Problem 46 - Elementary Algebra

Question

For Problems 41-60, simplify each of the complex fractions. $$ \frac{4+\frac{3}{5}}{\frac{1}{3}-2} $$

Step-by-Step Solution

Answer

The simplified form is \(-\frac{69}{25}\).

Step 1: Simplify the Numerator

The numerator is \(4 + \frac{3}{5}\). To simplify, convert 4 into a fraction with 5 as the denominator. This gives \(\frac{20}{5}\). Add \(\frac{20}{5}\) and \(\frac{3}{5}\) to get \(\frac{23}{5}\).

Step 2: Simplify the Denominator

The denominator is \(\frac{1}{3} - 2\). Convert 2 into a fraction with 3 as the denominator, which results in \(\frac{6}{3}\). Subtract \(\frac{6}{3}\) from \(\frac{1}{3}\) to get \(-\frac{5}{3}\).

Step 3: Form the Simplified Fraction

Insert the simplified numerator \(\frac{23}{5}\) and the simplified denominator \(-\frac{5}{3}\) into the original complex fraction. This forms the expression \(\frac{\frac{23}{5}}{-\frac{5}{3}}\).

Step 4: Simplify the Complex Fraction

Simplify the complex fraction by multiplying \(\frac{23}{5}\) by the reciprocal of \(-\frac{5}{3}\), which is \(-\frac{3}{5}\). This gives \(\frac{23}{5} \times -\frac{3}{5} = -\frac{69}{25}\).

Step 5: Final Simplification

The fraction \(-\frac{69}{25}\) is already in its simplest form as the numerator and denominator have no common factors.

Key Concepts

Fraction Simplification

Simplifying fractions is an essential skill in math. It's about expressing a fraction in the simplest form, with the numerator and denominator having no common factors other than 1. For the given problem, simplifying involves both the complex fraction as a whole and its components separately.
The complex fraction typically has a fraction in the numerator, a fraction in the denominator, or both. You first simplify these inner fractions. By solving them, you express the larger, complex fraction in a form that's easier to manage.

• To simplify, convert whole numbers to fractions using a common denominator.
• Perform basic operations: addition, subtraction, multiplication, and division.
• Always check if the resulting fraction can be further simplified by finding common factors.

Simplify by multiplying by the reciprocal of the denominator. Through these processes, you'll obtain a simplified expression that's much cleaner to work with.

Algebraic Expressions

In this exercise, algebraic expressions deal with operations involving fractions. They include numbers, variables, and operation symbols but here, we focus primarily on numeric fractions.
Fractions can be viewed as algebraic expressions because they represent division. The key to simplifying them lies in treating these operations like any other fraction operation.

• Fractional operations can be seen as algebraic manipulations.
• To simplify, make fractions share a common denominator, or express them for easier comparison and calculations.
• Understand how operations like addition and subtraction affect the overall fraction and its components.

Algebra is about finding the unknown. Even when working solely with numbers, you're using algebraic principles to arrive at the simplest, most revealing expression of the problem. That’s what fraction simplification is all about.

Numerator and Denominator

A fraction is made up of two essential parts: the numerator and the denominator. The numerator is the top part and represents how many parts you have. The denominator, on the bottom, shows the total number of equal parts in a whole.
When working with complex fractions, you often have smaller fractions as either the numerator or the denominator or sometimes both. This means you need to simplify these smaller parts first before tackling the overall division problem.

• Convert whole numbers into fractions to align numerators with their corresponding denominators.
• Perform operations to adjust its value, like addition in the numerator or subtraction in the denominator.
• Always simplify smaller fractions first, then the whole fraction.

The journey from simple fractions through complex fractions is simply expanding and contracting numerators and denominators for clarity and simplicity. This is the heart of understanding and managing fractions effectively.