Problem 30 - Prealgebra

Question

Find the LCD for each of the following; then use the methods developed in this section to add or subtract as indicated. $$\frac{x}{3}+\frac{1}{5}$$

Step-by-Step Solution

Answer

The sum of the fractions is \( \frac{5x + 3}{15} \).

Step 1: Identify the Denominators

The fractions given in the problem are \( \frac{x}{3} \) and \( \frac{1}{5} \). The denominators are 3 and 5.

Step 2: Find the Least Common Denominator (LCD)

The least common denominator is the smallest number that both denominators can divide into without leaving a remainder. For the numbers 3 and 5, the LCD is 15 because 15 is the smallest number that both 3 and 5 divide evenly into.

Step 3: Convert the Fractions to Have the LCD

To convert \( \frac{x}{3} \) to a fraction with a denominator of 15, multiply both the numerator and denominator by 5. This gives \( \frac{5x}{15} \). Similarly, to convert \( \frac{1}{5} \) to have a denominator of 15, multiply both the numerator and denominator by 3, resulting in \( \frac{3}{15} \).

Step 4: Add the Fractions

Now that both fractions have the same denominator, you can add them directly: \( \frac{5x}{15} + \frac{3}{15} = \frac{5x + 3}{15} \). This is the simplified form of the sum.

Key Concepts

Adding Fractions

When you encounter fractions with different denominators, adding them might seem a bit tricky initially. The key to adding fractions is to have a common ground, which in mathematical terms means having a common denominator. This ensures coherence and measurement consistency across the fractions.

To add fractions, follow these steps:

• Identify the denominators of the fractions you plan to add.
• Determine the Least Common Denominator (LCM) that both denominators can divide into without leaving a remainder.
• Transform each fraction by adjusting the numerator and the denominator so that each fraction has this least common denominator.
• Add only the numerators of the transformed fractions, keeping the common denominator unchanged.

With the denominators aligned, adding becomes straightforward. You simply sum the numerators and retain the denominator as is. It's like finding a common language both fractions understand!

Rational Expressions

Rational expressions are like fractions, but they involve polynomials in the numerator and the denominator. Handling them can be as simple as basic arithmetic operations, as long as you follow the right steps.

For operations involving rational expressions:

• Treat them similar to regular fractions. Look for the least common denominator (LCD) when adding or subtracting.
• Factor both the numerator and the denominator when possible to simplify.
• Be cautious of restrictions on variables, as these can influence the domain of the expression.

Finding a common denominator is crucial here to accurately perform addition or subtraction, just like in simple fractions. With rational expressions, though, their complexity comes from the variables present. Identifying the LCD helps align terms, making subsequent operations simpler to handle.

Simplifying Fractions

Simplifying fractions is about reducing them to their simplest form while maintaining their value. It's like cleaning up an untidy room while keeping everything functional.

To simplify a fraction:

• Determine the greatest common divisor (GCD) of the numerator and the denominator.
• Divide both the numerator and the denominator by this GCD.
• Ensure the fraction is in its lowest terms.

For example, in the fraction \(\frac{5x + 3}{15}\), once you've added your fractions with the least common denominator, check if there's a common factor in the numerator and denominator. If no common factor exists, the fraction is already in simplest form. Simplifying is key to making fractions or rational expressions easier to understand and work with, while conserving all the original relationships.