Problem 54 - Algebra and Trigonometry

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: Define the Harmonic Mean Formula

The harmonic mean of two numbers, say \(a\) and \(b\), is given by the formula: \[ HM = \frac{2ab}{a + b} \]

Step 2: Substitute the Given Numbers

Insert the numbers into the harmonic mean formula: \(a = 3\) and \(b = 5\). This gives: \[ HM = \frac{2 \times 3 \times 5}{3 + 5} \]

Step 3: Calculate the Denominator

Find the sum of the numbers in the denominator: \(3 + 5 = 8\).

Step 4: Calculate the Numerator

Find the product of the numbers times 2 in the numerator: \(2 \times 3 \times 5 = 30\).

Step 5: Compute the Harmonic Mean

Divide the numerator by the denominator: \(\frac{30}{8} = 3.75\). Therefore, the harmonic mean of 3 and 5 is 3.75.

Key Concepts

Understanding Reciprocals

The concept of reciprocals is foundational to understanding the harmonic mean. Simply put, the reciprocal of a number is one divided by that number. For example, the reciprocal of 3 is \( \frac{1}{3} \), and the reciprocal of 5 is \( \frac{1}{5} \). Reciprocals are important in various mathematical operations, especially when dealing with fractions. They transform division into multiplication, which can simplify many calculations.
In the context of the harmonic mean, we first find the reciprocals of the numbers we are working with. Then, we use these reciprocals to find the average, which is the next critical step in calculating the harmonic mean. Understanding how reciprocals work helps in connecting seemingly different numbers under a common process, making complex calculations easier and often quicker.

The Mathematical Formula for Harmonic Mean

The harmonic mean involves a specific mathematical formula, which allows us to find an average type that is particularly meaningful in various applications, like calculating averages for rates or ratios. Given two numbers, \(a\) and \(b\), the harmonic mean \(HM\) is calculated using the formula:

• \( HM = \frac{2ab}{a + b} \)

This formula takes the product of the numbers (multiplied by 2) and divides it by their sum. It is derived from averaging the reciprocals of the numbers and then taking the reciprocal of that result.
This formula is very useful when simple averages do not adequately represent the scenario, especially when dealing with quantities like speeds, efficiencies, or densities, where neither arithmetic nor geometric averages would provide an accurate reflection.
Applying this formula with specific numbers, like in our exercise, simplifies problems that might initially seem difficult to solve analytically.

Diving into the Concept of Average

Average is a fundamental concept in mathematics and statistics used to express a central or common value for a set of numbers. Typically, there are different types of averages including arithmetic, geometric, and harmonic.
In the exercise we are dealing with the harmonic mean, a specific type of average that is calculated using the reciprocals of the numbers. What makes the harmonic mean distinct is its specialization for situations where lower values are more significant than higher ones, a common scenario in rates and ratios.

• An arithmetic average sums up the numbers and divides by their count.
• A geometric average multiplies numbers together and then takes the root of the product.
• A harmonic average focuses on the reciprocals of the numbers.

This makes the harmonic mean very sensitive to small values and more effective in appropriately representing a dataset in certain contexts, unlike some other mean calculations which might overweight larger values. Knowing which type of average to use in which situation is instrumental in achieving accurate and meaningful results.