Problem 33 - Prealgebra

Question

Simplify each complex fraction as much as possible. [Examples 4–7] $$\frac{1+\frac{2}{3}}{1-\frac{2}{3}}$$

Step-by-Step Solution

Answer

The simplified form of the complex fraction is 5.

Step 1: Understand the Structure

The complex fraction \( \frac{1+\frac{2}{3}}{1-\frac{2}{3}} \) consists of two parts: a numerator \(1+\frac{2}{3}\) and a denominator \(1-\frac{2}{3}\). Our goal is to simplify this expression to a single simple fraction.

Step 2: Simplify the Numerator

First, let's simplify the numerator \(1 + \frac{2}{3}\). Convert the 1 into a fraction with a common denominator: \(1 = \frac{3}{3}\). So, we have: \(\frac{3}{3} + \frac{2}{3} = \frac{5}{3}\).

Step 3: Simplify the Denominator

Now, let's simplify the denominator \(1 - \frac{2}{3}\). Convert the 1 into a fraction with a common denominator: \(1 = \frac{3}{3}\). So, we have: \(\frac{3}{3} - \frac{2}{3} = \frac{1}{3}\).

Step 4: Divide the Fractions

Rewrite the complex fraction using the simplified numerator and denominator: \(\frac{\frac{5}{3}}{\frac{1}{3}}\). To divide by a fraction, multiply by its reciprocal: \(\frac{5}{3} \times \frac{3}{1} = \frac{5 \times 3}{3 \times 1} = 5\).

Step 5: Simplify the Result

The simplified result of the original complex fraction \(\frac{1+\frac{2}{3}}{1-\frac{2}{3}}\) is \(5\).

Key Concepts

Understanding Numerator and Denominator

When working with fractions, it's important to grasp the concepts of the numerator and the denominator. These are the two main parts of a fraction.

• The numerator is the top number of a fraction. It represents the number of parts we have.
• The denominator is the bottom number. It shows the total number of equal parts the whole is divided into.

In the context of complex fractions, like \( \frac{1+\frac{2}{3}}{1-\frac{2}{3}} \), each of these components can itself be a fraction or an expression. Here, both the numerator and denominator are composite expressions:- Numerator: \( 1 + \frac{2}{3} \) - Denominator: \( 1 - \frac{2}{3} \)When simplifying, our goal is to transform these components into simpler forms with a common denominator, helping us proceed to simplify the whole complex fraction.

Simplifying Fractions

Fraction simplification involves reducing a fraction to its simplest form where the numerator and denominator have no common factors other than 1.In our exercise, the first step in simplifying the complex fraction \( \frac{1+\frac{2}{3}}{1-\frac{2}{3}} \) is to manage each separate part. We simplify both the numerator \( 1 + \frac{2}{3} \) and the denominator \( 1 - \frac{2}{3} \) by finding a common denominator.For the numerator:- Convert 1 to a fraction with a denominator of 3: \( 1 = \frac{3}{3} \).- Add: \( \frac{3}{3} + \frac{2}{3} = \frac{5}{3} \).For the denominator:- Convert 1 similarly: \( 1 = \frac{3}{3} \).- Subtract: \( \frac{3}{3} - \frac{2}{3} = \frac{1}{3} \).The simplified expressions now form a new fraction \( \frac{\frac{5}{3}}{\frac{1}{3}} \). This simplification helps us proceed to solve the complex fraction by treating it as a division of simpler fractions.

Reciprocal of a Fraction

Understanding the reciprocal of a fraction is key for operations like division involving fractions. The reciprocal of a fraction is simply obtained by swapping its numerator and denominator.For example, if you have the fraction \(\frac{a}{b}\), its reciprocal is \(\frac{b}{a}\).When dealing with the complex fraction \( \frac{\frac{5}{3}}{\frac{1}{3}} \), divide by multiplying by the reciprocal of the denominator fraction. This means turning \( \frac{1}{3} \) into \( \frac{3}{1} \). Then multiply:- \( \frac{5}{3} \times \frac{3}{1} = \frac{5 \times 3}{3 \times 1} = 5 \).This operation not only simplifies the complex fraction but also shows the power of using reciprocals in fraction division. It is essential for solving complex fractions effectively.