Problem 58 - 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 \( \frac{15}{4} \).

Step 1: Understand the Formula for the Harmonic Mean

The harmonic mean \( H \) of two numbers \( a \) and \( b \) is given by the formula: \[ H = \frac{2}{\frac{1}{a} + \frac{1}{b}} \]. This means we first need to find the reciprocals of the two numbers, then find the average of these reciprocals, and finally take the reciprocal of that average.

Step 2: Calculate the Reciprocals of the Numbers

To find the harmonic mean of 3 and 5, start by calculating the reciprocals of these numbers. The reciprocal of 3 is \( \frac{1}{3} \) and the reciprocal of 5 is \( \frac{1}{5} \).

Step 3: Find the Average of the Reciprocals

Next, calculate the average of the reciprocals. This can be done by adding the reciprocals and dividing by 2: \( \frac{1}{3} + \frac{1}{5} = \frac{5}{15} + \frac{3}{15} = \frac{8}{15} \). Then, the average is \( \frac{\frac{8}{15}}{2} = \frac{8}{30} = \frac{4}{15} \).

Step 4: Calculate the Harmonic Mean

The harmonic mean is the reciprocal of the average reciprocal calculated in the last step. Thus, the harmonic mean is \( \frac{1}{\frac{4}{15}} = \frac{15}{4} \).

Key Concepts

Reciprocals

A reciprocal of a number is like an inverse that, when multiplied by the original number, results in 1. For instance, the reciprocal of a number like 3 is represented as \( \frac{1}{3} \). When you multiply 3 and \( \frac{1}{3} \), you get 1, demonstrating their reciprocal relationship. Understanding reciprocals is essential in many mathematical operations, such as finding the harmonic mean.

They can seem tricky at first, but here’s a little trick to remember:

• For any non-zero integer \( n \), the reciprocal is \( \frac{1}{n} \).
• For fractions, swap the numerator and denominator. For example, the reciprocal of \( \frac{2}{3} \) is \( \frac{3}{2} \).

Knowing how to find reciprocals quickly will aid in calculating various means and averages, particularly in this exercise about the harmonic mean.

Average of Fractions

When you need to find the average of fractions, you essentially follow the same steps as you would for any numbers: add them up and divide by the number of fractions. However, there’s an added step due to fractions' nature.

Consider two fractions, \( \frac{1}{3} \) and \( \frac{1}{5} \). To add them, they must share a common denominator. In this case, the least common denominator is 15. Convert these fractions:

• \( \frac{1}{3} \) becomes \( \frac{5}{15} \)
• \( \frac{1}{5} \) becomes \( \frac{3}{15} \)

Now, add them: \( \frac{5}{15} + \frac{3}{15} = \frac{8}{15} \). Divide the sum by 2 (since there are two fractions): \( \frac{8}{30} \). Simplify it to \( \frac{4}{15} \). Understanding this makes it easier to calculate means that involve fractional values.

Step-by-Step Solution

Solving problems step-by-step allows us to grasp each element of the process clearly, ensuring there's no confusion about where a final result comes from. For the harmonic mean of numbers 3 and 5, walk through the procedure rather than skipping to the answer.

First, find the reciprocals \( \frac{1}{3} \) and \( \frac{1}{5} \). Next, compute the average of these reciprocals by finding their sum, \( \frac{8}{15} \), and dividing by 2. Convert this division into multiplication for ease, knowing \( \frac{8}{30} = \frac{4}{15} \). Finally, to determine the harmonic mean, take the reciprocal of this average: \( \frac{1}{\frac{4}{15}} = \frac{15}{4} \). This orderly approach not only gives the correct harmonic mean but also reinforces your understanding of each mathematical step.