Problem 84 - Elementary and Intermediate Algebra

Question

Perform the operations and, if possible, simplify. $$ \frac{4}{3}+\frac{7}{2} $$

Step-by-Step Solution

Answer

The sum is \(\frac{29}{6}\).

Step 1: Determine the Least Common Denominator (LCD)

To add fractions, we need a common denominator. The denominators in the fractions \(\frac{4}{3}\) and \(\frac{7}{2}\) are 3 and 2. The least common denominator (LCD) of 3 and 2 is 6.

Step 2: Convert Fractions to Equivalent Fractions with the LCD

Convert \(\frac{4}{3}\) into a fraction with a denominator of 6 by multiplying both the numerator and denominator by 2, resulting in \(\frac{8}{6}\). Similarly, convert \(\frac{7}{2}\) by multiplying both the numerator and denominator by 3, resulting in \(\frac{21}{6}\).

Step 3: Add the Fractions

Now that both fractions have the same denominator, add their numerators: \(8 + 21 = 29\). The sum is \(\frac{29}{6}\).

Step 4: Simplify the Resulting Fraction

Since 29 and 6 have no common factors other than 1, \(\frac{29}{6}\) is already in its simplest form.

Key Concepts

Understanding the Least Common Denominator

When adding fractions, it's crucial that they have the same denominator. This uniformity allows us to add the numerators directly. But what happens when the denominators are different, like in \( \frac{4}{3} \) and \( \frac{7}{2} \)? Here comes the concept of the Least Common Denominator (LCD).

The LCD is the smallest number that both denominators can divide into without leaving a remainder. Finding the LCD involves:

1. Listing the multiples of each denominator.
2. Identifying the smallest common multiple.

For the fractions \( \frac{4}{3} \) and \( \frac{7}{2} \), we list the multiples of 3 (3, 6, 9, 12...) and 2 (2, 4, 6, 8...). The first common multiple is 6, making it our LCD. Using an LCD standardizes the fractions, preparing them for addition. This step is foundational in fraction operations as it simplifies complex additions into manageable steps.

Equivalent Fractions Explained

Behind the scenes of fraction addition lies the concept of equivalent fractions. An equivalent fraction keeps the same overall value while having a different numerator and denominator. This transformation is perfectly legitimate if you multiply or divide both the numerator and denominator by the same number.

For example, turning \( \frac{4}{3} \) into an equivalent fraction with a denominator of 6 involves:

• Deciding the factor that will turn 3 into the LCD, which is 6.
• Multiplying both the numerator and denominator by this factor (2 in this case).

This results in \( \frac{8}{6} \), an equivalent fraction to \( \frac{4}{3} \). Repeating this with \( \frac{7}{2} \) involves multiplying both aspects by 3 to obtain \( \frac{21}{6} \). These processes do not change the value of the fractions—only their form. Such flexibility underpins not just addition, but most operations involving fractions.

Simplifying Fractions Made Easy

Once you've added fractions and obtained a result, say \( \frac{29}{6} \), the next step is to simplify if possible. Simplifying a fraction involves reducing it to its lowest terms. A fraction is in its simplest form when the numerator and denominator have no common factors other than 1.

You can check for simplification through:

• Finding the greatest common factor (GCF) of the numerator and denominator.
• Dividing both the numerator and denominator by this GCF.

In \( \frac{29}{6} \), since 29 is a prime number and does not share any factors with 6 other than 1, it is already simplified. This concept ensures the neatest and most efficient representation of fractions, essential for clarity in mathematical communication.