# Harmonic Mean

The **harmonic mean** \( f_{\text{HM}} \) of a list of values \( a_1, \ldots, a_k \) is the mean with the property that the reciprocal of the mean is equal to the average of the reciprocals.

#### Contents

## Definition

Suppose one is given data that are to be weighted by their reciprocals. In that case, one might consider a mean with the property that \( k \) times the reciprocal of the mean equals the sum of the reciprocals of the values:

\[ \frac{k}{f_{\text{HM}}} = \frac 1{a_1} + \frac 1{a_2} + \frac 1{a_3} + \cdots + \frac 1{a_k}. \]

This leads to the **harmonic mean** defined as

\[ f_{\text{HM}} = \frac{k}{\frac 1{a_1} + \frac 1{a_2} + \frac 1{a_3} + \cdots + \frac 1{a_k}}.\]

What is the harmonic mean of \( 1, \frac{3}{2}, 3? \)

We have

\[ \frac{3}{1 + \frac{2}{3} + \frac{1}{3}} = \frac{3}{2}.\ _\square \]

Ben, the ultimate hero, spends 10 minutes to solve a math problem.

The pretty and smarter Gwen only needs 6 minutes to come up with an answer.

Kevin, on the other hand, doesn't attend any school, so he takes 15 minutes for the same task.

What is the average time per person required for this team to solve a math problem?

The head chef **A** can cook one dish in 4 minutes.

His fellow chef **B** can finish in 8 minutes,

A young chef **C** has an unknown capability.

Then two more chefs are hired into the team.

Chef **D** can cook one dish in 6 minutes.

Chef **E** can cook one dish in 9 minutes.

If the average cooking time per person before and after the new hiring is unchanged, in how many minutes can chef **C** cook one dish?

## Properties

- The reciprocal of the harmonic mean is the arithmetic mean of the reciprocals.
- By the QM-AM-GM-HM inequality, the harmonic mean is smaller than either the arithmetic mean or geometric mean and is the smallest of the classical (Pythagorean) means.