Problem 62 - Elementary Algebra

Question

Add or subtract as indicated and express your answers in simplest form. (Objective 3) $$\frac{11}{9 y}-\frac{8}{15 y}$$

Step-by-Step Solution

Answer

\( \frac{31}{45y} \)

Step 1: Identify the Problem

We are given the problem \( \frac{11}{9y} - \frac{8}{15y} \). The task is to subtract these two fractions and express the result in its simplest form.

Step 2: Determine a Common Denominator

The denominators of the fractions are \(9y\) and \(15y\). Since both have \(y\) in common, we focus on \(9\) and \(15\). The least common multiple (LCM) of \(9\) and \(15\) is \(45\). Thus, the common denominator is \(45y\).

Step 3: Rewrite each Fraction with the Common Denominator

We need to rewrite each fraction with the denominator \(45y\):\[ \frac{11}{9y} = \frac{11 \times 5}{9y \times 5} = \frac{55}{45y} \]\[ \frac{8}{15y} = \frac{8 \times 3}{15y \times 3} = \frac{24}{45y} \]

Step 4: Perform the Subtraction

Now that the fractions have a common denominator, subtract them:\[ \frac{55}{45y} - \frac{24}{45y} = \frac{55 - 24}{45y} = \frac{31}{45y} \]

Step 5: Simplify the Result

The fraction \( \frac{31}{45y} \) cannot be simplified further since 31 is a prime number and does not divide 45. Thus, \( \frac{31}{45y} \) is the simplest form.

Key Concepts

Common Denominator

When working with algebraic fractions, we often need to find a common denominator. The common denominator allows us to add or subtract fractions more easily by aligning their bases.

In the exercise given, you are tasked with subtracting two fractions: \( \frac{11}{9y} \) and \( \frac{8}{15y} \). To do so, both fractions must share the same denominator.

• First, identify the different denominators. Here, they are \( 9y \) and \( 15y \).
• Since both denominators include \( y \), focus on the numbers: 9 and 15.
• To find their common ground, calculate the least common multiple (LCM) of 9 and 15.
• Multiply each fraction by the necessary value to reach this LCM, bolstered by the \( y \) already in place, resulting in a mutual denominator of \( 45y \).

By having each fraction rewritten with a shared \( 45y \) denominator, subtraction becomes simpler and more straightforward.

Simplifying Fractions

Once you perform operations with fractions, like adding or subtracting, the next step is often simplifying the result. Simplifying means to express a fraction in its lowest terms, where the numerator and denominator have no more common factors other than 1.

Let’s look at our example. After rewriting the fractions and performing the subtraction, you end up with \( \frac{31}{45y} \). Here's how to check if simplification is possible:

• Examine the numerator, 31. Note that 31 is a prime number, meaning its only divisors are 1 and itself.
• Check the denominator, 45, which can be broken down into its prime factors.
• Since 31 and 45 have no common factors apart from 1, \( \frac{31}{45y} \) is already in its simplest form.

With no further simplification needed, this fraction represents the solution in clean, reduced form.

Least Common Multiple

The least common multiple (LCM) is an essential concept in algebra, especially when dealing with fractions. The LCM of two numbers is the smallest positive number that is a multiple of both. This concept ensures a common denominator for operations like addition and subtraction.

To find the LCM of 9 and 15 in our exercise, consider the following steps:

• List the multiples of 9 (e.g., 9, 18, 27, 36, 45, ...).
• List the multiples of 15 (e.g., 15, 30, 45, 60, ...).
• Identify the first number that appears in both lists — in this case, 45.
• 45 is thus the LCM, effectively serving as the common backbone upon which the other numbers can align.

Choosing the LCM as the mutual denominator simplifies calculations, offering consistency across fractions and easing operations like subtraction in this specific context.