Problem 44 - Elementary Algebra

Question

Add or subtract as indicated and express your answers in simplest form. (Objective 3) $$\frac{6 x}{9}+\frac{7 x}{12}$$

Step-by-Step Solution

Answer

\(\frac{5x}{4}\)

Step 1: Find the Least Common Denominator (LCD)

To add fractions, we need a common denominator. We start by finding the Least Common Multiple (LCM) of the denominators 9 and 12. The prime factorizations are 9 = 3^2 and 12 = 2^2 × 3. The LCM is 2^2 × 3^2 = 36, so the LCD is 36.

Step 2: Rewrite Fractions with the LCD

Convert each fraction to an equivalent fraction with the common denominator of 36. For \(\frac{6x}{9}\), multiply both the numerator and the denominator by 4 to get \(\frac{24x}{36}\). For \(\frac{7x}{12}\), multiply both the numerator and the denominator by 3 to get \(\frac{21x}{36}\).

Step 3: Add the Fractions

Now that both fractions have the same denominator, add the numerators while keeping the denominator the same. \(\frac{24x}{36} + \frac{21x}{36} = \frac{24x + 21x}{36} = \frac{45x}{36}\).

Step 4: Simplify the Fraction

Simplify \(\frac{45x}{36}\) by finding the greatest common divisor (GCD) of the numerator and the denominator. The GCD of 45 and 36 is 9. Divide both the numerator and the denominator by 9 to get \(\frac{5x}{4}\).

Key Concepts

Understanding the Least Common Denominator

When adding or subtracting algebraic fractions, the least common denominator (LCD) is crucial. This denominator is the smallest number that both original denominators divide into evenly. It's like finding a common ground for two different entities so you can combine them easily.
To find the LCD, identify the least common multiple (LCM) of the denominators. Let's see how this works: if you have 9 and 12 as denominators, break them down to their prime factors:

• 9 = 3²
• 12 = 2² × 3

Compare these factors and take the highest power of each different prime number appearing. For instance, we take 2² from 12 and 3² from 9. Multiply these: 2² × 3² = 36. Thus, 36 is your LCD.
Using the LCD helps align fractions to show the same division base, making the addition or subtraction process smoother and clearer.

Mastering Simplifying Fractions

Simplifying fractions is about reducing them to their simplest form so they are easier to work with. This means you'll be expressing the fraction without changing its value but with the smallest possible whole numbers in the numerator and denominator.
Begin by checking for common factors between the numerator and the denominator. Can they both be divided by any number? A simplified fraction is complete when the numerator and the denominator share no further common factors besides 1.
For example, take \(\frac{45x}{36}\). We simplify by determining the greatest common divisor of 45 and 36, which is 9. Divide both the numerator and the denominator by 9:

• Numerator: 45x ÷ 9 = 5x
• Denominator: 36 ÷ 9 = 4

This leaves us with the simplified expression \(\frac{5x}{4}\). Simplification not only makes fractions tidier but also makes further operations, like addition or subtraction, much more manageable.

Exploring the Greatest Common Divisor

The greatest common divisor (GCD) is the largest number that divides two or more numbers without leaving a remainder. It's an essential tool when simplifying fractions, as it helps reduce the fraction to its lowest terms.To find the GCD:

• List the factors of each number.
• Identify the largest factor they have in common.

In the case of 45 and 36, their factors are:

• 45: 1, 3, 5, 9, 15, 45
• 36: 1, 2, 3, 4, 6, 9, 12, 18, 36

The greatest common factor here is 9. This means you can divide both 45 and 36 by 9 to simplify the fraction \(\frac{45}{36}\) to \(\frac{5}{4}\).
Using the GCD not only simplifies calculations but ensures accuracy in mathematical operations by keeping values precise and uncomplicated.