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




Recent Articles

  1. How to estimate fractions

    Dec 11, 17 08:45 AM

    Learn how to estimate fractions with this easy to follow lesson.

    Read More

New math lessons

Your email is safe with us. We will only use it to inform you about new math lessons.

            Follow me on Pinterest





Recent Lessons

  1. How to estimate fractions

    Dec 11, 17 08:45 AM

    Learn how to estimate fractions with this easy to follow lesson.

    Read More

  Our Top Pages

Formula for percentage

Compatible numbers

Basic math test

Basic math formulas

Types of angles

Math problem solver

Algebra word problems

Surface area of a cube

Finding the average 

Scale drawings

Tough Algebra Word Problems.

If you can solve these problems with no help, you must be a genius!


Everything you need to prepare for an important exam!

K-12 tests, GED math test, basic math tests, geometry tests, algebra tests. 


Real Life Math Skills

Learn about investing money, budgeting your money, paying taxes, mortgage loans, and even the math involved in playing baseball.

 Recommended

Scientific Notation Quiz

Types of Angles Quiz

Graphing Slope Quiz

Adding and Subtracting Matrices Quiz  

Factoring Trinomials Quiz 

Solving Absolute Value Equations Quiz  

Order of Operations Quiz