Problem 11 - Elementary and Intermediate Algebra

Question

Simplify each complex fraction. See Example \(1 .\) $$ \frac{\frac{x}{2}}{\frac{6}{5}} $$

Step-by-Step Solution

Answer

The simplified form is \( \frac{5x}{12} \).

Step 1: Identify the Components

The complex fraction \( \frac{\frac{x}{2}}{\frac{6}{5}} \) consists of a numerator \( \frac{x}{2} \) and a denominator \( \frac{6}{5} \). Our task is to simplify this fraction.

Step 2: Convert Complex Fraction to Multiplication

To simplify the complex fraction, we change the division (which the fraction bar represents in the complex fraction) into multiplication by the reciprocal of the denominator. So, \( \frac{\frac{x}{2}}{\frac{6}{5}} \) becomes \( \frac{x}{2} \times \frac{5}{6} \).

Step 3: Multiply the Fractions

Now, multiply the numerators and the denominators. The product will be \( \frac{x \times 5}{2 \times 6} \) which equals \( \frac{5x}{12} \).

Step 4: Simplify the Fraction

In this step, we check if the fraction \( \frac{5x}{12} \) can be simplified further. Since there are no common factors between 5 and 12, \( \frac{5x}{12} \) is already in simplest form.

Key Concepts

Simplifying Fractions

When we talk about simplifying fractions, we mean making them easier to work with without changing their value. This is done by dividing both the numerator and the denominator by their greatest common factor (GCF).
For example, consider the fraction \( \frac{8}{12} \). The GCF of 8 and 12 is 4. By dividing both the numerator and denominator by 4, the simplified fraction is \( \frac{2}{3} \).
In our complex fraction problem \( \frac{5x}{12} \), the numbers 5 and 12 have no common factors (other than 1), meaning it cannot be simplified further. It's essential to confirm this step to ensure the fraction is in its simplest form. Simplifying fractions is a crucial skill in working with complex fractions.

Multiplying Fractions

Multiplying fractions is a straightforward process that involves two main steps: multiplying the numerators together and multiplying the denominators together.
Let's say we have two fractions, \( \frac{a}{b} \) and \( \frac{c}{d} \). To multiply them, you simply do the following:

• Numerator: Multiply the top numbers (\(a \times c\)).
• Denominator: Multiply the bottom numbers (\(b \times d\)).

This gives us a new fraction \( \frac{a \times c}{b \times d} \).
In our problem, when we converted the complex fraction to a multiplication problem, we arrived at \( \frac{x}{2} \times \frac{5}{6} \). After carrying out the multiplication, this resulted in \( \frac{5x}{12} \). Multiplying fractions is an important tool for dealing with complex algebraic expressions.

Reciprocal of Fractions

The concept of a reciprocal is fundamental in simplifying complex fractions. The reciprocal of a fraction is created by flipping its numerator and denominator.
For example, the reciprocal of \( \frac{2}{3} \) is \( \frac{3}{2} \).
In the context of our complex fraction problem, the denominator was \( \frac{6}{5} \). By taking the reciprocal, we turned it into \( \frac{5}{6} \), allowing us to change the division of fractions into a multiplication problem.
The use of reciprocals is crucial for converting complex fractions into simpler multiplication problems, making it easier to handle.