Problem 102 - Introductory Algebra for College Students

Question

perform the indicated operation. Where possible, reduce the answer to its lowest terms. $$\frac{1}{2}+\frac{2}{3}$$

Step-by-Step Solution

Answer

\(\frac{7}{6}\)

Step 1: Identify the least common denominator

The two fractions are \(\frac{1}{2}\) and \(\frac{2}{3}\). The denominators of these fractions are 2 and 3, respectively. The least common denominator (LCD) is the least common multiple (LCM) of 2 and 3. The LCM of 2 and 3 is 6. Therefore, the LCD of these fractions is 6.

Step 2: Rewrite the fraction in terms of the least common denominator

To rewrite each fraction in terms of the LCD, multiply each fraction by the factor which corresponds to the LCD divided by the denominator of the fraction. Multiply \(\frac{1}{2}\) by \(\frac{3}{3}\) and \(\frac{2}{3}\) by \(\frac{2}{2}\), to get the fractions \(\frac{3}{6}\) and \(\frac{4}{6}\), respectively.

Step 3: Perform the addition operation

Add the two fractions now that they have the same denominator. \( \frac{3}{6} + \frac{4}{6} = \frac{7}{6}\)

Step 4: Reduce to the simplest form, if possible

The resulting fraction, \(\frac{7}{6}\), is already in lowest terms, as 7 and 6 have no common factor other than 1; hence, it cannot be simplified down further.

Key Concepts

Least Common Denominator

When adding or subtracting fractions, it's essential to have the same denominator. This denominator is known as the least common denominator (LCD). The LCD ensures that the fractions are expressed in a way that makes it easier to perform these operations. Without an LCD, attempting to add fractions is like trying to combine apples and oranges; it just doesn't work properly.

To find the LCD, identify the denominators of the fractions involved. For instance, if dealing with \( \frac{1}{2} \) and \( \frac{2}{3} \), the denominators are 2 and 3. The LCD is the smallest number that both denominators can evenly divide into. This will allow both fractions to be rewritten with this common denominator, simplifying the addition process overall.

Least Common Multiple

Understanding the least common multiple (LCM) is crucial as it helps us determine the least common denominator. The LCM of two numbers is the smallest number that both numbers can multiply into without a remainder. For adding fractions, the LCM of the denominators becomes the least common denominator.

Let's take the denominators 2 and 3. Here's how you find the LCM:

• List multiples of 2: 2, 4, 6, 8, 10, ...
• List multiples of 3: 3, 6, 9, 12, 15, ...
• The smallest number common to both lists of multiples is 6.

So, the LCM, and thus the LCD for our fractions, is 6. By rewriting the fractions using this common number, we simplify the process of adding them together.

Simplifying Fractions

Once you have performed the addition operation with your fractions, the result may not always be in its simplest form. Simplifying a fraction involves ensuring that the numerator and the denominator share no common factors other than 1. If they do, you can divide both by their greatest common factor.

In our case, after adding \( \frac{3}{6} \) and \( \frac{4}{6} \), we end up with \( \frac{7}{6} \). This fraction is already simplified because the number 7 is a prime number, and the greatest common factor of 7 and 6 is only 1. Therefore, there's no need to simplify as it can't be reduced further.

When simplifying fractions, remember:

• Check for common factors of the numerator and the denominator.
• If found, divide both by the greatest common factor to simplify.
• If none exist other than 1, the fraction is already in its simplest form.