Problem 24 - College Algebra with Modeling and Visualization

Question

Simplify. $$ \frac{5}{9}+\frac{2}{9} $$

Step-by-Step Solution

Answer

\( \frac{7}{9} \)

Step 1: Identify Like Terms

Notice that the fractions \( \frac{5}{9} \) and \( \frac{2}{9} \) have the same denominator of 9. This means they are like terms and can be combined using simple addition.

Step 2: Add Numerators

Since the denominators are the same, add the numerators: \( 5 + 2 = 7 \). Keep the denominator the same.

Step 3: Write the Combined Fraction

Combine the numerators over the common denominator to get \( \frac{7}{9} \). This is the simplified result of the addition.

Key Concepts

Like Terms

In the world of fractions, like terms are fractions that have the same denominator. It's like having the same type of fruit in a basket; they can be easily grouped together. When fractions are like terms, they can be added or subtracted directly because their denominators are the same.
The denominator of a fraction represents the number of equal parts into which a whole is divided. When fractions have the same denominators, it indicates that the parts are of the same size.
This is why fractions like \( \frac{5}{9} \) and \( \frac{2}{9} \) are considered like terms. Since both share the common denominator of 9, we can add them without needing any adjustments to the denominators.

Common Denominators

The concept of common denominators is crucial when dealing with the addition or subtraction of fractions. A common denominator between two or more fractions means that they can be compared or combined directly.
For example, if two fractions like \( \frac{5}{9} \) and \( \frac{2}{9} \) already have common denominators, they can be added straightforwardly. There's no need for any conversion or finding a least common multiple.
Having common denominators simplifies the process because you only need to work with the numerators. This reduces confusion and helps in arriving at the solution more quickly.

Simplification of Fractions

Simplification of fractions involves reducing a fraction to its simplest form or finding an equivalent fraction with the smallest possible numerator and denominator.
In our example, after combining the numerators, the result was \( \frac{7}{9} \). We need to check if this fraction can be simplified further. A fraction is in its simplest form when the numerator and the denominator cannot be reduced any further, i.e., they have no common factors other than 1.
For \( \frac{7}{9} \), since 7 and 9 have no common factors other than 1, this fraction is already in its simplest form. This step ensures that the final output is as concise and clear as possible.