Problem 70 - Precalculus: Mathematics 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 \( \frac{15}{4} \).

Step 1: Identify the Given Values

We are given two numbers: 3 and 5. We need to find their harmonic mean.

Step 2: Understand the Formula

The harmonic mean of two numbers, say \( a \) and \( b \), is calculated with the formula: \[H = \frac{2ab}{a + b}\] where \( a = 3 \) and \( b = 5 \) in this case.

Step 3: Plug Values into the Formula

Substitute \( a = 3 \) and \( b = 5 \) into the harmonic mean formula: \[H = \frac{2 \times 3 \times 5}{3 + 5}\] Simplifying this gives: \[H = \frac{30}{8}\]

Step 4: Simplify the Expression

Now, simplify \( \frac{30}{8} \): \[H = \frac{30 \div 2}{8 \div 2} = \frac{15}{4}\] So the harmonic mean of 3 and 5 is \( \frac{15}{4} \).

Key Concepts

Arithmetic Mean

The arithmetic mean is a widely used concept that defines the average of a set of numbers. To find the arithmetic mean of two numbers, you need to sum the numbers together and then divide the result by the count of the numbers. For example, to find the arithmetic mean of 3 and 5:

• Add the numbers: 3 + 5 = 8
• Divide by the number of values, which is 2: \[\frac{8}{2} = 4.\]

This gives us an arithmetic mean of 4. It's essential because it provides a simple measure of central tendency, expressing where the center of a dataset is located. Unlike the harmonic mean, the arithmetic mean is especially useful when dealing with quantities related by simple addition.

Reciprocal

A reciprocal is a unique concept in mathematics, closely tied to fractions and division. The reciprocal of a number is the value that, when multiplied by the original number, results in 1. For instance, the reciprocal of 3 is \[\frac{1}{3}\]because: \(3 \times \frac{1}{3} = 1\).
Understanding reciprocals is crucial for finding harmonic means since the harmonic mean formula includes the reciprocals of the numbers involved. When calculating the harmonic mean, you first need to find the reciprocal of each number, which is then averaged and plugged into the formula. This concept allows one to transform a division problem into a multiplication one, simplifying many mathematical expressions and calculations.

Fraction Simplification

Fraction simplification is a foundational skill in mathematics, facilitating easier interpretation and comparison of fractional values. The process involves reducing a fraction to its simplest form, which means making the numerator and denominator as small as possible while keeping the value the same. For instance, simplify \(\frac{30}{8}\)by dividing both the numerator and the denominator by their greatest common divisor, which is 2:

• Numerator: 30 ÷ 2 = 15
• Denominator: 8 ÷ 2 = 4

Thus, \(\frac{30}{8} = \frac{15}{4}.\)Simplifying fractions makes calculations easier and results more interpretable, and it's crucial while working out the harmonic mean of numbers. Recognizing when fractions can be reduced helps to ensure that final answers are neatly represented and easily understood.