Formula

Still-water speed and current from the two legs

Arithmetic • Time, Speed and Distance • Boats and Streams

Formula

b = \frac{u + v}{2}, c = \frac{u - v}{2}

b

is the speed in still water and

c

the speed of the current, the downstream speed is

u = b + c

and the upstream speed is

v = b - c

; hence

b = \frac{u + v}{2}

and

c = \frac{u - v}{2}

How to use it

1Turn each leg into a speed first, by dividing its distance by its time, before using either formula.
2Half the sum is the boat and half the difference is the current, which is the standard sum-and-difference pair.
3The same relations govern an aircraft flying with a tailwind and against a headwind.

A boat covers

36

km downstream in

3

hours and

20

km upstream in

4

hours. What is the speed of the current?

Step-by-step

1
$u = \frac{36}{3} = 12$ km/hour and $v = \frac{20}{4} = 5$ km/hour
Why: Each leg yields one effective speed.
2
$c = \frac{12 - 5}{2} = 3.5$ km/hour

Answer

3.5

km/hour

Watch out

The formulas take speeds, not distances: halving a difference of distances is the standard wrong move. The upstream speed

b - c

must be positive, or the boat makes no headway at all.

Builds on

Also known as

boats and streams, downstream and upstream speed, speed of current, still water speed