Problem 29 - Elementary and Intermediate Algebra

Question

Simplify each complex fraction. See Examples 3 or \(5 .\) $$ \frac{\frac{2}{3}+1}{\frac{1}{3}+1} $$

Step-by-Step Solution

Answer

The simplified form of the complex fraction is \( \frac{5}{4} \).

Step 1: Simplify the Numerator

The numerator of the given complex fraction is \( \frac{2}{3} + 1 \). To simplify this expression, first convert the whole number 1 into a fraction with the same denominator as \( \frac{2}{3} \). This is done by rewriting 1 as \( \frac{3}{3} \). Thus, the expression becomes \( \frac{2}{3} + \frac{3}{3} \). Add the fractions by combining the numerators (2 + 3) over the common denominator: \( \frac{5}{3} \).

Step 2: Simplify the Denominator

The denominator of the complex fraction is \( \frac{1}{3} + 1 \). Similarly, rewrite the whole number 1 as a fraction \( \frac{3}{3} \). Now, add the fractions: \( \frac{1}{3} + \frac{3}{3} = \frac{4}{3} \).

Step 3: Simplify the Complex Fraction

Now that both the numerator and the denominator are single fractions, we can simplify \( \frac{\frac{5}{3}}{\frac{4}{3}} \). This is equivalent to multiplying by the reciprocal of the denominator. Thus, \( \frac{5}{3} \) is divided by \( \frac{4}{3} \) which is equivalent to \( \frac{5}{3} \times \frac{3}{4} \).

Step 4: Multiply and Simplify

Perform the multiplication: \( \frac{5}{3} \times \frac{3}{4} \). Multiply the numerators: \( 5 \times 3 = 15 \), and multiply the denominators: \( 3 \times 4 = 12 \). Hence, the result is \( \frac{15}{12} \).

Step 5: Simplify the Result

Reduce \( \frac{15}{12} \) to its simplest form by finding the greatest common divisor of 15 and 12, which is 3. Divide both the numerator and the denominator by 3: \( \frac{15}{12} = \frac{15 \div 3}{12 \div 3} = \frac{5}{4} \).

Key Concepts

Fraction Simplification

To tackle complex fractions, understanding fraction simplification is key. A complex fraction is essentially a fraction where the numerator, the denominator, or both contain a fraction themselves. The goal is to simplify these complex fractions so that calculations become easier.

Fraction simplification involves reducing fractions to their simplest form. This process requires:

• Finding a common denominator for fractions within the expression.
• Combining numerators using basic arithmetic operations like addition or subtraction.
• Ensuring that the final result is fully simplified by dividing both the numerator and the denominator by their greatest common divisor (GCD).

To simplify the complex fraction \( \frac{\frac{2}{3}+1}{\frac{1}{3}+1} \), you begin by rewriting integers to fractions with a matching denominator. This allows you to combine and simplify them easily.

Numerator and Denominator Simplification

Simplifying the numerator and the denominator of a complex fraction separately is an essential step in solving the problem. Let's delve into how this is done. For our complex fraction, we follow these steps:

• Convert whole numbers into fractions with the same denominator as the existing fraction parts. For example, rewrite the 1 in the numerator and the denominator as \(\frac{3}{3}\).
• Add the fractions in both the numerator and the denominator. In the exercise, the numerator becomes \(\frac{5}{3}\) and the denominator becomes \(\frac{4}{3}\).

Numerator and denominator simplification involves rewriting, adding, and combining like terms to ensure each part of the fraction is a single simplified fraction itself.

Reciprocal Multiplication

Once you have simplified the numerator and denominator into single fractions, the next challenge is to handle the division of these fractions. This is where reciprocal multiplication comes in. In reciprocal multiplication, instead of dividing one fraction by another, you multiply the first fraction by the reciprocal of the second. The reciprocal of a fraction is achieved by swapping its numerator and denominator. For the example \( \frac{\frac{5}{3}}{\frac{4}{3}} \), multiply \(\frac{5}{3}\) by the reciprocal of \(\frac{4}{3}\), which is \(\frac{3}{4}\).

This translates mathematically as:

• \( \frac{5}{3} \times \frac{3}{4} = \frac{15}{12} \)

Always remember to simplify the resulting fraction by finding the greatest common divisor. Here, 3 divides both the numerator and denominator, resulting in a final simplified form of \(\frac{5}{4}\).

Reciprocal multiplication changes the landscape of complex fraction problems, making them straightforward and easier to manage.