Problem 67 - Beginning and Intermediate Algebra

Question

Find each difference. $$ \frac{1}{2}-\left(-\frac{1}{4}\right) $$

Step-by-Step Solution

Answer

The answer is \(\frac{3}{4}\).

Step 1: Identify the Operation

The given problem is to find the difference between two fractions. The expression is \(\frac{1}{2} - \big(-\frac{1}{4} \big)\). Note that subtracting a negative number is equivalent to adding its positive.

Step 2: Rewrite the Expression

Rewrite the expression by eliminating the negative sign. \(\frac{1}{2} - \big( -\frac{1}{4} \big) = \frac{1}{2} + \frac{1}{4}\).

Step 3: Find a Common Denominator

To add the fractions, find a common denominator. The denominators are 2 and 4. The least common denominator is 4.

Step 4: Convert Fractions to Common Denominator

Convert \(\frac{1}{2}\) to a fraction with denominator 4. \(\frac{1}{2} = \frac{2}{4}\). Now the expression is \(\frac{2}{4} + \frac{1}{4}\).

Step 5: Add the Fractions

Add the two fractions with the same denominator: \(\frac{2}{4} + \frac{1}{4} = \frac{3}{4}\).

Key Concepts

Common Denominator

To add or subtract fractions, they must have the same denominator. The denominator is the bottom number of the fraction and tells us how many equal parts the whole is divided into.
Finding a common denominator means finding a number that both denominators can divide into without leaving a remainder. This number is often the least common multiple (LCM) of the original denominators.
In the exercise, we have denominators 2 and 4. The LCM of 2 and 4 is 4. We then convert \(\frac{1}{2}\) to \(\frac{2}{4}\) to give both fractions the same denominator. This step is crucial for proper addition or subtraction.

Adding Fractions

Once the fractions have a common denominator, adding them becomes straightforward. You simply add the numerators (the top numbers) and keep the denominator the same.
In our exercise, after converting \(\frac{1}{2}\) to \(\frac{2}{4}\), we add it to \(\frac{1}{4}\).
This looks like: \(\frac{2}{4} + \frac{1}{4} = \frac{3}{4}\)
The crucial part here is not to add the denominators, only the numerators. The denominator indicates the size of the parts we are adding, which stays the same.

Negative Numbers

Subtracting a negative number can be tricky but simpler than it sounds. The rule is straightforward: subtracting a negative number is the same as adding the positive of that number.
In our example, \(\frac{1}{2} - \big(-\frac{1}{4}\big)\) converts to \(\frac{1}{2} + \frac{1}{4}\).
This subtraction turns into an addition because a negative sign followed by another negative sign becomes positive.
This concept helps simplify many problems that involve both negative and positive numbers.