Problem 78 - Prealgebra

Question

The following problems review addition and subtraction with fractions and mixed numbers. $$\frac{1}{2}+\frac{1}{4}$$

Step-by-Step Solution

Answer

The sum of \( \frac{1}{2} + \frac{1}{4} \) is \( \frac{3}{4} \).

Step 1: Identify the Least Common Denominator (LCD)

To add fractions, they must have the same denominator. First, identify the least common denominator (LCD) of the fractions \( \frac{1}{2} \) and \( \frac{1}{4} \). The denominators are 2 and 4. The smallest number that both 2 and 4 divide into evenly is 4. Thus, the LCD is 4.

Step 2: Convert to Equivalent Fractions

Convert \( \frac{1}{2} \) to an equivalent fraction with the denominator of 4. Multiply both the numerator and the denominator of \( \frac{1}{2} \) by 2 to get \( \frac{2}{4} \). Now, both fractions \( \frac{2}{4} \) and \( \frac{1}{4} \) have the same denominator.

Step 3: Add the Fractions

With the same denominators, you can add the numerators directly. \( 2 + 1 = 3 \). So add \( \frac{2}{4} + \frac{1}{4} = \frac{3}{4} \).

Step 4: Simplify the Answer

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

Key Concepts

Understanding the Least Common Denominator

When adding or subtracting fractions, it's essential to work with the same denominator, also known as the Least Common Denominator (LCD). It facilitates directly adding or subtracting the numerators without any confusion. The LCD is the smallest number that all the denominators can evenly divide into. For instance, in the problem \(\frac{1}{2} + \frac{1}{4}\), the denominators are 2 and 4.

• 2 can divide evenly into 4.
• The smallest number both divide into is 4, making it the LCD.

By converting each fraction to this common base, you ensure a streamlined process for addition or subtraction. This step is crucial for making accurate calculations with fractions.

Navigating Mixed Numbers

Mixed numbers consist of both a whole number and a fraction. While the original problem does not directly involve mixed numbers, understanding them is critical for more complex problems. When dealing with mixed numbers, the whole number part remains separate, but the fractional parts must have a common denominator for operations such as addition or subtraction.Suppose you were adding \(1\frac{1}{2} + \frac{1}{4}\). You handle it by:

• Keeping the whole number intact.
• Ensuring fractions have the least common denominator.
• Adding the fractions separately.

This approach simplifies mixed numbers and integrates smoothly with fraction operations. Mixed numbers can easily be converted into improper fractions if necessary for more straightforward calculations.

Simplifying Fractions Effectively

Once you perform addition or subtraction, simplifying the resulting fraction is a key step. A fraction is simplified when the numerator (top number) and the denominator (bottom number) have no common factors other than 1. Let's consider the fraction obtained from \(\frac{1}{2} + \frac{1}{4}\), which is \(\frac{3}{4}\). This fraction is already simplified because:

• The numerator is 3.
• The denominator is 4.
• The only common factor is 1, meaning \(\frac{3}{4}\) is in its simplest form.

Simplifying is crucial as it presents the answer in the most reduced form, making it easier to understand and verify. Always check whether further division is possible for both the numerator and denominator.