Problem 51 - Prealgebra

Question

Find the LCD for each of the following; then use the methods developed in this section to add or subtract as indicated. $$\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{6}$$

Step-by-Step Solution

Answer

The result is \(\frac{5}{4}\).

Step 1: Identify the Denominators

The fractions involved are \( \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \text{and} \frac{1}{6} \). The denominators are 2, 3, 4, and 6.

Step 2: Find the Least Common Denominator (LCD)

The least common denominator is the smallest number that each of the denominators can divide into without a remainder. For 2, 3, 4, and 6, we need the least common multiple (LCM).\[ \text{LCM}(2, 3, 4, 6) = 12 \]

Step 3: Convert Each Fraction to the LCD

Change each fraction so that each has the denominator of 12. Calculate the new numerators:\[\frac{1}{2} = \frac{6}{12}, \quad \frac{1}{3} = \frac{4}{12}, \quad \frac{1}{4} = \frac{3}{12}, \quad \frac{1}{6} = \frac{2}{12}\]

Step 4: Add the Fractions

Now that each fraction has the same denominator, you can add the numerators together while keeping the denominator the same.\[\frac{6}{12} + \frac{4}{12} + \frac{3}{12} + \frac{2}{12} = \frac{15}{12}\]

Step 5: Simplify the Resulting Fraction

The fraction \( \frac{15}{12} \) can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 3.\[ \frac{15 \div 3}{12 \div 3} = \frac{5}{4} \]

Key Concepts

Least Common Multiple (LCM)

When working with fractions, a fundamental concept is finding the least common multiple (LCM) of the denominators involved. The LCM is the smallest multiple that is exactly divisible by each of the denominators.
This is important when adding or subtracting fractions, as it allows us to rewrite the fractions with a common denominator.

• To find the LCM of a set of numbers, list the multiples of each number.
• Identify the smallest multiple that is shared among all the numbers.

For instance, with the numbers 2, 3, 4, and 6, we start by listing out the multiples. The smallest number that appears in all sets is 12, making it the LCM, and thus the least common denominator (LCD) required for the fractions \( \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \text{and} \frac{1}{6} \).
This step is crucial as it sets the stage for adding these fractions accurately.

Adding Fractions

Adding fractions typically involves a few key steps, especially when the denominators are different. After determining the least common denominator (LCD) using the LCM, you convert each fraction so they have the same denominator.
Here's how you do it:

• Adjust each fraction to have the LCD as its denominator.
• To do this, multiply the numerator and denominator of each fraction by the same number.

This process ensures that all denominators are alike, simplifying the addition process.
For example:

• Convert \( \frac{1}{2} \) to \( \frac{6}{12} \), \( \frac{1}{3} \) to \( \frac{4}{12} \), and so on.
• Once all fractions have the same denominator, simply add their numerators. The denominators remain unchanged.

As we did in the exercise, \( \frac{6}{12} + \frac{4}{12} + \frac{3}{12} + \frac{2}{12} = \frac{15}{12} \).
This combined fraction is now ready for further steps, if needed.

Simplifying Fractions

After adding fractions, the resulting sum must sometimes be simplified. Simplifying a fraction involves reducing it to its smallest possible form. This is done by dividing the numerator and the denominator by their greatest common divisor (GCD).

• Identify the GCD of the numerator and the denominator.
• Divide both the numerator and the denominator by the GCD.

For \( \frac{15}{12} \), the greatest common divisor is 3. By dividing both 15 and 12 by 3, we simplify the fraction to \( \frac{5}{4} \).
This fraction is already in its simplest form since there are no further common factors between the numerator and the denominator.
Simplifying fractions plays an essential role in not only making them easier to understand but also in ensuring they are correctly presented in their simplest terms.