Problem 3 - Intermediate Algebra

Question

To simplify the following complex fraction, it is multiplied by what form of 1? $$ \frac{\frac{4}{t^{2}}+\frac{b}{t}}{\frac{3 b}{t}}=\frac{\frac{4}{t^{2}}+\frac{b}{t}}{\frac{3 b}{t}} \cdot \frac{t^{2}}{t^{2}} $$

Step-by-Step Solution

Answer

Multiply by \( \frac{t^{2}}{t^{2}} \) to simplify the complex fraction.

Step 1: Understanding the Problem

To simplify a complex fraction, we commonly multiply by a form of 1 that helps to eliminate the fractions within the numerator and the denominator. The challenge is identifying the correct form of 1 to use.

Step 2: Identifying the Denominators

In the complex fraction, the entire denominator is \( \frac{3b}{t} \), while there are two terms in the numerator: \( \frac{4}{t^{2}} \) and \( \frac{b}{t} \). To simplify, aim to make these terms easier to handle by eliminating the denominators.

Step 3: Choosing the Form of 1

To eliminate the smallest denominator \( t^{2} \) in the numerator, use \( \frac{t^{2}}{t^{2}} \) as the form of 1, because multiplying by \( t^{2} \) will cancel out both \( t^{2} \) and \( t \) in the fractions.

Step 4: Multiplying by the Form of 1

Multiply both the numerator and denominator of the complex fraction by \( \frac{t^{2}}{t^{2}} \), leading to: \[\frac{\left(\frac{4}{t^{2}}+\frac{b}{t}\right)\cdot t^{2}}{\frac{3b}{t}\cdot t^{2}}. \]This step helps to simplify expressions by removing fractions within the complex fraction.

Step 5: Simplifying the Expressions

Distribute \( t^{2} \) across the terms in the numerator and simplify: \[\frac{\left(\frac{4t^{2}}{t^{2}}+\frac{bt^{2}}{t}\right)}{\frac{3bt^{2}}{t}} = \frac{4 + bt}{3bt}.\]After simplification, we no longer have small denominators, completing the exercise.

Key Concepts

Simplification

Simplification is a crucial skill in mathematics because it makes expressions easier to understand and solve. When simplifying complex fractions, the goal is to reduce the expression to its simplest form.

A complex fraction is one where the numerator, the denominator, or both are also fractions. Simplifying often involves eliminating these smaller fractions to create a simpler expression.

Here’s how you can simplify complex fractions effectively:

• Identify a common denominator: Look for a common factor that can cancel out the smaller denominators within the complex fraction. In our original exercise, we used \( \frac{t^2}{t^2} \) because it canceled both \( t^2 \) and \( t \).
• Multiply through: Apply your chosen form of 1 to both the numerator and the denominator of the complex fraction. This step clears the underlying fractions.
• Simplify the results: After multiplying, you often need to distribute and combine like terms to fully simplify the expression.

By mastering simplification techniques, you can make tackling complex problems much more manageable.

Fractions

Fractions represent parts of a whole or division of any quantity. A fraction is expressed as \( \frac{a}{b} \), where \( a \) is the numerator, and \( b \) is the denominator.

Understanding how fractions work aids in various mathematical operations, such as addition, subtraction, multiplication, and division. Here is how these operations generally work with fractions:

• Addition/Subtraction: Fractions must have a common denominator to be added or subtracted. Convert the fractions, perform the operation, and adjust the result if needed.
• Multiplication: Multiply the numerators together for the new numerator, and the denominators together for the new denominator. Simplify if possible.
• Division: Invert the second fraction and multiply. This is known as "multiplying by the reciprocal.”

In complex fractions, like the one in the original problem, understanding these basic operations is vital for simplification. Initially, learn to manipulate fractions before handling their complex counterparts.

Algebraic Expressions

An algebraic expression is a mathematical phrase that involves variables, numbers, and operations. Examples include \( 4x + 3 \), or more complex forms like \( \frac{4}{t^2} + \frac{b}{t} \). In algebra, understanding these expressions is essential as they allow you to describe relationships between quantities.

For manipulation of algebraic expressions, keep these tips in mind:

• Identify like terms: These are terms in an expression that have the same variables raised to the same power. You can combine them by adding or subtracting their coefficients.
• Use distributive property: This property lets you multiply a sum by a term outside the parenthesis, as seen with \( t^2(\frac{4}{t^2} + \frac{b}{t}) \).
• Apply division and multiplication effectively: To reduce expressions, use division or multiplication to clear fractions or to consolidate terms.

Mastering the handling of algebraic expressions equips you to address various algebra problems, including simplifying complex fractions.