Problem 50 - Intermediate Algebra

Question

Simplify each complex fraction. $$ -\frac{m}{\frac{1}{m}-\frac{1}{n}+\frac{1}{t}} $$

Step-by-Step Solution

Answer

The simplified complex fraction is \(-\frac{m^2nt}{nt - mt + mn}\).

Step 1: Identify the Complex Fraction

The given expression is a complex fraction:\[-\frac{m}{\frac{1}{m} - \frac{1}{n} + \frac{1}{t}}\]Our goal is to simplify this fraction by eliminating the nested fractions in the denominator.

Step 2: Find a Common Denominator

To eliminate the fractions in the denominator, find a common denominator for the terms \(\frac{1}{m}\), \(\frac{1}{n}\), and \(\frac{1}{t}\). The common denominator is the product \(m \times n \times t\).

Step 3: Rewrite the Denominator with Common Denominator

Rewrite each term in the denominator using the common denominator. The denominator becomes:\[\frac{nt}{mnt} - \frac{mt}{mnt} + \frac{mn}{mnt}\] which simplifies to\[\frac{nt - mt + mn}{mnt}\]

Step 4: Simplify the Complex Fraction

Substitute the simplified denominator back into the original expression:\[-\frac{m}{\frac{nt - mt + mn}{mnt}}\]To simplify, multiply by the reciprocal of the denominator:\[-m \times \frac{mnt}{nt - mt + mn}\] which simplifies to:\[-\frac{m^2nt}{nt - mt + mn}\]

Step 5: Final Simplified Expression

The final simplified expression for the given complex fraction is:\[-\frac{m^2nt}{nt - mt + mn}\]

Key Concepts

Simplifying Fractions

Simplifying fractions refers to the process of reducing a fraction to its simplest form. This means ensuring the numerator and the denominator have no common divisors other than 1. It involves canceling out the greatest common divisor (GCD) between the two parts. To simplify any fraction:

• Identify the numerator and the denominator.
• Find the greatest common divisor (GCD) of the numbers.
• Divide the numerator and the denominator by their GCD.

This procedure keeps the value of the fraction the same but increases the ease of understanding or comparing it. Simplifying is a crucial skill in algebra, particularly when dealing with algebraic expressions or complex fractions, as it increases clarity and aids in further manipulation.

Common Denominator

A common denominator is essential when dealing with fractions that you're adding, subtracting, or simplifying, especially when fractions are nested within each other like in complex algebraic expressions. For the fractions \(\frac{1}{m}\), \(\frac{1}{n}\), and \(\frac{1}{t}\), the smallest common denominator is their product \(m \times n \times t\). Here's why this technique is useful:

• It converts different denominators to a shared base, making the fractions comparable or combinable.
• It allows for easy subtraction or addition of fractions by aligning them under a single common base.

By finding the common denominator and rewriting each fraction with it, we transform the complex fractions into a simpler form, facilitating further simplification.

Algebraic Expressions

Algebraic expressions consist of variables, numbers, and operations (such as addition, subtraction, multiplication, and division). They are the building blocks of algebra, and understanding them is fundamental to solving complex problems. In an expression like \(-\frac{m}{\frac{1}{m} - \frac{1}{n} + \frac{1}{t}}\), we deal with multiple operations and variables that pose a simplification challenge:

• First, handle operations inside parentheses or fractions.
• Utilize techniques like finding a common denominator to simplify.
• Manipulate the expressions per algebraic rules for addition, subtraction, etc.

Mastering algebraic expressions enables students to deconstruct and solve complex problems, whether they be solving equations or simplifying intricate fractions like the one given.