Problem 13 - Elementary and Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

The simplified form is \( \frac{x^2}{y} \).

Step 1: Understand the Problem

We are given a complex fraction that looks like \( \frac{\frac{x}{y}}{\frac{1}{x}} \). This is a fraction where both the numerator and the denominator are themselves fractions. Our goal is to simplify this expression to a simpler form.

Step 2: Simplify the Numerator and Denominator

The numerator of the complex fraction is \( \frac{x}{y} \) and the denominator is \( \frac{1}{x} \). We will simplify the expression by rewriting the division of fractions as multiplication. Remember that dividing by a fraction is the same as multiplying by its reciprocal.

Step 3: Write Division as Multiplication

We rewrite the complex fraction \( \frac{\frac{x}{y}}{\frac{1}{x}} \) as a multiplication problem: \( \frac{x}{y} \times \frac{x}{1} \). Here, \( \frac{1}{x} \) is replaced by its reciprocal, which is \( \frac{x}{1} \).

Step 4: Multiply the Fractions

To multiply the fractions, multiply the numerators together and the denominators together. So we have: \( \frac{x \times x}{y \times 1} = \frac{x^2}{y} \).

Step 5: State the Final Simplified Form

The simplified form of the complex fraction \( \frac{\frac{x}{y}}{\frac{1}{x}} \) is \( \frac{x^2}{y} \).

Key Concepts

Simplifying Fractions

Simplifying fractions makes them easier to work with and understand. When you simplify a fraction, you're reducing it to its simplest form by dividing the numerator and the denominator by their greatest common factor (GCF).
Consider the fraction \( \frac{4}{8} \). The GCF of 4 and 8 is 4. So, when we divide both the numerator and the denominator by 4, we get \( \frac{1}{2} \), which is the simplified form.

• Check for a common number that divides into both numerator and denominator.
• Divide both by this number to simplify.
• Continue until the numerator and denominator are as small as possible, but still whole numbers.

The principle of simplifying fractions also applies to complex fractions, where simplification means reducing the entire expression to be more manageable.

Multiplication of Fractions

Multiplying fractions might seem tricky at first, but it's quite straightforward once you get the hang of it. You simply multiply straight across the numerators and denominators.
For example, if you have the fractions \( \frac{2}{3} \) and \( \frac{4}{5} \), you multiply the numerators (2 and 4) to get 8, and the denominators (3 and 5) to get 15.
The result is \( \frac{8}{15} \).

• Multiply the numerators together.
• Multiply the denominators together.
• If possible, simplify the resultant fraction.

This simple rule makes calculations easy. When dealing with complex fractions, breaking them down into multiplication helps in their simplification, as you've seen in the example.

Reciprocal of a Fraction

The reciprocal of a fraction is what you get when you swap its numerator and denominator. It's like flipping the fraction over. For example, the reciprocal of \( \frac{3}{4} \) is \( \frac{4}{3} \).
This concept is crucial when dividing fractions, as dividing by a fraction is the same as multiplying by its reciprocal.
For instance:

• The reciprocal of \( \frac{5}{6} \) is \( \frac{6}{5} \).
• The reciprocal of \( \frac{7}{2} \) is \( \frac{2}{7} \).

It simplifies the process of dividing fractions and handling complex fractions. Understanding reciprocals allows you to restate division problems as multiplication, streamlining many fractional calculations. This principle was used in the step-by-step solution to simplify the complex fraction.