Harmonic mean
The harmonic mean (HM) of n numbers ( x
_{1}, x
_{2}, x
_{3}, x
_{4}, x
_{5}, ... , x
_{n} ) is given by the formula below.
HM =
n
/
1/x_{1} + 1/x_{2} + 1/x_{3} + 1/x_{4} + 1/x_{5} + ... + 1/x_{n}
Notice that n is the number of numbers.
For 2 numbers, HM =
2
/
1/x_{1} + 1/x_{2}
For 3 numbers, HM =
3
/
1/x_{1} + 1/x_{2} + 1/x_{3}
Calculate the harmonic mean
Example #1:
Find the HM of 3 and 4
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
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 t
_{1} be the time it took to travel the first half of the total distance
Let d be the first half of the total distance.
Let t
_{2} be the time it took to travel the second half of the total distance
Let d be the second half of the total distance.
Total time = t
_{1} + t
_{2} = 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

Jun 08, 17 01:52 PM
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