Problem 54 - Precalculus: Mathematical for Calculus

Question

The harmonic mean of two numbers is the reciprocal of the average of the reciprocals of the two numbers. Find the harmonic mean of 3 and 5

Step-by-Step Solution

Answer

The harmonic mean of 3 and 5 is 3.75.

Step 1: Understand the formula

The harmonic mean for two numbers \(a\) and \(b\) is calculated using the formula \(HM = \frac{2}{\frac{1}{a} + \frac{1}{b}}\). This means the harmonic mean is twice the reciprocal of the sum of the reciprocals of the two numbers.

Step 2: Calculate the reciprocals

Calculate the reciprocal of each number: for number 3, the reciprocal is \(\frac{1}{3}\), and for number 5, it is \(\frac{1}{5}\).

Step 3: Find the average of the reciprocals

Add the reciprocals obtained in Step 2: \(\frac{1}{3} + \frac{1}{5}\). The sum is \(\frac{5}{15} + \frac{3}{15} = \frac{8}{15}\).

Step 4: Calculate the reciprocal of the average

Since the harmonic mean is based on the reciprocal of the sum, find the reciprocal of \(\frac{8}{15}\), which is \(\frac{15}{8}\).

Step 5: Apply the harmonic mean formula

Substitute back into the harmonic mean formula: \(HM = \frac{2}{\frac{8}{15}} = 2 \times \frac{15}{8}\). Simplifying, we get \(HM = \frac{30}{8} = \frac{15}{4}\).

Step 6: Simplify the result

Simplify \(\frac{15}{4}\) to a decimal, which is 3.75.

Key Concepts

Reciprocal

In mathematics, the reciprocal of a number is what you get when you flip the number over. Think of it as the number you multiply with the original number to get 1. For example, the reciprocal of 3 is \( \frac{1}{3} \). When you multiply 3 by its reciprocal, \((3 \, \times \, \frac{1}{3})\), you end up with 1. This concept is a key building block for understanding how averages and means work, especially the harmonic mean.

Some properties of reciprocals include:

• The reciprocal of a fraction is obtained by swapping its numerator and denominator.
• Zero does not have a reciprocal because division by zero is undefined.
• The reciprocal of 1 is still 1, because \((1 \, \times \, 1)\) equals 1.

Average of Reciprocals

The average of reciprocals involves finding the reciprocals first and then calculating their average. This is a little different from the regular average (or arithmetic mean) that we learn early on. Instead of summing up the numbers and dividing by the count, you first convert the numbers to their reciprocals.

For example, with two numbers, 3 and 5:

• The reciprocal of 3 is \( \frac{1}{3} \).
• The reciprocal of 5 is \( \frac{1}{5} \).

To find the average of these two reciprocals, \( \frac{1}{3} \) and \( \frac{1}{5} \, \) you add them together: \( \frac{1}{3} \, + \, \frac{1}{5} \, = \, \frac{8}{15} \).

A significant distinction is that this method is focused more on the position of each number relative to 1, instead of their straight arithmetic values.

Formula for Harmonic Mean

The formula for the harmonic mean provides a way to find the mean of a set of numbers based on their reciprocals. The formula for two numbers, say \( a \) and \( b \, \) is expressed as: \[HM = \frac{2}{\frac{1}{a} + \frac{1}{b}}\] Using this formula helps to manage and balance rates and ratios in practical scenarios like speeds or densities rather than values like prices or sums.

Let's break it down with an example:

• First, calculate the reciprocals of the numbers. For 3 and 5, they are \( \frac{1}{3} \) and \( \frac{1}{5} \) respectively.
• Sum these reciprocals: \( \frac{1}{3} + \frac{1}{5} = \frac{8}{15} \).
• The harmonic mean is then twice the reciprocal of this sum: \( HM = \frac{2}{\frac{8}{15}} = 2 \times \frac{15}{8} \).
• Simplifying gives \( HM = \frac{30}{8} = \frac{15}{4} = 3.75\).

The harmonic mean provides us with a different perspective on the dataset, often used when dealing with rates and times. It considers the total reciprocal value, giving more significance to smaller numbers within the set.