Mathematical game: calculating the average speed of cars

If you drive 100 kilometers at an average speed of 50 km/h then the following 100 kilometers at an average speed of 100 km/h, what is your average over the 200 kilometres? It seems logical to answer 75 km/h, the average between 50 and 100, but is it so sure?

Before answering, let’s think a bit. The important thing is to know in how long the 200 kilometers were covered. The first part of the trip took two hours and the second one hour, so the whole thing was three hours. The average speed is therefore 200/3 = 66.66 km/h, a little less than predicted by a hasty calculation.

The average we used is the harmonic meandifferent from the arithmetic mean.

What is the average speed of the two cars?

As the Paris/Marseille motorway is divided into eight consecutive sections of 100 kilometers each, we recorded the average speeds of two cars (A and B) making this route:

What is the fastest car? What is its average speed?

Responnse :

Car A. To demonstrate this, we are interested in the travel time of each part. For A, this gives successively: 100 / 50, 100 / 100, 100 / 120, etc. The total time is therefore:

(100/50) + (100/100) + (100/120) + (100/130) + (100/80) + (100/100)+ (100/120) + (100/60) i.e. 1459 / 156. Speed ​​is equal to the ratio of 800 to this number, i.e. 85.54 km/h. For B, we find 83.6 km/h. A is therefore faster than B, the opposite of what we obtain using the average arithmetic.