Problem 23 - Elementary and Intermediate Algebra

Question

Simplify each complex fraction. See Examples 2 or \(4 .\) $$ \frac{\frac{1}{4}+\frac{1}{y}}{\frac{y}{3}-\frac{1}{2}} $$

Step-by-Step Solution

Answer

The simplified complex fraction is \( \frac{6(y+4)}{8y^2 - 12y} \).

Step 1: Identify the Numerator and the Denominator

In the given complex fraction, the entire fraction in the numerator is \( \frac{1}{4} + \frac{1}{y} \) and the entire fraction in the denominator is \( \frac{y}{3} - \frac{1}{2} \). Our goal is to simplify this expression.

Step 2: Find a Common Denominator for the Numerator

For the expression \( \frac{1}{4} + \frac{1}{y} \), the denominators are 4 and \( y \). The common denominator is \( 4y \). Rewrite each term with this common denominator: \( \frac{y}{4y} + \frac{4}{4y} = \frac{y+4}{4y} \).

Step 3: Find a Common Denominator for the Denominator

For the expression \( \frac{y}{3} - \frac{1}{2} \), the denominators are 3 and 2. The common denominator is 6. Rewrite each term with this common denominator: \( \frac{2y}{6} - \frac{3}{6} = \frac{2y-3}{6} \).

Step 4: Divide the Fraction by Multiplying with the Reciprocal

Now that we have the simplified numerator \( \frac{y+4}{4y} \) and the simplified denominator \( \frac{2y-3}{6} \), divide the entire expression by multiplying with the reciprocal of the denominator. Thus, \( \frac{\frac{y+4}{4y}}{\frac{2y-3}{6}} = \frac{y+4}{4y} \times \frac{6}{2y-3} \).

Step 5: Simplify the Expression

Multiply across the numerators and the denominators: \( \frac{(y+4) \cdot 6}{4y(2y-3)} = \frac{6(y+4)}{8y^2 - 12y} \).Check for any common factors to simplify further; since there are no common factors, this is the simplified expression.

Key Concepts

Numerator and Denominator

A complex fraction is essentially a fraction in which the numerator, denominator, or both contain fractions themselves. Understanding the role of the numerator and denominator is critical because they dictate how we simplify these structures. Here, the numerator is the expression \( \frac{1}{4} + \frac{1}{y} \), a sum of two simple fractions.
- In any fraction, the numerator is the top part, representing 'how many parts' are taken.
- The denominator, conversely, is the bottom part, showing 'into how many equal parts' the whole is divided.
In our complex fraction, the denominator is \( \frac{y}{3} - \frac{1}{2} \), a difference of simple fractions. It's essential to treat the terms in these expressions carefully to combine and simplify them accurately. By identifying these roles, we set the stage for finding common denominators, a pivotal next step.

Common Denominator

To add or subtract fractions, finding a common denominator is necessary. This allows us to combine fractions with different denominators into a single fraction. In the expression from the numerator, \( \frac{1}{4} + \frac{1}{y} \), the denominators are 4 and \( y \).
- The least common denominator (LCD) here is the product of these: \( 4y \).
- We adjust each term to have this common denominator: changing \( \frac{1}{4} \) to \( \frac{y}{4y} \) and \( \frac{1}{y} \) to \( \frac{4}{4y} \).
This process allows us to combine the terms as \( \frac{y+4}{4y} \). Similarly, in the denominator of the fraction \( \frac{y}{3} - \frac{1}{2} \), the common denominator is found to be 6. By rewriting \( \frac{y}{3} \) as \( \frac{2y}{6} \) and \( \frac{1}{2} \) as \( \frac{3}{6} \), they are easily combined into \( \frac{2y-3}{6} \).
Using common denominators allows complex fractions to be expressed in a simplified form suitable for further operations.

Reciprocal

A reciprocal is essentially flipping the numerator and the denominator of a fraction. When simplifying complex fractions, dividing by a fraction is equivalent to multiplying by its reciprocal. This is a simple yet powerful concept used in the exercise's final steps.
Once the complex fraction is expressed as \( \frac{\frac{y+4}{4y}}{\frac{2y-3}{6}} \), you multiply the numerator by the reciprocal of the denominator, \( \frac{6}{2y-3} \).
This changes the problem to a multiplication: \( \frac{y+4}{4y} \times \frac{6}{2y-3} \). The concepts of multiplication are easier to navigate than division with complex fractions.
- By multiplying across the numerators and across the denominators, you end up with \( \frac{6(y+4)}{8y^2 - 12y} \).
The reciprocal is a key tool in transforming division problems into multiplication, offering a straightforward path to simplification.