Problem 61 - Prealgebra

Question

Add or subtract the following fractions, as indicated. $$\frac{2}{3}+\frac{8}{9}$$

Step-by-Step Solution

Answer

The result is \(\frac{14}{9}\) or \(1\frac{5}{9}\).

Step 1: Identify the Least Common Denominator (LCD)

To add fractions, they must have the same denominator. The denominators here are 3 and 9. The smallest number that both 3 and 9 divide into evenly is 9. Hence, the least common denominator (LCD) is 9.

Step 2: Convert Fractions to Equivalent Fractions with the LCD

We need to convert \(\frac{2}{3}\) to a fraction with a denominator of 9. To do this, multiply both the numerator and the denominator of \(\frac{2}{3}\) by 3: \(\frac{2\times3}{3\times3} = \frac{6}{9}\). The second fraction \(\frac{8}{9}\) already has a denominator of 9, so it remains unchanged.

Step 3: Add the Fractions

Now that the fractions have the same denominator, add the numerators: \(\frac{6}{9} + \frac{8}{9} = \frac{6+8}{9} = \frac{14}{9}\).

Step 4: Simplify the Resulting Fraction, if necessary

The fraction \(\frac{14}{9}\) is in its lowest terms because 14 and 9 have no common factors other than 1. It can also be expressed as a mixed number: \(1\frac{5}{9}\), since 14 divided by 9 is 1 with a remainder of 5.

Key Concepts

Least Common Denominator (LCD)

When adding fractions, one essential step is to ensure they have the same denominator. This makes the process straightforward as it allows us to add directly across the numerators. But how do we find this common denominator? By determining the Least Common Denominator (LCD). The least common denominator is the smallest number that the denominators can both divide into without leaving a remainder.

In the example of adding \(\frac{2}{3}\) and \(\frac{8}{9}\), we look at the denominators, which are 3 and 9. We need to find the smallest number both can divide into evenly. Since 9 can be divided by both itself and 3 evenly, we conclude that 9 is the LCD. This is an important foundational skill in fraction arithmetic.

Equivalent Fractions

The next part of adding fractions is to convert them to equivalent fractions with the same denominator. This step is necessary to ensure we can add their numerators effectively. Equivalent fractions are different fractions that name the same amount. You create them by multiplying both the numerator and the denominator by the same number.

For instance, to convert \(\frac{2}{3}\) to have a denominator of 9 (as we found in our LCD step), we multiply both the numerator and the denominator by 3 (since \(9 \div 3 = 3\)). This changes \(\frac{2}{3}\) into \(\frac{6}{9}\), making it compatible for addition with \(\frac{8}{9}\) since both now share the same denominator. Understanding how to form equivalent fractions is key to solving problems involving addition or subtraction of fractions.

Mixed Numbers

The last step involves simplifying your answer. Sometimes, this might mean converting an improper fraction into a mixed number. Improper fractions occur when the numerator is larger than the denominator, as seen with \(\frac{14}{9}\). To convert it into a mixed number, divide the numerator by the denominator.

In this example, 14 divided by 9 equals 1 with a remainder of 5. This converts \(\frac{14}{9}\) into the mixed number \(1\frac{5}{9}\). A mixed number combines a whole number with a proper fraction and often makes the result more intuitive to understand. Recognizing when to convert and how to properly express your answers as mixed numbers enhances your ability to communicate mathematical solutions clearly.