Problem 36 - 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 \(\frac{1}{z}\) Eight is the harmonic mean of 20 and what number?

Step-by-Step Solution

Answer

The number is 5.

Step 1: Understanding the Harmonic Mean Formula

To find the harmonic mean of two numbers, we need to understand that the harmonic mean \( x \) of numbers \( y \) and \( z \) is given by the relation \( \frac{1}{x} = \frac{1}{2} \left( \frac{1}{y} + \frac{1}{z} \right) \). In this problem, \( x = 8 \) and \( y = 20 \). We need to find \( z \).

Step 2: Substitute Known Values

Substituting \( x = 8 \) and \( y = 20 \) into the harmonic mean equation gives us \( \frac{1}{8} = \frac{1}{2} \left( \frac{1}{20} + \frac{1}{z} \right) \).

Step 3: Clear the Fraction

Multiply both sides of the equation by 2 to eliminate the fraction on the right-hand side: \( \frac{2}{8} = \frac{1}{20} + \frac{1}{z} \). Simplify \( \frac{2}{8} \) to get \( \frac{1}{4} \).

Step 4: Reorganize the Equation

Rearrange the equation to isolate \( \frac{1}{z} \): \( \frac{1}{z} = \frac{1}{4} - \frac{1}{20} \).

Step 5: Simplify the Expression

Find a common denominator for \( \frac{1}{4} \) and \( \frac{1}{20} \). The common denominator is 20. Rewrite \( \frac{1}{4} \) as \( \frac{5}{20} \). Then the equation becomes \( \frac{1}{z} = \frac{5}{20} - \frac{1}{20} \).

Step 6: Perform the Subtraction

Subtract \( \frac{1}{20} \) from \( \frac{5}{20} \) to get \( \frac{4}{20} \), which simplifies to \( \frac{1}{5} \). So, \( \frac{1}{z} = \frac{1}{5} \).

Step 7: Solve for z

Since \( \frac{1}{z} = \frac{1}{5} \), we can say that \( z = 5 \).

Key Concepts

Fractions

Fractions are a way to represent parts of a whole. They consist of a numerator on top, which tells us how many parts we have, and a denominator on the bottom, which tells us into how many parts the whole is divided. For example, in the fraction \( \frac{1}{4} \), the numerator is 1 and the denominator is 4, representing one part out of four.

Understanding fractions is essential in solving mathematical problems, especially when dealing with equations involving fractions. When adding or subtracting fractions, it's crucial to have a common denominator. This means the bottom numbers must be the same, enabling you to easily perform operations. If not already common, you need to find a least common multiple to proceed.

Fractions appear in a lot of different contexts, such as measurements in recipes or expressing probabilities. Mastery in dealing with them can make complex mathematical tasks feel much more manageable.

Equation Solving

Equation solving is like solving a puzzle. It is about finding the value of an unknown variable that makes the equation true. You start with an equation, like \( \frac{1}{8} = \frac{1}{2} \left( \frac{1}{20} + \frac{1}{z} \right) \), and perform operations to isolate the unknown variable (in this case, \( z \)).

We can simplify equations by performing the same operation on both sides, such as adding, subtracting, multiplying, or dividing. Solving equations often involves rearranging them to bring all terms involving the unknown to one side and everything else on the other.

• Step-by-step simplification is key. Start by simplifying the equation as much as possible.
• Next, isolate the variable you're solving for by systematically undoing operations in reverse order.
• Finally, check your solution by substituting back into the original equation to ensure it holds true.

Equation solving helps develop strong analytical skills and is a fundamental skill in mathematics, physics, and engineering.

Common Denominator

Finding a common denominator is an essential skill when working with fractions. It allows you to perform addition or subtraction on fractions that have different denominators.

To find the common denominator:

• Identify the denominators you are working with (e.g., 4 and 20 in the problem).
• Find the smallest number that both denominators divide into. This is your least common denominator (LCD). For 4 and 20, the LCD is 20.
• Convert each fraction to an equivalent fraction with the common denominator. For instance, \( \frac{1}{4} \) becomes \( \frac{5}{20} \).

Having a common denominator simplifies the process of combining fractions, ensuring that you're only adding or subtracting numerators. This technique is crucial for solving problems involving the harmonic mean and many other mathematical concepts.