Problem 54 - Elementary and Intermediate Algebra

Question

Simplify each complex fraction. $$ \begin{aligned} &1\\\ &\frac{\frac{1}{m}-\frac{1}{n}}{\frac{m}{n}-\frac{n}{m}} \end{aligned} $$

Step-by-Step Solution

Answer

The simplified form of the complex fraction is \( \frac{-1}{m+n} \).

Step 1: Understand the Structure

The given expression is \( \frac{\frac{1}{m} - \frac{1}{n}}{\frac{m}{n} - \frac{n}{m}} \). Complex fractions contain fractions in their numerators, denominators, or both. Here, both the numerator and the denominator are separate fractions.

Step 2: Simplifying the Numerator

The numerator is \( \frac{1}{m} - \frac{1}{n} \). To combine these into a single fraction, find a common denominator, which is \( mn \). This yields: \[ \frac{n}{mn} - \frac{m}{mn} = \frac{n-m}{mn}. \]

Step 3: Simplifying the Denominator

The denominator is \( \frac{m}{n} - \frac{n}{m} \). To combine these, find a common denominator, which is also \( mn \). This gives: \[ \frac{m^2}{mn} - \frac{n^2}{mn} = \frac{m^2-n^2}{mn}. \]

Step 4: Division of Fractions

To simplify the complex fraction \( \frac{\frac{n-m}{mn}}{\frac{m^2-n^2}{mn}} \), multiply by the reciprocal of the denominator: \[ \frac{n-m}{mn} \times \frac{mn}{m^2-n^2}. \] The \( mn \) terms cancel out, simplifying to: \[ \frac{n-m}{m^2-n^2}. \]

Step 5: Simplify Further

Recognize that \( m^2-n^2 \) is a difference of squares, which can be factored into \( (m-n)(m+n) \). Thus, the expression becomes: \[ \frac{n-m}{(m-n)(m+n)}. \]

Step 6: Simplify the Signs

Noticing the order of \( m-n \) and \( n-m \) can simplify the fraction. \( n-m = -(m-n) \). Therefore, substitute this in to get: \[ \frac{-(m-n)}{(m-n)(m+n)}. \]

Step 7: Final Simplification

The \( -(m-n) \) and \( (m-n) \) cancel each other as \( \frac{-(m-n)}{(m-n)} = -1 \). The overall expression simplifies to: \[ \frac{-1}{m+n}. \]

Key Concepts

Simplifying Fractions

Simplifying fractions is the process of reducing fractions to their simplest form, where the numerator and denominator have only 1 as their common factor. Here's how you can approach simplifying fractions easily:

• Identify the greatest common divisor (GCD) of the numerator and the denominator. You can use prime factorization or a simple division method for this.
• Divide both the numerator and the denominator by the GCD to get the simplest form of the fraction.

When dealing with complex fractions, as in the exercise, start by simplifying the individual fractions within the complex fraction. Then, combine these simpler fractions into a single fraction by finding a common denominator.
This approach ensures you reduce the complexity one step at a time, which makes the overall problem more manageable. After simplifying the smaller fractions, focus on how they interact with each other within the complex fraction. Understanding the basic rules of operations with fractions, such as multiplication, division, and reduction, can significantly aid in simplifying complex fractions.

Difference of Squares

The difference of squares is a useful algebraic identity that can simplify expressions involving quadratic terms. This principle states that:\[ a^2 - b^2 = (a-b)(a+b) \]Whenever you see a subtraction between squares, you can factor them into their binomial components. This technique applies directly to the complex fraction's denominator in the given exercise:

• The expression \( m^2 - n^2 \) is a difference of squares.
• It can be rewritten as \( (m-n)(m+n) \).

This step is crucial, as factoring using the difference of squares simplifies the structure of the expression and can often reveal cancelation opportunities. With the identity in hand, you can reframe the problem in a much simpler form. This simplification often makes it easier to spot further reductions or cancelations, focusing on more straightforward algebraic manipulations.
Using the difference of squares not only simplifies calculations but also enhances understanding of the underlying algebraic relationships.

Common Denominator

The concept of a common denominator is essential when simplifying fractions, especially when adding or subtracting them. It refers to having the same number as the denominator for multiple fractions, which makes addition and subtraction straightforward. To find a common denominator:

• List the denominators of each fraction involved.
• Determine the least common multiple (LCM) of these denominators.
• Convert each fraction so that they all share this common denominator by multiplying both numerator and denominator by suitable numbers.

In the exercise, both numerator \( \frac{1}{m} - \frac{1}{n} \) and denominator \( \frac{m}{n} - \frac{n}{m} \) of the complex fraction required a common denominator: \( mn \).
Setting a common baseline allowed easy subtraction of the fractions, reducing them into single simpler fractions like \( \frac{n-m}{mn} \) and \( \frac{m^2-n^2}{mn} \).
Once fractions are expressed with a common denominator, it becomes a straightforward process to simplify, allowing focus on the numerator and on making efficient cancellations. Simplifying complex fractions becomes much more systematic with this approach, minimizing errors and enhancing understanding of the relationships between components.