Average speed on a round trip:
why 40 and 60 km/h do not average to 50

The trap in adding two speeds and halving

You drive to a town at 30 km/h and return at 60 km/h. Most people say the average is 45 km/h. It is not. You spent far longer at the slow speed, so the slow leg weighs more in the overall journey than the fast one.

Average speed has a single definition that never changes: total distance divided by total time. A plain average of two speeds equals that only when the two legs take the same amount of time. When the legs cover the same distance, they do not take the same time, and the arithmetic mean breaks.

The same confusion appears in cycling, running and flight planning. A tailwind outbound and headwind back does not cancel: the time lost into the wind exceeds the time gained with it, for the same reason as the 30 and 60 km/h example. Pilots allow for exactly this effect when computing fuel for a round trip.

Building the answer from distance and time

Let each leg have distance d, with speeds a and b. The first leg takes d ÷ a and the second takes d ÷ b. The whole trip covers 2d in d/a + d/b.

The distance d cancels, which is why the shortcut needs no distance at all. Whether the route is 10 km or 1,000 km, driving it each way at 30 and 60 km/h gives the same average. The result of 2 × 30 × 60 ÷ 90 is exactly 40 km/h, a figure that surprises people who expected 45.

It helps to see the same result through reciprocals. The reciprocal of a speed is a pace, the time taken per kilometre. Paces can be averaged ordinarily when the distances are equal, because time simply adds. At 30 km/h the pace is 2 minutes per kilometre and at 60 km/h it is 1 minute, so the average pace is 1.5 minutes per kilometre, which converts back to 40 km/h. The harmonic mean is nothing more than an arithmetic average taken in pace terms and then turned back into a speed.

Worked example: 120 km each way at 40 and 60 km/h

The plain average would have been 50 km/h, which would imply a journey time of 4.8 hours, about 12 minutes shorter than reality. For a route that you plan to schedule against, such an optimistic error matters.

Three or more equal-distance legs

With n legs of equal length, the generalisation takes n over the sum of the reciprocal speeds. For three equal legs at 40, 50 and 60 km/h, the reciprocals add to 0.025 + 0.020 + 0.01667 = 0.06167, and 3 ÷ 0.06167 is about 48.6 km/h.

The same expression applies to any rate where the amount of work done in each stage is the same: litres per minute of several pumps filling equal tanks, or pages per hour of readers who each read the same chapter. In each case the thing that is equal is the numerator of the rate, and time is what differs.

Which average goes with which situation

The last row is the general case and the safest habit. Log the distance and time of every stage, add each column, and divide. The harmonic mean is a special case of that rule, which is why remembering the principle beats memorising the formula.

Fuel economy and unequal distances

The same logic governs any rate measured per unit of something consumed. A car does 12 km per litre in city driving and 20 km per litre on the highway. If a trip has 100 km of each, the fuel used is 100 ÷ 12 + 100 ÷ 20 = 8.33 + 5.00 = 13.33 litres for 200 km, so the average is 15 km per litre. The shortcut 2 × 12 × 20 ÷ 32 gives the same 15, while the plain average of 16 would flatter the car.

When distances are not equal, use total distance over total time directly. Suppose 60 km is covered at 30 km/h and 180 km at 90 km/h. The times are 2 h and 2 h, the total is 240 km in 4 h, and the average is 60 km/h. The harmonic shortcut, which assumes equal distances, would have given 45 km/h, so it applies only when each leg is the same length.

A good habit when a problem mentions two speeds is to ask what is held fixed. If it is distance, time must differ and the harmonic mean applies. If it is time, distance differs and the plain average applies. Everything else is total distance divided by total time.

Practical limits

Real roads add signals, gradients and traffic that change speed continuously, so the two-speed model is a simplification. It is still an excellent sanity check for trip planning and for reading speed claims such as the one on a fitness tracker, where an out-and-back run with a faster return may report a different average than the pace data suggests.

A general mean calculator can give you the arithmetic figure to compare against, and a trip-time tool can convert the resulting average into an arrival time. Neither replaces the key step of choosing the correct type of average for your data.