Problem 27 - Elementary and Intermediate Algebra

Question

Simplify each complex fraction. See Examples 2 or \(4 .\) $$ \frac{\frac{4}{c}-\frac{c}{6}}{\frac{2}{c}} $$

Step-by-Step Solution

Answer

The complex fraction simplifies to \( \frac{24 - c^2}{12} \).

Step 1: Simplify the Numerator

The numerator of the complex fraction is \( \frac{4}{c} - \frac{c}{6} \). To simplify this, find a common denominator. The common denominator for \( c \) and 6 is 6c. Rewriting these fractions: - \( \frac{4}{c} = \frac{4 \cdot 6}{c \cdot 6} = \frac{24}{6c} \)- \( \frac{c}{6} = \frac{c \cdot c}{6 \cdot c} = \frac{c^2}{6c} \)Therefore, the simplified numerator is \( \frac{24 - c^2}{6c} \).

Step 2: Simplify the Denominator

The denominator of the complex fraction is \( \frac{2}{c} \). It is already simplified, but ensure it has the same denominator of 6c, so multiply both the numerator and denominator by 6:\[ \frac{2}{c} = \frac{2 \cdot 6}{c \cdot 6} = \frac{12}{6c} \] Now, the denominator is \( \frac{12}{6c} \).

Step 3: Simplify the Complex Fraction

Now simplify the entire complex fraction:\[ \frac{\frac{24 - c^2}{6c}}{\frac{12}{6c}} = \frac{24 - c^2}{6c} \times \frac{6c}{12} \]The \( 6c \) terms cancel: \( \frac{24 - c^2}{12} \)Therefore, the complex fraction simplifies to \( \frac{24 - c^2}{12} \).

Step 4: Final Check

Verify all simplifications were done correctly. Check that cancellation and arithmetic operations do not alter the final expression. Correctly simplifying gives the final answer as \( \frac{24 - c^2}{12} \), ensuring all common denominators were appropriately managed.

Key Concepts

Numerator Simplification

Simplifying the numerator of a complex fraction involves finding a common denominator for the smaller fractions within the numerator. This step is crucial for allowing the subtraction of the fractions to take place effectively. In the case where the numerator is \( \frac{4}{c} - \frac{c}{6} \), the best approach is to determine the least common denominator. Both \( c \) and 6 share a common denominator, which is 6c. Here's how you can rewrite each fraction:

• \( \frac{4}{c} = \frac{4 \cdot 6}{c \cdot 6} = \frac{24}{6c} \)
• \( \frac{c}{6} = \frac{c \cdot c}{6 \cdot c} = \frac{c^2}{6c} \)

With a shared denominator of 6c, these can now be subtracted directly: \( \frac{24}{6c} - \frac{c^2}{6c} = \frac{24 - c^2}{6c} \). This formula is simpler and ready for further simplification within the larger fraction.

Denominator Simplification

Simplifying the denominator in complex fractions often involves similar steps to the numerator but may already appear simple, as shown in our example. The fraction \( \frac{2}{c} \) has a straightforward structure, but to align it with the numerator's denominator, it must be converted into a common form. Multiply both the numerator and denominator by the missing elements needed to match the least common denominator, and this will make further calculations seamless. By multiplying \( \frac{2}{c} \) by \( \frac{6}{6} \), you adjust the fraction effortlessly:\[ \frac{2}{c} = \frac{2 \cdot 6}{c \cdot 6} = \frac{12}{6c} \]This ensures both the numerator and denominator of the complex fraction share the same base, simplifying further calculations and making the final steps more direct.

Fractions in Algebra

Algebra often requires the manipulation and simplification of fractions, which can be a challenging task for students. Complex fractions are those where either the numerator, the denominator, or both, contain a fraction themselves. Successfully simplifying these involves several critical steps:

• Identifying common denominators for both components of the fraction.
• Performing operations like addition or subtraction once a unified denominator is established.
• Simplifying the overall fraction by eliminating common terms and reducing the expression wherever possible.

In the exercise provided, complex fractions require attention both to arithmetic skills and the understanding of algebraic structures. By consistently recognizing how fractions integrate into equations and adjustments, students can enhance both their computational skills and algebraic thinking. Successfully simplifying such fractions unveils cleaner, more workable equations, allowing for clarity and precision in problem-solving scenarios.