Problem 43 - Elementary Algebra

Question

Add or subtract as indicated and express your answers in simplest form. (Objective 3) $$\frac{2 x}{6}+\frac{3 x}{5}$$

Step-by-Step Solution

Answer

\( \frac{14x}{15} \)

Step 1: Identify the operation

The expression given is \( \frac{2x}{6} + \frac{3x}{5} \). This is an addition problem with two fractions.

Step 2: Find a common denominator

The denominators of the fractions are 6 and 5. To add these fractions, we need a common denominator. The least common multiple (LCM) of 6 and 5 is 30. So, we will convert each fraction to have a denominator of 30.

Step 3: Convert fractions to have the common denominator

Convert \( \frac{2x}{6} \) to an equivalent fraction with a denominator of 30 by multiplying the numerator and the denominator by 5: \( \frac{2x \times 5}{6 \times 5} = \frac{10x}{30} \).Convert \( \frac{3x}{5} \) similarly by multiplying the numerator and the denominator by 6:\( \frac{3x \times 6}{5 \times 6} = \frac{18x}{30} \).

Step 4: Add the fractions

Now, add the fractions \( \frac{10x}{30} + \frac{18x}{30} \). Since they have the same denominator, you just add the numerators:\( \frac{10x + 18x}{30} = \frac{28x}{30} \).

Step 5: Simplify the fraction

Simplify \( \frac{28x}{30} \) by dividing both the numerator and denominator by their greatest common divisor, which is 2:\( \frac{28x \div 2}{30 \div 2} = \frac{14x}{15} \). This is the expression in its simplest form.

Key Concepts

Least Common Denominator

When adding or subtracting fractions, the denominators need to match. This is where the Least Common Denominator (LCD) comes into play. The least common denominator is the smallest number that is a multiple of each of the denominators. With denominators like 6 and 5, finding the smallest number they both divide into evenly is crucial for simplifying the process of addition or subtraction. To find the LCD:

• List the multiples of each denominator.
• Identify the smallest multiple common to both lists.

For example, with the fractions \( \frac{2x}{6} \) and \( \frac{3x}{5} \):

• Multiples of 6: 6, 12, 18, 24, 30, 36, ...
• Multiples of 5: 5, 10, 15, 20, 25, 30, ...

Here, 30 is the least common multiple. Thus, it's our least common denominator for both fractions. Using the LCD allows us to rewrite the fractions with matching denominators, enabling easy addition or subtraction.

Simplifying Fractions

Simplifying fractions helps in expressing the fraction in its most reduced form, making it easier to understand and compare. A fraction is simplified when the numerator and denominator have no common factors other than 1. In other words, when you cannot divide them further, except by 1.After performing arithmetic operations on fractions, you may end up with a fraction that can be reduced. For the fraction result \( \frac{28x}{30} \), the simplification process is as follows:

• Find the greatest common divisor (GCD) of both the numerator and the denominator. Here, the GCD is 2.
• Divide both numerator (28) and denominator (30) by the GCD (2).
• The simplified result is \( \frac{14x}{15} \).

Simplifying helps keep numbers manageable and prepares them for additional calculations. Plus, a simplified fraction is usually more acceptable in mathematical communication.

Common Denominator

Before adding or subtracting fractions, you need a common denominator for them. Having a common denominator means the fractions share the same base, allowing direct addition or subtraction of the numerators. Without a common denominator, trying to add or subtract fractions is like trying to add apples and oranges.For the problem \( \frac{2x}{6} + \frac{3x}{5} \), the common denominator is 30, as identified earlier as the least common denominator. Here's how you convert each fraction:

• Convert \( \frac{2x}{6} \) to have a denominator of 30 by multiplying both the numerator and denominator by 5, resulting in \( \frac{10x}{30} \).
• Convert \( \frac{3x}{5} \) by multiplying both the numerator and the denominator by 6, resulting in \( \frac{18x}{30} \).

Now that both fractions share a common denominator, they can be easily added, yielding \( \frac{10x + 18x}{30} \). This process underlines the importance of finding a common denominator for simplifying fraction operations.