Problem 37 - Introductory and Intermediate Algebra for College Students 4th

Question

Perform the indicated subtraction. $$\frac{1}{2}-\frac{1}{4}$$

Step-by-Step Solution

Answer

\(\frac{1}{4}\)

Step 1: Identify the Common Denominator

Find a common denominator. In this case, the denominators are 2 and 4. For these numbers, the common denominator is 4, because 2 * 2 = 4.

Step 2: Rewrite the Fractions Using the Common Denominator

Rewrite the fraction so that they both have the common denominator. The first fraction becomes \(\frac{2}{4}\), where as the second fraction stays as \(\frac{1}{4}\).

Step 3: Subtracting the Fractions

Now, subtract the fractions: \(\frac{2}{4}-\frac{1}{4}=\frac{1}{4}\)

Key Concepts

Understanding Fractions

Fractions represent parts of a whole. When you see a fraction like \( \frac{1}{2} \), the number above the line is called the numerator, and the number below the line is the denominator.
The numerator tells you how many parts you have, and the denominator tells you how many equal parts the whole is divided into. For example, if you eat one slice of a pizza that's cut into eight slices, you've eaten \( \frac{1}{8} \) of the pizza. Fractions can represent numbers less than one, equal to one, or greater than one. Fractions like \( \frac{2}{2} \) or \( \frac{8}{8} \) are equal to 1 because the numerator and denominator are the same. When the numerator is larger than the denominator, the fraction represents a number greater than 1.
Fractions are a crucial concept in mathematics because they help us describe numbers that are not whole, allowing for more precise calculations in various everyday contexts.

Finding a Common Denominator

Before you can subtract fractions, both of them must have the same denominator. The denominator at the bottom of the fraction describes how many equal parts the total is divided into. This is why matching these numbers is essential.
But how do you find a common denominator? You look for the least common multiple (LCM) of the two denominators. In our exercise, the fractions \( \frac{1}{2} \) and \( \frac{1}{4} \) were presented. The denominators are 2 and 4, and the LCM of these numbers is 4 because 4 is the lowest number that both 2 and 4 can divide into without a remainder.
The next step is to convert each fraction to have this common denominator. Since \( \frac{1}{2} \) needs to be changed to have a denominator of 4, you multiply both the top and bottom by 2, converting it to \( \frac{2}{4} \). The fraction \( \frac{1}{4} \) already has the denominator we need. Now, both fractions are ready for subtraction.

Subtracting Fractions with the Same Denominator

Once both fractions share a common denominator, subtracting them becomes straightforward. If you have \( \frac{2}{4} \) and \( \frac{1}{4} \), you simply subtract the numerators, leaving the denominator unchanged. Here's the subtraction process step-by-step:

• First, subtract the numerator of the second fraction from the numerator of the first fraction: \( 2 - 1 = 1 \).
• The common denominator stays the same, which is 4.
• Thus, \( \frac{2}{4} - \frac{1}{4} = \frac{1}{4} \).

This method is effective because it's like dealing only with the portion or number of parts, while the denominator continues to show how many equal parts each whole is divided into. Always remember, subtract only the numerators when the denominators are common, and keep the denominator the same. This concept not only applies to subtraction but also makes sense when adding fractions. By keeping these steps in mind, you'll find it easier to work with fractions in your math challenges.