Problem 6 - Elementary and Intermediate Algebra

Question

Fill in the blanks. $$ \text { Consider the complex fraction: } \frac{\frac{1}{y}-\frac{1}{3}}{\frac{5}{6}+\frac{1}{y}} $$ a. What is the LCD of all the rational expressions in the complex fraction? b. To simplify the complex fraction using method \(2,\) it should be multiplied by what form of \(1 ?\)

Step-by-Step Solution

Answer

a. The LCD is \(6y\). b. Multiply by \(\frac{6y}{6y}\).

Step 1: Identify the fractions

In the complex fraction \( \frac{\frac{1}{y}-\frac{1}{3}}{\frac{5}{6}+\frac{1}{y}} \), we have two fractions in both the numerator and the denominator: \( \frac{1}{y} \) and \( \frac{1}{3} \) in the numerator; \( \frac{5}{6} \) and \( \frac{1}{y} \) in the denominator.

Step 2: Determine the LCD

The denominators in the complex fraction are \( y \), \( 3 \), and \( 6 \). To find the least common denominator (LCD) of these denominators, we take the least common multiple of these numbers. \( y \) is a variable, \( 3 \) and \( 6 \) are constants. Thus, the LCD is the product of \( y \) and the LCM of \( 3 \) and \( 6 \), which is \( 6 \). Therefore, the LCD is \( 6y \).

Step 3: Choose the form of 1 to multiply

To simplify the complex fraction using Method 2, we need to multiply by a form of 1 that consists of the LCD over itself. Thus, we multiply the complex fraction by \( \frac{6y}{6y} \). This choice is based on the LCD determined in the previous step.

Key Concepts

Least Common Denominator

When working with fractions, especially complex ones, identifying the least common denominator (LCD) is crucial. The LCD ensures that all fractions involved are expressed with the same denominator, making addition, subtraction, or even simplification possible.

To find the LCD, we first examine all the denominators in the fractions we’re dealing with. For instance, the complex fraction \(\frac{\frac{1}{y}-\frac{1}{3}}{\frac{5}{6}+\frac{1}{y}}\) has denominators \(y\), \(3\), and \(6\).

• First, consider the numerical denominators: \(3\) and \(6\). The least common multiple (LCM) of these numbers is \(6\) because \(6\) is the smallest number divisible by both \(3\) and \(6\).
• Next, incorporate the variable from the other denominator, which is \(y\).
• Hence, the LCD is a combination of both the LCM and the variable, resulting in \(6y\).

Understanding and finding the LCD allows us to rewrite each fraction involved with the same denominator, simplifying work and making calculations easier.

Rational Expressions

Rational expressions are similar to fractions but they involve polynomials. In other words, a rational expression is any expression that can be written in the form \(\frac{P(x)}{Q(x)}\), where \(P(x)\) and \(Q(x)\) are polynomials and \(Q(x)\) is not equal to zero.

Rational expressions behave just like numeric fractions. You can perform operations such as addition, subtraction, multiplication, and division on them, but they often require more steps due to the presence of variables.

In a complex fraction like \(\frac{\frac{1}{y}-\frac{1}{3}}{\frac{5}{6}+\frac{1}{y}}\), each term is an example of a rational expression. Simplifying these requires finding a common denominator and rewriting each fraction in terms of this denominator, as we did in the previous section, using the LCD to streamline the entire fraction.

It's important to simplify these expressions as much as possible, as this process makes it easier to interpret and solve equations involving rational expressions.

Simplifying Fractions

Simplifying fractions means reducing them to their simplest or most basic form. This simplification makes fractions easier to understand and work with.

When simplifying, the goal is to cancel any common factors between the numerator and the denominator.

In the case of the complex fraction \(\frac{\frac{1}{y}-\frac{1}{3}}{\frac{5}{6}+\frac{1}{y}}\), once we've determined the LCD as \(6y\), we can simplify the overall fraction by multiplying the entire expression by \(\frac{6y}{6y}\). Doing so clears the denominators within the fractions, transforming them into more manageable terms:

• The fraction \(\frac{1}{y}\) becomes \(6\), due to the multiplication by \(\frac{6y}{y}\).
• The fraction \(\frac{1}{3}\) becomes \(2y\), as a result of the multiplication.
• The entire complex fraction transforms into a simpler form.

This method retains the value of the expression but changes its form, making the problem much easier to tackle. Simplification is a powerful tool that not only makes calculations easier but is key to understanding and solving more complex problems.