Problem 47 - Algebra for College Students

Question

Simplify each complex fraction. $$ \frac{\frac{3}{x}-\frac{2}{y}}{\frac{4}{y}-\frac{7}{x y}} $$

Step-by-Step Solution

Answer

The simplified form is \( \frac{3y - 2x}{4x - 7} \).

Step 1: Identify the Complex Fraction

A complex fraction is a fraction where the numerator and/or the denominator contains a fraction. In this case, we have the complex fraction \( \frac{\frac{3}{x} - \frac{2}{y}}{\frac{4}{y} - \frac{7}{xy}} \) where the numerators and denominators contain two fractions each.

Step 2: Find a Common Denominator for the Numerator

To combine the fractions in the numerator \( \frac{3}{x} - \frac{2}{y} \), find a common denominator. The least common denominator (LCD) of \( x \) and \( y \) is \( xy \). Rewrite each fraction: \[ \frac{3}{x} = \frac{3y}{xy}, \quad \frac{2}{y} = \frac{2x}{xy} \]Thus, the numerator becomes: \[ \frac{3y - 2x}{xy} \]

Step 3: Find a Common Denominator for the Denominator

Similarly, for the denominator \( \frac{4}{y} - \frac{7}{xy} \), the common denominator is \( xy \). Express each fraction with this denominator: \[ \frac{4}{y} = \frac{4x}{xy}, \quad \frac{7}{xy} = \frac{7}{xy} \]Thus, the denominator becomes: \[ \frac{4x - 7}{xy} \]

Step 4: Simplify the Entire Complex Fraction

Now that the complex fraction is: \[ \frac{\frac{3y - 2x}{xy}}{\frac{4x - 7}{xy}} \]To simplify, divide the numerator by the denominator:\[ \frac{3y - 2x}{xy} \times \frac{xy}{4x - 7} \]Cancel out \( xy \) as it appears in both the numerator and denominator:\[ \frac{3y - 2x}{4x - 7} \]

Step 5: Final Simplified Fraction

The complex fraction is now simplified completely to: \( \frac{3y - 2x}{4x - 7} \) as all terms have been consolidated and simplified.

Key Concepts

Simplification

Simplifying a complex fraction involves transforming it into a simpler expression without altering its value. A complex fraction is one that has fractions in either its numerator, denominator, or both. To simplify, we focus on:

• Combining fractions in both the numerator and the denominator into single expressions.
• Finding a common denominator for these fractions.
• Converting the complex fraction into a simpler one by division.

Start by simplifying the individual fractions in the numerator and denominator, then use these simplified expressions to rewrite the complex fraction. Remember, the goal of simplification is to express the fraction in its most reduced form for ease of understanding and further calculations.

Common Denominator

Finding a common denominator is a crucial step in adding or subtracting fractions. It involves finding the least common multiple (LCM) of the denominators involved. For fractions \( \frac{a}{b} \) and \( \frac{c}{d} \), the common denominator would be the product \( bd \).
This commonality allows us to rewrite each fraction with the same base, thus making arithmetic operations straightforward.
In the exercise, we considered the fractions \( \frac{3}{x} - \frac{2}{y} \) in the numerator and \( \frac{4}{y} - \frac{7}{xy} \) in the denominator. The common denominator for both sets was \( xy \).

• By multiplying each fraction by a form of 1 (like adding the missing terms to match the denominator), we can rewrite them as equivalent expressions with \( xy \) as the denominator.
• This step is essential for simplifying complex fractions because it allows consolidating multiple terms into a single fraction for easier simplification.

Thus, finding a common denominator is a key technique in handling complex fractions efficiently.

Fraction Division

Once you have a common denominator and have rewritten the complex fraction, the next step is fraction division. Division of fractions can be simplified by multiplying by the reciprocal. This involves flipping the fraction you wish to divide by and then changing the division sign to multiplication.
For example, for the complex fraction \( \frac{3y - 2x}{xy} / \frac{4x - 7}{xy} \), you first rewrite it using multiplication by the reciprocal of the denominator:

• Keep the first fraction as is: \( \frac{3y - 2x}{xy} \).
• Flip the second fraction: \( \frac{xy}{4x - 7} \).
• Change the division sign between them to multiplication.

Now perform the multiplication of the numerators and the denominators. Cancel any common factors before carrying out the multiplication for simplicity.
In this exercise, the \( xy \) terms in the numerator and denominator cancel out, leaving the final simplified fraction: \( \frac{3y - 2x}{4x - 7} \). This method streamlines the process and yields the simplest form of the complex fraction.