Problem 5 - Algebra 2 and Trigonometry

Question

In \(3-12,\) multiply and express each product in simplest form. In each case, list any values of the variables for which the fractions are not defined. $$ \frac{4 y}{5 x} \cdot \frac{x}{8 y} $$

Step-by-Step Solution

Answer

The product is \(\frac{1}{10}\), undefined for \(x = 0\) or \(y = 0\).

Step 1: Multiply the numerators

To find the product of two fractions, we first multiply their numerators together. Here, the numerators are \(4y\) and \(x\). Therefore, \(4y \cdot x = 4yx\).

Step 2: Multiply the denominators

Next, we multiply the denominators of the fractions. The denominators are \(5x\) and \(8y\). Thus, we have \(5x \cdot 8y = 40xy\).

Step 3: Write the combined fraction

The combined fraction from the previous calculations is \(\frac{4yx}{40xy}\).

Step 4: Simplify the fraction

To simplify \(\frac{4yx}{40xy}\), we notice that \(4yx\) and \(40xy\) share common factors. Cancel out the common terms \(x\) and \(y\), resulting in \(\frac{4}{40} = \frac{1}{10}\).

Step 5: Determine values where the expression is undefined

Fractions are undefined where the denominator equals zero. For \(\frac{4y}{5x} \), it is undefined when \(x = 0\), and for \(\frac{x}{8y} \), it is undefined when \(y = 0\). Therefore, the original expression is undefined when \(x = 0\) or \(y = 0\).

Key Concepts

Simplifying Fractions

Simplifying fractions is about reducing a fraction to its smallest equivalent form. This means making both the numerator (top number) and the denominator (bottom number) smaller by dividing them by their greatest common factor (GCF). For example, let's consider \( \frac{8}{12} \).
First, find the GCF of 8 and 12, which is 4. Then, divide both numbers by 4:

• \( \frac{8}{4} = 2 \)
• \( \frac{12}{4} = 3 \)

So, \( \frac{8}{12} \) simplifies to \( \frac{2}{3} \). In the original exercise, we simplified \( \frac{4yx}{40xy} \) by canceling out the common terms \( x \) and \( y \) from both the numerator and the denominator, ultimately reducing the fraction to \( \frac{1}{10} \).
Simplifying makes fractions easier to work with and understand. Always look for common factors to make the fraction as simple as possible.

Undefined Fractions

Undefined fractions occur when the denominator of a fraction is zero. This makes the fraction division invalid, as division by zero is not defined in mathematics. Think of dividing a pizza into zero parts – it doesn't make sense!
In our exercise, fractions like \( \frac{4y}{5x} \) become undefined when \( x = 0 \), and similarly, \( \frac{x}{8y} \) becomes undefined when \( y = 0 \).
Always check the denominator:

• If it would make the denominator zero, the fraction is undefined.

Understanding where fractions are undefined helps avoid errors in mathematical operations.

Common Factors

Common factors are numbers that evenly divide two or more given numbers. Finding common factors is crucial when simplifying fractions, as they point out numbers that can cancel each other out.
Take the expression in the problem: \( \frac{4yx}{40xy} \).

• The terms \( x \) and \( y \) appear in both the numerator and denominator, allowing us to cancel them out.
• This leaves us with \( \frac{4}{40} \), which can be simplified further as they have a greatest common factor of 4.

By dividing both by 4, we simplify \( \frac{4}{40} \) to \( \frac{1}{10} \).
Knowing how to find and use common factors makes fraction multiplication and division much easier.