Problem 22 - Elementary Algebra

Question

Simplify the complex fraction \(\frac{4-\frac{3}{1}}{4+\frac{3}{x}}\).

Step-by-Step Solution

Answer

Question: Simplify the complex fraction \(\frac{4-\frac{3}{1}}{4+\frac{3}{x}}\). Answer: \(\frac{x}{4x + 3}\)

Step 1: Simplify the numerator and denominator

First, we simplify the numerator and denominator of the complex fraction separately: Numerator: \(4 - \frac{3}{1} = 4 - 3 = 1\) Denominator: No simplification required The simplified complex fraction is now \(\frac{1}{4+\frac{3}{x}}\).

Step 2: Create a common denominator for the numerator and denominator

In the denominator, we have two terms with different denominators: \(4\) and \(\frac{3}{x}\). We need to find a common denominator and combine these two terms to simplify further. The lower part of the denominator can be written as \(\frac{4x}{x} + \frac{3}{x}\).

Step 3: Simplify the resulting fraction

Now, we combine the terms in the denominator and simplify the fraction: Denominator: \(\frac{4x}{x} + \frac{3}{x} = \frac{4x + 3}{x}\) So the simplified complex fraction is \(\frac{1}{\frac{4x + 3}{x}}\). To further simplify, we use the rule \(\frac{a}{\frac{b}{c}} = \frac{a\cdot c}{b}\) and multiply the numerator and denominator by \(x\) to eliminate the fraction in the denominator: Result: \(\frac{1 \cdot x}{4x + 3} = \frac{x}{4x + 3}\). The final simplified complex fraction is \(\frac{x}{4x + 3}\).

Key Concepts

Numerator Simplification

When dealing with complex fractions, the first step is often to simplify the numerator. In our example, the numerator is \(4 - \frac{3}{1}\). To simplify this, recognize that \(\frac{3}{1}\) equals 3. So, the expression becomes \(4 - 3\).

This further simplifies to 1, making our numerator very straightforward! Simplifying the numerator separately ensures clarity and ease when working with the next steps of the fraction.

Denominator Simplification

In complex fractions, examining the denominator is just as crucial as the numerator. In this instance, the original denominator is \(4 + \frac{3}{x}\). It appears to contain terms with different bases.

Unlike the numerator which initially required simplification, the original denominator does not need immediate simplification. However, later steps do require handling these terms for further calculation. It's okay if an immediate simplification isn't obvious; the focus is on maintaining consistency with the available terms.

Common Denominator

To combine fractions within either the numerator or denominator, finding a common denominator is essential. Here, the denominator is \(4 + \frac{3}{x}\), where you have two terms: a whole number and a fractional term.

To combine these effectively, you rewrite 4 as \(\frac{4x}{x}\). This way, both terms in the denominator have a common denominator, \(x\).

This step is crucial for simplifying the fraction further and ensures all terms are standardized for easy calculation.

Fraction Simplification

Once the denominator is expressed with a common denominator, you simplify the complex fraction by combining it. In our example, \(\frac{4x}{x} + \frac{3}{x}\) is combined to \(\frac{4x + 3}{x}\).

Now we have \(\frac{1}{\frac{4x + 3}{x}}\). At this point, simplify further by employing the rule \(\frac{a}{\frac{b}{c}} = \frac{a\cdot c}{b}\).

This means you multiply the entire fraction by \(x\) to eliminate the fraction within the denominator, yielding \(\frac{x}{4x + 3}\). Remember, simplification is about making expressions easier to work with and understand.