Problem 40 - BIG IDEAS MATH Algebra 2: Common Core Student Edition 2015

Question

Simplify the complex fraction. \(\frac{15-\frac{2}{x}}{\frac{x}{5}+4}\)

Step-by-Step Solution

Answer

The simplified form of the given complex fraction \(\frac{15-\frac{2}{x}}{\frac{x}{5}+4}\) is \(\frac{15x-2}{x+20}\).

Step 1: Clear Fraction in the Numerator

The aim is to eliminate the fraction within the numerator which is \(15-\frac{2}{x}\). This can be achieved by finding a common denominator. In this case, the common denominator between \(15\) (which can be considered as \(\frac{15}{1}\)) and \(\frac{2}{x}\) is \(x\). So, we multiply \(15\) by \(x\) which results in the numerator becoming \(15x - 2\).

Step 2: Clear Fraction in the Denominator

The next step is to eliminate the fraction within the denominator which is \(\frac{x}{5}+4\). Again, by finding a common denominator. Here, the common denominator between \(\frac{x}{5}\) and \(4\) (which can be considered as \(\frac{4}{1}\)) is \(5\). As a result, we multiply \(4\) by \(5\), resulting in the denominator becoming \(x + 20\).

Step 3: Simplify the complex fraction

Now you simplify the complex fraction with the newly obtained numerator \(15x - 2\) and denominator \(x + 20\). Therefore the simplified complex fraction is \(\frac{15x-2}{x+20}\).

Key Concepts

Numerator and Denominator

In complex fractions, the numerator and denominator can themselves contain fractions. It is important to know how to handle these internal fractions effectively. Let's consider the given problem:

• The numerator is \(15-\frac{2}{x}\), where \(15\) is essentially \(\frac{15}{1}\).
• Similarly, the denominator is \(\frac{x}{5}+4\), where \(4\) can be seen as \(\frac{4}{1}\).

The strategy is to simplify these by treating the numerator and denominator just like any other fraction. When handling complex fractions, always start by identifying the fractional parts both in the numerator and denominator. This will help you know your starting point in the simplification.

Common Denominator

Finding a common denominator is crucial when simplifying complex fractions because it allows you to combine fractions into a single expression. Here's how we find it in the given problem:

• For the numerator \(15-\frac{2}{x}\), the common denominator between \(15\) (considered as \(\frac{15}{1}\)) and \(\frac{2}{x}\) is \(x\). We rewrite the numerator as \(15x - 2\).
• In the denominator \(\frac{x}{5} + 4\), the common denominator between \(\frac{x}{5}\) and \(4\) (considered as \(\frac{4}{1}\)) is \(5\). Thus, the denominator becomes \(x + 20\).

Finding a common denominator is like finding common ground between numbers; it makes adding or subtracting fractions possible. This step will transform each part into simpler expressions that can be easily reduced.

Simplification Process

After obtaining common denominators, the next step is the simplification process, where we aim to reduce our complex fraction into its simplest form. The process is as follows:

• With the simplified numerator as \(15x - 2\) and the simplified denominator as \(x + 20\), construct the new complex fraction: \(\frac{15x-2}{x+20}\).
• If possible, simplify further by factoring or canceling out terms. In our problem, the fraction can't be simplified further because there are no common factors between \(15x-2\) and \(x+20\).

Simplifying complex fractions often involves multiple layers of simplification. Always check your work by ensuring that the numerator and denominator have no further common factors that can be canceled away. This will ensure you've achieved the most simplified form possible.