Problem 17 - Prealgebra

Question

Find each of the following products. (Multiply.) $$\frac{x}{y} \cdot \frac{y}{z} \cdot \frac{z}{x}$$

Step-by-Step Solution

Answer

The product is 1.

Step 1: Identify the Fractions

We start by identifying the fractions given in the problem: \( \frac{x}{y} \), \( \frac{y}{z} \), and \( \frac{z}{x} \).

Step 2: Multiply the Numerators

Multiply the numerators of the fractions: \( x \times y \times z \). The product of the numerators is \( x \cdot y \cdot z \).

Step 3: Multiply the Denominators

Multiply the denominators of the fractions: \( y \times z \times x \). The product of the denominators is \( y \cdot z \cdot x \).

Step 4: Form the Combined Fraction

Combine the products from Step 2 and Step 3 to form a single fraction: \( \frac{x \,y \,z}{y \,z \,x} \).

Step 5: Simplify the Fraction

Notice that the terms \( x \), \( y \), and \( z \) in the numerator and denominator are identical, allowing for cancellation: \( \frac{x \, y \, z}{y \, z \, x} = 1 \).

Step 6: Finalize the Result

After simplification, the final result of the product of fractions is 1.

Key Concepts

Simplifying Fractions

Simplifying fractions is a crucial skill when working with fractions. It makes calculations easier and results clearer. The goal is to express the fraction in its simplest form, where the numerator and denominator have no common divisors except 1. For instance, if we have \(\frac{4}{8}\), both 4 and 8 can be divided by 4, the greatest common divisor (GCD). By dividing both the numerator and denominator by their GCD, we get the simplified fraction \(\frac{1}{2}\).

In the given example from the exercise, notice how the expression \(\frac{x \, y \, z}{y \, z \, x}\) simplifies elegantly. Since each element in the numerator matches an element in the denominator, we can "cancel" them out. This is another form of simplification, where the factors are eliminated one by one based on identical terms on both sides. The fraction simplifies to 1 as all variables and numbers "cancel out." This demonstrates the power of algebraic simplification: ensuring fractions are as straightforward and concise as possible.

Identifying Numerators and Denominators

Understanding numerators and denominators is essential for working with fractions. The numerator is the top part of a fraction, indicating how many parts we have. The denominator is the bottom part, showing the total number of equal parts or the size of each part we are considering. For example, in the fraction \(\frac{3}{4}\), 3 is the numerator, and 4 is the denominator.

In the exercise, we see fractions like \(\frac{x}{y}\), \(\frac{y}{z}\), and \(\frac{z}{x}\). Here, each fraction's numerator and denominator provide clues about how the elements in the expression are related. The exercise uses these to multiply fractions, keeping track of which terms appear on the top vs. the bottom.

• The numerator tells you "how much" you have.
• The denominator tells you "out of what" – this helps in understanding the fraction's value or portion.

Correctly identifying these parts is the first step in managing any fraction-based problem, making it possible to move onto operations like multiplication or division.

Fraction Multiplication Steps

Multiplying fractions involves several straightforward steps. Here's a quick guide to ensure you get it right every time.

Step 1: Multiply the Numerators
Start by multiplying all of the numerators together. For the exercise, the numerators are \(x\), \(y\), and \(z\), resulting in \(x \cdot y \cdot z\). This step is crucial in determining the numerator of the resulting product.
Step 2: Multiply the Denominators
Next, do the same with the denominators: multiply them all together. In our example, multiply \(y\), \(z\), and \(x\) to get \(y \cdot z \cdot x\). This becomes the denominator of your final fraction.
Step 3: Form and Simplify the Result
Combine the product of the numerators over the product of the denominators to create a single fraction: \(\frac{x \, y \, z}{y \, z \, x}\). The last crucial step is simplification, where you eliminate common factors in both the numerator and denominator.

• This gives you \(\frac{x \, y \, z}{y \, z \, x} = 1\).
• Remember, if every term in the numerator appears in the denominator, they "cancel out," simplifying to 1.

Master these steps, and multiplying fractions becomes a simple task, even when variables are involved.