If you drive out at 60 mph and come back the same way at 40 mph, your average speed is 48 mph, not 50. The plain average fails because you spend more time on the slow leg. Average speed is always total distance divided by total time.
How to calculate it with a concrete trip
Pick a length for each direction. With 60 miles the numbers come out clean, and further down you will see that any other length gives the same answer.
Worked example: 60 miles out at 60 mph and 60 miles back at 40 mph
A round trip has two equal legs of 60 miles. What is the average speed for the whole trip?
- Time out. 60 ÷ 60 = 1 h.
- Time back. 60 ÷ 40 = 3/2 h, that is, 1.5 h.
- Totals. 60 + 60 = 120 miles in 1 + 3/2 = 5/2 h.
- Divide. 120 ÷ 5/2 = 120 × 2/5 = 48.
Result: 48 mph.
Check: 48 mph for 2.5 h covers 120 miles, the real distance. At 50 mph it would be 125 miles.
When the numbers come out as fractions of an hour, the dividing fractions calculator does the last step, and fraction of an hour to minutes turns 5/2 h into 150 min.
The short formula: the harmonic mean
When both legs are the same length, the distance cancels and what is left is the harmonic mean of the two speeds a and b: 2ab ÷ (a + b), or equivalently 2 ÷ (1/a + 1/b). With 60 and 40, 2 × 60 × 40 = 4,800 and 4,800 ÷ 100 = 48. With a distance d in each direction, the total time is d/60 + d/40 = d/24, and 2d ÷ (d/24) = 48 for any d: it makes no difference whether the trip is 10 miles or 600 miles. Adding d/60 and d/40 uses the common denominator 120; the adding fractions calculator handles those steps.
English Wikipedia says that the harmonic mean is sometimes used for ratios and rates such as speeds, and that when each leg covers the same distance, the average speed is the harmonic mean of the leg speeds. Each term 1/a is the time it takes to cover each mile at that speed, so the formula averages the times per mile and then inverts the result.
When the arithmetic mean does work
If what repeats is the time and not the distance, the mean of the speeds is correct. One hour at 30 mph and another at 90 mph cover 120 miles in 2 h: 60 mph, the same as (30 + 90) ÷ 2. With the same pair and equal distances, the average is 2 × 30 × 90 ÷ 120 = 45 mph. The question that decides it is which quantity is equal in both legs. To average other quantities with equal weight, use the average fractions calculator.
Table of harmonic means
| Out (mph) | Back (mph) | Real (mph) | Simple average |
|---|---|---|---|
| 90 | 60 | 72 | 75 |
| 120 | 60 | 80 | 90 |
| 80 | 20 | 32 | 50 |
| 100 | 50 | 200/3 = 66.6… | 75 |
| 50 | 50 | 50 | 50 |
The further apart the two speeds are, the further the arithmetic mean drifts from the real one, and they only match when the two speeds are equal. The 100 and 50 case gives 200/3, a repeating decimal (66.6… with the 6 repeating); keep the fraction while you work and round at the end. The guide on exact and repeating decimals explains when that happens.
The puzzle: after going 30 mph, averaging 60 is impossible
An average of 60 mph over 2d miles takes a total time of 2d ÷ 60 = d/30 hours. But driving a distance d at 30 mph already uses d ÷ 30 = d/30 hours. No time is left for the return, so no finite speed can do it. If the trip out were at 40 mph, d/30 − d/40 = d/120 hours would be left, and the return would have to be at 120 mph.
What this formula does not cover
It works when the two legs have the same length and the speed is constant on each. If the distances differ or there are stops, add up the actual times and divide the total distance by the total time. The guide on speed, distance and time with fractions works through those cases. Faucet problems use reciprocals in a similar way, but there the rates are added; you will find them in work-rate problems.
Source: English Wikipedia, article Harmonic mean (consulted October 17, 2026).