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

Mar 06, 17 11:44 AM
Learn to divide using repeated subtraction with this easy to follow math lesson
Read More
New math lessons
Your email is safe with us. We will only use it to inform you about new math lessons.