Problem 45 - Prealgebra

Question

The problems below review some basic concepts of addition of fractions and mixed numbers. Add each of the following and reduce all answers to lowest terms. $$\frac{1}{2}+\frac{1}{4}$$

Step-by-Step Solution

Answer

\(\frac{1}{2} + \frac{1}{4} = \frac{3}{4}\)

Step 1: Find a Common Denominator

To add the fractions \(\frac{1}{2}\) and \(\frac{1}{4}\), we need to ensure they have the same denominator. The denominators are 2 and 4. The least common denominator of 2 and 4 is 4.

Step 2: Convert Fractions to Equivalent Fractions

Convert \(\frac{1}{2}\) to an equivalent fraction with a denominator of 4. Multiply the numerator and denominator by 2: \(\frac{1}{2} = \frac{1 \times 2}{2 \times 2} = \frac{2}{4}\). Now we have \(\frac{2}{4}\) and \(\frac{1}{4}\).

Step 3: Add the Fractions

Now that both fractions have the same denominator, simply add the numerators. \(\frac{2}{4} + \frac{1}{4} = \frac{2+1}{4} = \frac{3}{4}\).

Step 4: Simplify the Fraction if Possible

The fraction \(\frac{3}{4}\) is already in its simplest form since there are no common factors between 3 and 4 other than 1.

Key Concepts

Understanding Common Denominator

When adding fractions, it's essential to ensure they share a common denominator. A common denominator is a shared multiple of the denominators of the fractions involved. This commonality allows us to compare or add fractions meaningfully. Think of it like speaking different dialects that need a common language to communicate. For our example of \(\frac{1}{2}+\frac{1}{4}\), the denominators are 2 and 4. To find the common denominator, identify the least common multiple (LCM) of these numbers. Here, the LCM of 2 and 4 is 4. By converting both fractions to have this shared denominator, we simplify the process of adding them. A common denominator is a foundational concept that helps in various operations involving fractions, including subtraction.

Equivalent Fractions: Making Them Match

Once you've identified the common denominator, the next step is to convert each fraction to an equivalent fraction that shares this common denominator. Equivalent fractions are different fractions that express the same value or proportion. For \(\frac{1}{2}\) and \(\frac{1}{4}\), we need to adjust \(\frac{1}{2}\) since it does not yet have the denominator of 4. To do this, multiply both the numerator and the denominator of \(\frac{1}{2}\) by 2. This yields \(\frac{2}{4}\). Now, both fractions are \(\frac{2}{4}\) and \(\frac{1}{4}\).

• Remember: Multiply both parts of the fraction by the same number to maintain equal value.
• The operation should create an identical denominator, permitting simple addition or subtraction.

Equivalent fractions are helpful in ensuring that operations involving fractions can proceed correctly, preserving accuracy in calculations.

The Simplification of Fractions

Once the fractions are added, you might need to simplify the resulting fraction. Simplifying a fraction means reducing it to its most basic form, where the numerator and denominator share no common factors other than 1. In our example, after adding \(\frac{2}{4}\) and \(\frac{1}{4}\), we get \(\frac{3}{4}\). Since there are no common factors between 3 and 4 (besides 1), \(\frac{3}{4}\) is already a simplified fraction.

• Check for common factors: Divide both numbers by the greatest common factor if possible.
• This step ensures that your answer is both concise and precise.

Simplification makes the fraction easier to understand and work with, especially when dealing with complex mathematical operations. It is a critical step to ensure clarity and accuracy in mathematics.