# What does harmonic mean consist of?

#### short answers for big questions

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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