# Harmonic mean

The harmonic mean (HM) of n numbers ( x1, x2, x3, x4, x5, ... , xn ) is given by the formula below.

HM =
n / 1/x1 + 1/x2 + 1/x3 + 1/x4 + 1/x5 + ... + 1/xn

Notice that n is the number of numbers.

For 2 numbers, HM =
2 / 1/x1 + 1/x2

For 3 numbers, HM =
3 / 1/x1 + 1/x2 + 1/x3

## Calculate the harmonic mean

Example #1:

Find the HM of 3 and 4

HM =
2 / 1/3 + 1/4

HM =
2 / 4/12 + 3/12

HM =
2 / 7/12

HM =
2 × 12 / 7

HM =
24 / 7
= 3.4285

Example #2:

Find the HM of 1, 2, 4, and 10

HM =
4 / 1/1 + 1/2 + 1/4 + 1/10

HM =
4 / 20/20 + 10/20 + 5/20 + 2/20

HM =
4 / 37/20

HM =
4 × 20 / 37

HM =
80 / 37
= 2.1621

## A linear motion problem that leads to the harmonic formula.

A car travels with a speed of 40 miles per hour for the first half of the way. Then, the car travels with a speed of 60 miles per hour for the second half of the way. What is the average speed?

Average speed =
total distance / total time

First notice that it is not possible to use directly the speed formula since we do not know for how long the car kept driving with a speed of 40 m/h and then 60 m/h. However, with some manipulation, we can still tackle the problem.

Let t1 be the time it took to travel the first half of the total distance

Let d be the first half of the total distance.

t1 =
d / 40

Let t2 be the time it took to travel the second half of the total distance

Let d be the second half of the total distance.

t2 =
d / 60

Total time = t1 + t2 = d/40 + d/60

Total distance = d + d = 2d

Now replace these in the formula

Average speed =
total distance / total time

Average speed =
2d / d/40 + d/60

Average speed =
2d / d(1/40 + 1/60)

Cancel d and the average speed =
2 / (1/40 + 1/60)

Now, you can see that it looks like we are calculating the harmonic mean for 2 numbers by using the formula above.

HM = average speed =
2 / (3/120 + 2/120)

HM = average speed =
2 / (5/120)

HM = average speed =
2 × 120 / 5

HM = average speed =
240 / 5
= 48 miles per hour

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