Problem 22 - Elementary and Intermediate Algebra

Question

Simplify each complex fraction. See Examples 2 or \(4 .\) $$ \frac{\frac{2}{3}-\frac{5}{2}}{\frac{2}{3}-\frac{3}{2}} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{11}{5} \).

Step 1: Identify the Complex Fraction

The given complex fraction is \( \frac{\frac{2}{3}-\frac{5}{2}}{\frac{2}{3}-\frac{3}{2}} \). This fraction consists of two fractional terms in both the numerator and the denominator. We first note that we will need a common denominator to simplify both the numerator and the denominator.

Step 2: Simplify the Numerator

The numerator is \( \frac{2}{3} - \frac{5}{2} \). To subtract these fractions, find the least common denominator (LCD). The LCD of 3 and 2 is 6. Convert \( \frac{2}{3} \) to \( \frac{4}{6} \) and \( \frac{5}{2} \) to \( \frac{15}{6} \). Then, subtract: \( \frac{4}{6} - \frac{15}{6} = \frac{-11}{6} \).

Step 3: Simplify the Denominator

The denominator is \( \frac{2}{3} - \frac{3}{2} \). Again, find the LCD, which is 6. Convert \( \frac{2}{3} \) into \( \frac{4}{6} \) and \( \frac{3}{2} \) into \( \frac{9}{6} \). Subtract these fractions: \( \frac{4}{6} - \frac{9}{6} = \frac{-5}{6} \).

Step 4: Divide the Simplified Fractions

Having simplified the numerator to \( \frac{-11}{6} \) and the denominator to \( \frac{-5}{6} \), divide the two fractions: \( \frac{-11}{6} \div \frac{-5}{6} \). This is equivalent to multiplying by the reciprocal: \( \frac{-11}{6} \times \frac{6}{-5} = \frac{-11 \times 6}{6 \times -5} \).

Step 5: Cancel the Common Factor

The \( 6 \) in the numerator and denominator cancels out, leaving \( \frac{-11}{-5} \), which simplifies to \( \frac{11}{5} \) because dividing two negative numbers yields a positive result.

Key Concepts

Simplifying Fractions

Simplifying fractions is an essential skill when working with complex fractions. In order to simplify, we need to ensure that we are working with an expression that is straightforward and without unnecessary complexities. Complex fractions are those in which either the numerator, the denominator, or both, consist of fractions themselves. This can often appear confusing.
To simplify such fractions, the first step is to find the least common denominator (LCD) for the fractions involved in both the numerator and the denominator of the complex fraction.

• Convert each fraction into an equivalent one with a common denominator.
• Perform the necessary operations, such as addition or subtraction, to simplify each individual part.

By doing this, we ensure that the fractions are reduced to their simplest forms, making further operations easier.

Least Common Denominator

Finding the least common denominator is essential when you are tasked with adding or subtracting fractions. It is the smallest number that each denominator divides into evenly, enabling the simplification process.
For example, if you have fractions with denominators of 3 and 2, the least common multiple of these numbers is 6, which becomes your LCD.

• Convert each fraction to an equivalent fraction with this new denominator.
• Multiply the numerator and the denominator of each fraction to achieve this common denominator.

This step allows all fractions to "speak the same language," making it easy to add or subtract them.

Fraction Division

Once the numerator and denominator of a complex fraction are simplified, the next step is to perform fraction division.
This involves dividing the two simplified fractions, but rather than dividing directly, we use the reciprocal method. Here's how it works:

• Take the reciprocal of the divisor (the fraction in the denominator).
• Multiply the simplified numerator by this reciprocal.

The reason we multiply by the reciprocal is that division of fractions is essentially multiplying by the reciprocal. After multiplication, simplify the resulting fraction by canceling any common factors, giving you the final simplified fraction.