Problem 37 - Precalculus Mathematics for Calculus

Question

Perform the multiplication or division and simplify. $$\frac{x / y}{z}$$

Step-by-Step Solution

Answer

The simplified form is \( \frac{x}{y z} \).

Step 1: Understand the Expression

The given expression is \( \frac{x/y}{z} \). This means we have \( x/y \) divided by \( z \). We can rewrite this expression as the fraction \( \frac{x}{y} \times \frac{1}{z} \).

Step 2: Multiply the Fractions

To find the product of \( \frac{x}{y} \) and \( \frac{1}{z} \), multiply the numerators and the denominators. So, \( \frac{x}{y} \times \frac{1}{z} = \frac{x \times 1}{y \times z} = \frac{x}{y z} \).

Step 3: Simplify the Fraction

The resulting expression is \( \frac{x}{y z} \). Check if there are any common factors between the numerator and the denominator for further simplification. Since there are none, the simplified form is \( \frac{x}{y z} \).

Key Concepts

Multiplication of Fractions

Multiplication of fractions is much simpler than it might seem at first. When you multiply fractions, you are essentially combining two numerical expressions into one. Instead of treating each number separately, you will first multiply all numerators together and then multiply all denominators together.
For example, if you have two fractions \( \frac{a}{b} \) and \( \frac{c}{d} \), the multiplication of these fractions would look like this:

• Multiply the numerators: Combine the tops of the fractions: \( a \times c = ac \).
• Multiply the denominators: Combine the bottoms of the fractions: \( b \times d = bd \).

Putting them together, the result of \( \frac{a}{b} \times \frac{c}{d} \) is the new fraction \( \frac{ac}{bd} \). This method remains consistent regardless of the complexity of the fractions you are working with. The goal is to simplify the resulting fraction by looking for common factors if possible.

Dividing Fractions

Dividing fractions might initially seem complicated, but there's a straightforward method to handle it. When you divide by a fraction, you're essentially multiplying by its reciprocal. The reciprocal of a fraction is simply switching its numerator and denominator.
Let's say you have to divide \( \frac{a}{b} \) by \( \frac{c}{d} \). Here's how you can manage this:

• Flip the second fraction: Take the reciprocal of \( \frac{c}{d} \), which becomes \( \frac{d}{c} \).
• Multiply the fractions: Instead of dividing, multiply \( \frac{a}{b} \) by \( \frac{d}{c} \).

The computation becomes \( \frac{a}{b} \times \frac{d}{c} = \frac{a \times d}{b \times c} \). This new fraction is your answer to the original division problem. Always remember this simple trick: Dividing by a fraction is the same as multiplying by its reciprocal.

Numerators and Denominators

When working with fractions, understanding numerators and denominators is key. A fraction consists of two main parts: the numerator and the denominator.
The numerator is the top part of the fraction, representing the number of parts we have. For instance, in \( \frac{3}{5} \), the numerator is 3, indicating three parts.
The denominator is the bottom portion, signifying the total number of equal parts that make up a whole. In \( \frac{3}{5} \), 5 is the denominator, showing the whole is divided into five equal parts.

• Simplification: Always check if both parts of the fraction have common factors. Simplifying means finding the greatest common factor (GCF) and then dividing both the numerator and denominator by it, reducing the fraction to its simplest form.
• Understand their roles: Numerators define how many, while denominators determine into how many segments the whole is divided.

Grasping these concepts makes working with fractions much more manageable. It helps in multiplying, dividing, and simplifying fractions efficiently.