstream and upstream speeds.
- The distance is the same in both directions, so the slower direction always takes longer.
- A boat makes headway against the current only while its still water speed is greater than the speed of the stream.
- If only a time each way is given, the two speeds have to be found from distance and time before anything else can be done.
Formulas to remember
How to work through these questions
- Find the downstream speed as distance divided by time, and the upstream speed the same way, before doing anything else.
- Add the two speeds and halve the result to get the speed of the boat in still water.
- Subtract the two speeds and halve the result to get the speed of the stream.
- When the still water speed and the stream are given instead, build the two journey speeds by adding and subtracting first.
- Check that the still water speed is greater than the stream speed. If it is not, the boat could not have travelled upstream.
Mistakes that cost marks
- Halving the difference but not the sum, or halving the sum but not the difference.
- Reporting the downstream speed as though it were the still water speed.
- Adding the stream speed where it should be subtracted, which counts the current twice.
- Mixing hours with minutes before dividing.
- Using one time for both directions when the two speeds are different.
Worked example
A boat covers 30 km downstream in 2 hours and the same 30 km upstream in 3 hours. Find the speed of the boat in still water.
- Downstream speed = 30 / 2 = 15 km/hr.
- Upstream speed = 30 / 3 = 10 km/hr.
- Speed in still water = (15 + 10) / 2 = 12.5 km/hr.
- Check: the stream is then (15 - 10) / 2 = 2.5 km/hr, and 12.5 + 2.5 = 15 while 12.5 - 2.5 = 10, which matches both legs.
Answer: 12.5 km/hr
Practice questions with answers
A few Boats and Streams questions with the full solution shown, so you can see
how the method is applied before you attempt the timed set.
Question 1
Going 84 km with the stream takes 6 hours, and returning 84 km against it takes 7 hours. Find the boat's speed in still water.
Answer: Option B — with explanation
Downstream speed = 84 / 6 = 14 km/hr, and upstream speed = 84 / 7 = 12 km/hr.
The speed in still water is the average of the two, because the stream helps on one leg and hinders on the other.
= (14 + 12) / 2 = 13 km/hr.
Common mistakes
- gave the downstream speed: 14
- gave the speed of the stream instead of the boat: 1
- took the harmonic mean of the two speeds: 12.92
Question 2
The speed of a boat downstream is 20 km/hr and upstream is 4 km/hr. Find the speed of the stream.
Answer: Option D — with explanation
The stream adds to the boat's own speed going downstream and subtracts from it going upstream.
So the speed of the stream is half the difference of the two speeds.
= (20 - 4) / 2 = 8 km/hr.
Common mistakes
- gave the downstream speed: 20
- halved the sum instead of the difference: 6
- forgot to halve the difference: 16
Question 3
A boat travels 270 km with the current in 6 hours. Given that the current is 8 km/hr, find the speed of the boat in still water.
Answer: Option D — with explanation
Downstream speed = 270 / 6 = 45 km/hr.
Downstream speed = boat + stream, so the boat's own speed = 45 - 8 = 37 km/hr.
Common mistakes
- added the stream speed instead of subtracting it: 53
- gave the speed of the stream: 8
- reported the downstream speed: 45
Question 4
Going 100 km with the stream takes 5 hours, and returning 100 km against it takes 10 hours. Find the boat's speed in still water.
Answer: Option A — with explanation
Downstream speed = 100 / 5 = 20 km/hr, and upstream speed = 100 / 10 = 10 km/hr.
The speed in still water is the average of the two, because the stream helps on one leg and hinders on the other.
= (20 + 10) / 2 = 15 km/hr.
Common mistakes
- gave the speed of the stream instead of the boat: 5
- took the harmonic mean of the two speeds: 13.33
- gave the downstream speed: 20
Question 5
A boat covers 195 km downstream in 5 hours. If the stream flows at 4 km/hr, find the speed of the boat in still water.
Answer: Option A — with explanation
Downstream speed = 195 / 5 = 39 km/hr.
Downstream speed = boat + stream, so the boat's own speed = 39 - 4 = 35 km/hr.
Common mistakes
- added the stream speed instead of subtracting it: 43
- reported the downstream speed: 39
- subtracted the stream speed twice over: 31
Frequently asked questions
How do I find the speed of the stream?
Halve the difference between the downstream and the upstream speed. If those are 15 km/hr and 10 km/hr, the stream flows at 2.5 km/hr.
What if only one direction is given?
You also need either the speed of the stream or the still water speed. With only one journey there are two unknowns and the question cannot be solved.
Why is the still water speed the average of the two journey speeds?
Because the stream adds exactly as much on the way downstream as it takes away on the way back. Averaging the two speeds makes those two effects cancel.
Can a boat travel upstream at the same speed as the stream?
No. If the stream is as fast as the boat, the boat makes no progress at all, and if the stream is faster it is swept backwards.
