Problem 28 - Intermediate Algebra

Question

Simplify each complex fraction. See Example 4. $$ \frac{\frac{3}{y^{2}}-\frac{4}{y}}{\frac{1}{y}+\frac{15}{y^{2}}} $$

Step-by-Step Solution

Answer

The simplified form is \( \frac{3-4y}{y+15} \).

Step 1: Simplify the Numerator

We begin by finding a common denominator for the numerator's fractions, which are \( \frac{3}{y^2} \) and \( \frac{4}{y} \). The common denominator is \( y^2 \). Rewrite \( \frac{4}{y} \) with this common denominator:\[ \frac{4}{y} = \frac{4y}{y^2} \]Thus, the numerator becomes:\[ \frac{3}{y^2} - \frac{4y}{y^2} = \frac{3 - 4y}{y^2} \]

Step 2: Simplify the Denominator

Next, we simplify the denominator fractions \( \frac{1}{y} \) and \( \frac{15}{y^2} \) by also finding a common denominator, which is \( y^2 \). Rewrite \( \frac{1}{y} \) with this common denominator:\[ \frac{1}{y} = \frac{y}{y^2} \]Thus, the denominator becomes:\[ \frac{y}{y^2} + \frac{15}{y^2} = \frac{y + 15}{y^2} \]

Step 3: Divide the Numerator by the Denominator

Now, divide the simplified numerator by the simplified denominator. Our fraction is:\[ \frac{\frac{3-4y}{y^2}}{\frac{y+15}{y^2}} \]Since both the numerator and the denominator have the same denominator \( y^2 \), they cancel out:\[ \frac{3-4y}{y + 15} \]

Step 4: Final Result

Now the complex fraction is simplified to a simple fraction:\[ \frac{3-4y}{y + 15} \]This is the most simplified form of the given complex fraction.

Key Concepts

Simplifying Fractions

Fractions can sometimes seem tricky, especially when you're dealing with complex fractions like \( \frac{\frac{3}{y^{2}}-\frac{4}{y}}{\frac{1}{y}+\frac{15}{y^{2}}} \). A complex fraction is one that has a fraction as either its numerator, its denominator, or both. The main goal when simplifying such fractions is to rewrite it in its simplest form. This often involves several steps:

• Find a common denominator for the terms in the numerator and the denominator separately. This is the key to combining the fractions in each part.
• Once you have one fraction for the numerator and one for the denominator, divide the numerator by the denominator.
• Always simplify the resulting fraction if possible by factoring and cancelling common terms.

Remember to take it step by step— focus on simplifying each part of the fraction thoroughly. Understanding each process makes the entire task more manageable, especially in more complicated algebraic expressions.

Algebraic Expressions

Algebraic expressions represent a combination of numbers, variables, and operations. In the given fraction \(\frac{3-4y}{y + 15}\), you’re dealing with an algebraic expression because it contains variables, \(y\), which change depending on the specific problem.

• These expressions can include operations such as addition, subtraction, multiplication, and division.
• When simplifying, you often need to manipulate the expression to a form that is easier to handle or more understandable, like factoring it or rearranging terms.

A strong grasp of algebraic expressions will help not just in simplifying fractions, but also in solving equations and understanding the broader concepts of algebra.

Rational Expressions

Much like a fraction, a rational expression is a ratio or quotient of two polynomials. They exhibit properties that blend features of both fractions and algebraic expressions. In our example, \(\frac{3-4y}{y+15}\) is a rational expression, as both the numerator and the denominator constitute polynomials.

• To simplify rational expressions, always look for common factors in the numerator and the denominator that can be cancelled out.
• If you can factor the numerator and the denominator, do so, and check if terms cancel out across the division line.

Understanding rational expressions is crucial, as they often appear in algebra and calculus problems. They support the solution of functions and enlarge your toolkit for handling various mathematical challenges.