Problem 7 - Beginning and Intermediate Algebra

Question

Subtract. See Examples 1 through 5 $$ \frac{1}{2}-\frac{1}{3} $$

Step-by-Step Solution

Answer

The result of \( \frac{1}{2} - \frac{1}{3} \) is \( \frac{1}{6} \).

Step 1: Find a Common Denominator

To subtract two fractions with different denominators, find a common denominator. The least common denominator of 2 and 3 is 6.

Step 2: Convert the Fractions

Convert each fraction to have the common denominator of 6. \[ \frac{1}{2} = \frac{3}{6} \quad \text{and} \quad \frac{1}{3} = \frac{2}{6} \]

Step 3: Subtract the Fractions

Now that the fractions are \( \frac{3}{6} \) and \( \frac{2}{6} \), subtract them: \[ \frac{3}{6} - \frac{2}{6} = \frac{1}{6} \]

Step 4: Simplify the Result

The fraction \( \frac{1}{6} \) is already in its simplest form, so no further simplification is needed.

Key Concepts

Least Common Denominator

In order to perform operations like subtraction on fractions, it's essential that they have the same denominator. This shared denominator is called the Least Common Denominator (LCD), which is the smallest number that is a multiple of both denominators. For example, when subtracting \( \frac{1}{2} \) and \( \frac{1}{3} \), their denominators are 2 and 3. The multiples of 2 are 2, 4, 6, 8, etc., and the multiples of 3 are 3, 6, 9, 12, etc. The least common number you find in both lists is 6, which is the LCD.

• Finding the LCD helps make the fractions compatible for addition or subtraction.
• It's similar to finding the least common multiple (LCM) in whole numbers.
• Having a single shared denominator means you can focus entirely on the numerators for the subsequent steps.

Simplifying Fractions

Simplifying fractions involves reducing the fraction to its simplest form, where the numerator and the denominator have no common factors other than 1. Although the fraction \( \frac{1}{6} \) in the example given is already simplified, knowing how to simplify fractions is crucial:

• First, find the greatest common divisor (GCD) of both the numerator and the denominator.
• Then, divide both the numerator and the denominator by this number.
• This process ensures your answer is in the simplest, cleanest form possible.

For instance, if you had a fraction like \( \frac{4}{8} \), the GCD is 4. Dividing both 4 and 8 by 4 gives you \( \frac{1}{2} \). This simplified form is easier to understand and work with in further calculations.

Fraction Conversion

Fraction conversion is a key skill when working with fractions, especially when it comes to adding or subtracting them. It involves changing fractions to have the same denominator, which facilitates easy calculation. In the example you saw,

• The first step was finding the LCD of 2 and 3, which was 6.
• Once the LCD is known, convert each fraction: \( \frac{1}{2} \) becomes \( \frac{3}{6} \) because multiplying both the numerator and denominator by 3 keeps the fraction's value the same.
• Similarly, \( \frac{1}{3} \) becomes \( \frac{2}{6} \) by multiplying both top and bottom by 2.

This conversion is essential for properly aligning fractions. It means you can easily perform operations like subtraction and addition since they share the same denominator. Always ensure the added or subtracted fractions are simplified in the end to maintain clarity and simplicity.