Problem 37 - Algebra 2

Question

STATISTICS For Exercises 36 and \(37,\) use the following information. A number \(x\) is the harmonic mean of \(y\) and \(z\) if \(\frac{1}{x}\) is the average of \(\frac{1}{y}\) and \(\frac{1}{z}\) What is the harmonic mean of 5 and 8\(?\)

Step-by-Step Solution

Answer

The harmonic mean of 5 and 8 is approximately 6.154.

Step 1: Understand the Harmonic Mean Definition

The harmonic mean of two numbers, \(y\) and \(z\), is given in such a way that \(\frac{1}{x}\) is the average of \(\frac{1}{y}\) and \(\frac{1}{z}\). This means \(\frac{1}{x} = \frac{1}{2} \left( \frac{1}{y} + \frac{1}{z} \right)\).

Step 2: Substitute Values Given

We substitute \(y = 5\) and \(z = 8\) into the harmonic mean formula. This gives us \(\frac{1}{x} = \frac{1}{2} \left( \frac{1}{5} + \frac{1}{8} \right)\).

Step 3: Calculate the Fractions Inside the Parentheses

Calculate \(\frac{1}{5}\) and \(\frac{1}{8}\): \(\frac{1}{5} = 0.2\) and \(\frac{1}{8} = 0.125\). Add them together: \(0.2 + 0.125 = 0.325\).

Step 4: Find the Average

Take the average of \(0.325\): \(\frac{1}{2} \times 0.325 = 0.1625\). So \(\frac{1}{x} = 0.1625\).

Step 5: Solve for x

To find \(x\), take the reciprocal of \(0.1625\), which gives \(x = \frac{1}{0.1625} = 6.153846153846154\).

Step 6: Round off the Answer

The harmonic mean is typically rounded to a few decimal places for simplicity. Thus, \(x \approx 6.154\).

Key Concepts

Understanding the Average in Harmonic Mean

The term 'average' commonly brings to mind the arithmetic mean, where you sum all values and divide by their number. However, the harmonic mean is a different kind of average that is particularly useful in specific situations like rates and ratios. It is calculated in a way that

• places more weight on smaller numbers,
• is often used when the numbers are rates or proportions,
• and provides a better measure of central tendency for certain types of data.

In the context of harmonic mean, the average is not about summing numbers, but rather about averaging reciprocal values. This changes how we think about combining numbers and is especially effective when dealing with situations like calculating average speed or density.

Reciprocal Numbers in Harmonic Mean

A reciprocal of a number is simply 1 divided by that number. For instance, the reciprocal of 5 is \(\frac{1}{5}\), which equals 0.2. In the harmonic mean, we take the reciprocal of each number we wish to average. These reciprocals are crucial because they invert the original values, reflecting a unique relationship that highlights smaller numbers.
When working with the harmonic mean, understanding reciprocals is essential as it forms the foundation of the calculation process. This step might be surprising if you're used to arithmetic means, where numbers are just added together. Here, by taking reciprocals, we transform the problem into one that focuses on the inverse of the data, providing insight into rates per unit rather than total sums.

Using Fractions in Calculations

Fractions are fundamental to understanding the harmonic mean because they're often the first step in converting numbers to their reciprocals. In our example, \(\frac{1}{5} = 0.2\) and \(\frac{1}{8} = 0.125\). These fractions help us work through the problem without having to deal immediately with decimals, allowing us to keep the mathematical process clear.
Fractions also allow for a clear presentation of parts of the whole, which is why they're used in harmonic means where individual rates or pieces of information are combined. By using fractions, you can easily add these parts together, convert them to decimals, and find the mean. Remember, when calculating harmonic means, being comfortable with fraction arithmetic will be a huge advantage. This concept is a cornerstone of achieving accurate results when dealing with inverse values.

Step-by-Step Calculation Process

To find the harmonic mean, follow these clear steps. First, find the reciprocals of each number you are averaging. Then add these reciprocal values together. In our example, \(\frac{1}{5} + \frac{1}{8} = 0.2 + 0.125 = 0.325\).
Next, find the average of these reciprocals by dividing the sum by the number of values. With our example, we have two numbers, so: \(\frac{1}{2} \times 0.325 = 0.1625\). This gives us \(\frac{1}{x} = 0.1625\).
Finally, to find \(x\), take the reciprocal of your result: \(x = \frac{1}{0.1625}\), which simplifies to \(x \approx 6.154\). Rounding may be applied depending on the level of precision required. By following these steps systematically, the often complex nature of harmonic means becomes straightforward and manageable.