Problem 36 - Algebra A Combined Function

Question

Add. See Examples 1 through 12,18, and 19. $$ -\frac{5}{9}+\frac{1}{3} $$

Step-by-Step Solution

Answer

The sum is \(-\frac{2}{9}\).

Step 1: Identify the Problem

You need to add two fractions: \(-\frac{5}{9}\) and \(\frac{1}{3}\).

Step 2: Find a Common Denominator

To add fractions, they must have the same denominator. The denominators here are 9 and 3. The least common denominator (LCD) of 9 and 3 is 9.

Step 3: Adjust Fractions to Common Denominator

The first fraction \(-\frac{5}{9}\) already has the correct denominator. The second fraction \(\frac{1}{3}\) needs to be converted to have a denominator of 9. Multiply both numerator and denominator of \(\frac{1}{3}\) by 3:\[\frac{1 \times 3}{3 \times 3} = \frac{3}{9}\]

Step 4: Add the Fractions

Now add the fractions with the common denominator:\[-\frac{5}{9} + \frac{3}{9} = \frac{-5 + 3}{9} = \frac{-2}{9}\]

Step 5: Simplify the Fraction (if necessary)

The fraction \(-\frac{2}{9}\) is already in its simplest form, as 2 and 9 have no common factors other than 1.

Key Concepts

Understanding the Least Common Denominator

When dealing with fractions, adding them directly can be complex unless they share the same denominator. The denominator is the bottom number of a fraction, and it signifies the total parts the whole is divided into. To add fractions effectively, finding a common denominator is crucial because it creates a uniform base for comparison.

For the fractions \(-\frac{5}{9}\) and \(\frac{1}{3}\), their denominators are 9 and 3, respectively. The least common denominator (LCD) is the smallest number that both denominators can divide into evenly.

To find the LCD:

• List the multiples of each denominator.
• Identify the smallest multiple common to both lists.

In our problem, the multiples of 3 are 3, 6, 9, 12, etc., and the multiples of 9 are 9, 18, 27, etc. The smallest common multiple is 9, making it the least common denominator.

This means both fractions can be adjusted to have 9 as a common base, which simplifies the process of adding them.

Simplifying Fractions

Simplifying fractions means reducing them to their simplest form where the numerator (top number) and denominator (bottom number) are as small as possible.

• Look for the greatest common factor (GCF) between the numerator and denominator.
• Divide both the numerator and denominator by the GCF.

In our exercise, after adding the fractions, we obtain \(-\frac{2}{9}\). To check if this fraction can be simplified:

• Find the GCF of 2 and 9, which is 1 because they have no common factors other than 1.

Since no further reduction is possible, \(-\frac{2}{9}\) is already in its simplest form.

Understanding simplification is vital because it makes fractions easier to understand and compare, which is especially useful in solving other math problems.

Handling Negative Fractions

Fractions can sometimes be negative, which simply means the value is less than zero. The negative sign can be on the numerator, the denominator, or outside the fraction. For consistency, it's typically written in front of the fraction.

• When adding or subtracting fractions, keep an eye on the negative signs.
• It influences whether you add or subtract the values in the numerators.

In the example \(-\frac{5}{9} + \frac{3}{9}\), the negative sign is on the first fraction. When adding these:

• Consider \(-5\) and \(3\) as the values you're combining.
• The result is \(-5 + 3 = -2\).

This negative result connects back to the original negative fraction. Recognizing and correctly handling negative fractions ensures accurate calculations and results in mathematical operations.