Problem 74 - Elementary Algebra

Question

For Problems \(73-76\), simplify each complex fraction. $$ 2-\frac{3 n}{1+\frac{4}{n}} $$

Step-by-Step Solution

Answer

\(-\frac{3n^2 - 2n - 8}{n+4}\)

Step 1: Simplify the Denominator

Start by simplifying the denominator of the fraction. We have:\[1 + \frac{4}{n}\]To simplify this, find a common denominator, which is \(n\). Rewrite \(1\) as \(\frac{n}{n}\):\[\frac{n}{n} + \frac{4}{n} = \frac{n+4}{n}\]

Step 2: Substitute Simplified Denominator into Original Expression

Now substitute the simplified denominator back into the original complex fraction:\[2 - \frac{3n}{\frac{n+4}{n}}\]

Step 3: Divide by a Fraction (Multiply by the Reciprocal)

To simplify \(\frac{3n}{\frac{n+4}{n}}\), multiply by the reciprocal of the denominator:\[\frac{3n}{\frac{n+4}{n}} = 3n \times \frac{n}{n+4} = \frac{3n^2}{n+4}\]

Step 4: Combine into a Single Expression

Now substitute back into the expression:\[2 - \frac{3n^2}{n+4}\]

Step 5: Express as a Single Fraction

To combine \(2\) and \(-\frac{3n^2}{n+4}\), rewrite \(2\) as a fraction with the same denominator:\[2 = \frac{2(n+4)}{n+4} = \frac{2n+8}{n+4}\]Thus, the expression becomes:\[\frac{2n+8}{n+4} - \frac{3n^2}{n+4}\]

Step 6: Subtract the Fractions

Since the denominators are the same, subtract the numerators:\[\frac{(2n+8) - 3n^2}{n+4} = \frac{-3n^2 + 2n + 8}{n+4}\]

Step 7: Final Simplified Expression

The simplified complex fraction is:\[-\frac{3n^2 - 2n - 8}{n+4}\]

Key Concepts

Understanding Algebra

Algebra is a branch of mathematics that uses symbols, typically letters, to represent numbers in equations and expressions. These symbols are known as variables and are powerful tools because they allow us to work with unknown quantities. Consider an algebraic expression like the one we are dealing with in the original exercise:

• The expression 0: \(2 - \frac{3n}{1+\frac{4}{n}}\) demonstrates the manipulation of variables and constants in a complex fraction.
• Variables like \( n \) can represent any number, and manipulating these symbols requires a solid understanding of algebraic principles.

Algebra helps us to form general rules about how numbers behave, which we use to solve problems. When simplifying complex fractions, the concepts of algebra are crucial. We need to know how to manage variables and constants effectively, which involves operations like addition, subtraction, multiplication, and division of algebraic terms.

Demystifying Fractions

Fractions represent parts of a whole. In algebra, fractions become even more significant as numbers in the form of fractions comprise expressions and equations. A fraction consists of a numerator (the top part) and a denominator (the bottom part). For example, in the fraction \(\frac{3n^2}{n+4}\), \(3n^2\) is the numerator and \(n+4\) is the denominator.

• Complex fractions, like the one in the exercise, involve a fraction within another fraction. They can look intimidating at first, but the key is to take it step by step.
• The goal is to simplify these fractions into a more manageable form by finding common denominators and using techniques like rewriting and reciprocals.

Working with fractions in algebra often requires finding a common denominator, which simplifies the operation by making the denominators the same. This common denominator is crucial when you want to add or subtract fractions. It's this understanding that allows us to simplify the expression effectively.

Rational Expressions Made Simple

Rational expressions are expressions that involve fractions with polynomials in the numerator, the denominator, or both. Simplifying a rational expression involves similar techniques used in simplifying numeric fractions, but with algebraic expressions.

• Our original exercise involved simplifying a rational expression, \(2 - \frac{3n}{\frac{n+4}{n}}\), which demonstrates typical challenges.
• Rational expressions need to be manipulated using multiplication of reciprocals, common denominators, or factoring to reduce them to their simplest form.

In the exercise, after rewriting the denominator, the rational expression was simplified by multiplying by the reciprocal of the denominator. This approach is a standard technique in algebra for transforming a complex fraction into a simpler expression. Rational expressions often result in polynomials that need additional simplification, like factoring, to find the simplest form. Mastery of these concepts ensures efficiency in dealing with any expression or problem involving rational expressions.