Problem 52 - Elementary and Intermediate Algebra

Question

Simplify each complex fraction. $$ \frac{18 n^{2}}{\frac{6 n}{13}} $$

Step-by-Step Solution

Answer

The simplified form of the complex fraction is \(39n\).

Step 1: Understand the Complex Fraction

A complex fraction has a fraction in either its numerator, denominator, or both. In this problem, there is a fraction in the denominator: \( \frac{6n}{13} \).

Step 2: Find the Reciprocal of the Denominator

The reciprocal of a fraction is obtained by swapping its numerator and denominator. Therefore, the reciprocal of \( \frac{6n}{13} \) is \( \frac{13}{6n} \).

Step 3: Multiply by the Reciprocal

To simplify the complex fraction, multiply the numerator by the reciprocal of the denominator: \( \frac{18n^2}{1} \times \frac{13}{6n} \).

Step 4: Multiply the Numerators and Denominators

Multiply the numerators and denominators of the resulting fractions: Numerator: \( 18n^2 \times 13 = 234n^2 \)Denominator: \( 1 \times 6n = 6n \)

Step 5: Simplify the Resulting Fraction

Divide both the numerator and the denominator by the common factor. Since both 234 and 6 can be divided by 6, we get:\( \frac{234n^2}{6n} = \frac{39n}{1} = 39n \).

Key Concepts

Understanding Reciprocals

To solve complex fractions efficiently, understanding reciprocals is crucial. A reciprocal is simply what you get when you "flip" a fraction. For example, if you start with a fraction like \( \frac{a}{b} \), its reciprocal would be \( \frac{b}{a} \). This transformation turns the roles of the numerator and denominator around.
Reciprocals are particularly helpful because multiplying a number by its reciprocal results in 1. This property is essential when you want to divide by a fraction, as dividing by a fraction is the same as multiplying by its reciprocal.

• Example: The reciprocal of \( \frac{3}{4} \) is \( \frac{4}{3} \).
• This makes calculation in complex fractions easier, such as changing divisions to multiplications.

The Simplification Process

Simplification is the process of making an expression easier to understand or solve. When simplifying complex fractions, this usually means transforming them into simpler fractions that are easier to compute or understand.

• First, you identify the complex part, for example, a fraction in the numerator or denominator.
• Then, you find the reciprocal of the fraction if it's in the denominator. This transforms the division into multiplication.
• Next step is to multiply the rest of the fractions. Simplify by canceling out any common factors between the numerator and the denominator.
• Finally, reduce the expression to its simplest form.

Simplification helps in solving algebraic problems more efficiently and in real-world scenarios, it's useful for understanding proportions or rates.

Fractions in Algebra

Fractions are a fundamental concept in algebra, providing a way to represent numbers that are not whole. In algebra, fractions can both represent operations and solutions. They are used in equations, expressions, and inequalities.
Working with fractions in algebra often involves:

• Finding a common denominator when adding or subtracting fractions.
• Multiplying fractions directly across numerators and denominators.
• Dividing fractions by multiplying by the reciprocal.
• Simplifying fractions by canceling common factors.

Understanding how to manipulate and simplify fractions is vital in algebra. It lays the groundwork for solving more complex problems and requires comfort with basic arithmetic operations, reciprocals, and factoring. This topic is foundational for progressing through more advanced mathematics.