Problem 21 - Prealgebra

Question

Simplify each complex fraction as much as possible. [Examples 4–7] $$\frac{\frac{2}{3}}{\frac{3}{4}}$$

Step-by-Step Solution

Answer

The simplified form is \( \frac{8}{9} \).

Step 1: Understand the Problem

We need to simplify the complex fraction \( \frac{\frac{2}{3}}{\frac{3}{4}} \). A complex fraction has a fraction in the numerator and denominator. Our task is to simplify it.

Step 2: Write as Division

A complex fraction can be rewritten as a division problem: \( \frac{2}{3} \div \frac{3}{4} \). This means we need to divide the fraction in the numerator by the fraction in the denominator.

Step 3: Division Rule for Fractions

To divide by a fraction, we multiply by its reciprocal. The reciprocal of \( \frac{3}{4} \) is \( \frac{4}{3} \). Thus, the expression becomes \( \frac{2}{3} \times \frac{4}{3} \).

Step 4: Multiply the Fractions

Multiply the numerators and denominators: \( \frac{2 \times 4}{3 \times 3} = \frac{8}{9} \). Now the fraction is simplified because both the numerator and denominator are integers with no common factors other than 1.

Key Concepts

Simplifying Fractions

Simplifying fractions is a process by which a fraction is reduced to its simplest form. This occurs when the numerator and the denominator have no common factors other than 1. By simplifying, we make fractions easier to understand and work with. Here's a simple way to go about it:

• Find the Greatest Common Factor (GCF): Determine the largest number that divides both the numerator and the denominator.
• Divide Both Parts: Divide the numerator and denominator by this GCF.
• Check Your Work: If there's no more simplifying possible, you've got the fraction in its simplest form.

Consider an example: In simplifying \( \frac{8}{12} \), we first find the GCF, which is 4. Dividing both the numerator (8) and the denominator (12) by 4, we get \( \frac{2}{3} \).
This fraction is now fully simplified.

Division of Fractions

Dividing fractions involves a simple but crucial step. Instead of directly dividing one fraction by another, we transform the operation into a multiplication.

• Invert the Second Fraction: Find the reciprocal of the divisor fraction. The reciprocal swaps the numerator and the denominator.
• Multiply the Fractions: Proceed by multiplying the first fraction (dividend) and the reciprocal of the second fraction (divisor).
• Simplify if Necessary: Check if your result can be simplified further.

For example, dividing \( \frac{2}{3} \) by \( \frac{3}{4} \) requires us to take the reciprocal of \( \frac{3}{4} \) (which is \( \frac{4}{3} \)) and multiply: \( \frac{2}{3} \times \frac{4}{3} = \frac{8}{9} \).
This yields the final simplified result.

Reciprocal of a Fraction

Understanding the reciprocal of a fraction is key to handling division of fractions neatly. A reciprocal is what you obtain when you interchange the numerator and the denominator of a fraction.

• Swap Places: For any fraction \( \frac{a}{b} \), the reciprocal is \( \frac{b}{a} \).
• Role in Division: In division of fractions, instead of directly dividing, we multiply by this reciprocal.
• Reciprocal of Whole Numbers: Remember, the reciprocal of a whole number x is \( \frac{1}{x} \).

This concept clarifies how division is essentially transformed into multiplication during fraction division steps. Taking \( \frac{3}{4} \) as an example, the reciprocal is \( \frac{4}{3} \), essential for multiplying with \( \frac{2}{3} \) in our original problem.
Mastering this step simplifies complex fractions effectively.