Problem 21 - Intermediate Algebra

Question

Perform the operations. See Example 1 . $$ \frac{1}{2}+\left(-\frac{1}{3}\right) $$

Step-by-Step Solution

Answer

\( \frac{1}{6} \)

Step 1: Find a Common Denominator

To add or subtract fractions, they need to have a common denominator. Here, the denominators are 2 and 3. The least common denominator (LCD) of 2 and 3 is 6.

Step 2: Convert Fractions to Equivalent Fractions

Convert \( \frac{1}{2} \) to an equivalent fraction with a denominator of 6:\[ \frac{1}{2} = \frac{1 \times 3}{2 \times 3} = \frac{3}{6} \]Convert \( -\frac{1}{3} \) to an equivalent fraction with a denominator of 6:\[ -\frac{1}{3} = \frac{-1 \times 2}{3 \times 2} = -\frac{2}{6} \]Now the fractions are \( \frac{3}{6} \) and \( -\frac{2}{6} \).

Step 3: Perform the Addition

Add the two fractions with a common denominator:\[ \frac{3}{6} + \left(-\frac{2}{6}\right) = \frac{3 - 2}{6} = \frac{1}{6} \] So, the sum of the fractions is \( \frac{1}{6} \).

Key Concepts

Finding Common Denominators

When adding or subtracting fractions, the first step we need to take is finding a common denominator for the fractions involved. The denominator is the bottom number of a fraction, and it shows how many equal parts the whole is divided into. A common denominator allows us to add or subtract fractions directly. To find a common denominator:

• Identify the denominators of the fractions. In our example, these are 2 and 3.
• Find the least common denominator (LCD), which is the smallest number that each denominator can divide evenly into. For 2 and 3, the least common multiple is 6.

Having a common denominator means we can easily convert fractions into forms that make them compatible for addition or subtraction. This step is crucial because it simplifies the whole process moving forward.

Equivalent Fractions

After finding a common denominator, the next step is to convert the original fractions into equivalent fractions with this shared denominator. An equivalent fraction simply represents the same value in a different form. We're changing the fractions so they have the same size pieces, which makes it easier for us to work with them.To convert a fraction to an equivalent fraction with the new denominator:

• Multiply both the numerator (top number) and the denominator of the fraction by the same number such that the denominator becomes the LCD.
• In our example, for \( \frac{1}{2} \), multiply the top and bottom by 3 to get \( \frac{3}{6} \). For \( -\frac{1}{3} \), multiply by 2 to change it to \( -\frac{2}{6} \).

By converting to equivalent fractions, we are rearranging these values into a form that shares a common denominator, making it possible for us to easily add or subtract them.

Adding and Subtracting Fractions

Now that both fractions have been converted to equivalent forms with a common denominator, adding or subtracting them becomes straightforward. Since the denominators are the same, you simply combine the numerators.Here's how it works:

• Write down the fractions with the common denominator (from the previous step).
• Add or subtract the numerators. The denominator stays the same.

In our example:

• The fractions are \( \frac{3}{6} \) and \( -\frac{2}{6} \).
• Perform the operation: \( \frac{3}{6} + \left(-\frac{2}{6}\right) = \frac{3 - 2}{6} = \frac{1}{6} \).

Thus, the result of the operation is \( \frac{1}{6} \). This method ensures accuracy and ease in calculations when working with fractions.