Let's suppose that we want to travel to Lisbon or to Oporto by car, whose distance is 300 km (so as to make the calculation easier). Let´s also assume that Pombal city stays in the middle of our way. If during the first half of the trip (150 km) we drive at 100 km/h and then from Pombal on we start driving at 150km/h, what is the mean of our speed? Lots of people will say that the mean is 125 km/h, since it corresponds to the mathematical mean.

**Arithmetic mean: ** `(100+150)/2 = 250/2 = 125`

But, it´s wrong! Take notice that this trip lasts for 2h30, this is, 2.5 hours. In fact it takes us 1.5 hours to drive to Pombal at a speed of 100km/h, during the first 150 km. To cover the last 150 km at a speed of 150 km/h we need more one hour, which means 2.5 hours. So, if we have travelled at a mean speed of 125 km/h, it would mean that we would have covered 312.5 km, because 125 X 2.5 = 312.5.

What is the mistake? In these situations we should use the harmonic mean as the most correct. What does it consist of? In a simple way, it means “The inverse of the inverse mean”. In this case, it is calculated as it follows:

**Harmonic mean:** `2/(1/100 + 1/150) = 2/(3/300 + 2/300) = 2/(5/300) = 600/5 = 120`

The mean speed of our trip was 120 km/h after all! Let's check if it makes sense: 120 X 2.5 = 300.

**Pay attention:** if the question was the following: In a 300 km trip from Lisbon to Oporto, you travelled half of the time at 100 km/h and the other half of the time at 150 km/h, What would be the mean of the trip? In this case you would have to use the Arithmetic Mean and the answer would be 125 km/h. Next time you travel, make sure your calculations are well done so as to know if it is worth to speed up.

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