The reciprocal of the arithmetic mean of the reciprocal of all numbers.
That sounds like a tongue twister as defined split into three steps very simple:
- All figures take reciprocal
- These reciprocal of the arithmetic mean calculation
- Previous calculations of taking the reciprocal
Example 1:
30 downstream speed, upstream speed is 20, the average speed is how much?
Speed = distance / time for the same distance, the same molecule.
Assumed that the distance is 1, then the denominator is downstream speed: 1/30, countercurrent to: 1/20.
Mean denominator is: (1/30 + 1/20) / 2
1 divided by the average molecular denominator then find the average speed: 2 / (1/30 + 1/20) * 30 * 2 = 20 / (30 + 20)
If you uphill speed is 2 meters per second, down the mountain as fast as 6 meters per second (assuming that go uphill and downhill is the same mountain road). So, the whole of your average speed is how much?
This trip is one of the primary issues in the easiest question wrong, a child who can not understand life and death issue. The answer is not four meters per second, but three meters per second. Let's assume that the whole is S m, the mountain time is S / 2, down time is the S / 6, the total distance round-trip is 2S, total time round is S / 2 + S / 6, and thus the whole average speed 2S / (S / 2 + S / 6) = 3.
In fact, it is easy to see that, if the speed of the front half of the distance is a, the distance is half rate is b, then the total average velocity should be less than (a + b) / 2. This is because you will spend more time on slow half away, so the average speed slow down. In fact, the total harmonic mean the average speed should be a and b, i.e., 2 / (1 / a + 1 / b), it is easy to prove that the harmonic mean is always less than or equal to the arithmetic mean.