Problem 86 - Elementary and Intermediate Algebra

Question

Perform the operations. $$ -\frac{1}{2}-\left(-\frac{1}{4}\right) $$

Step-by-Step Solution

Answer

The result of the operation is \(-\frac{1}{4}\).

Step 1: Identify the Operation

This exercise involves subtraction of two fractions. The expression given is \(-\frac{1}{2}-\left(-\frac{1}{4}\right)\). We need to carefully handle the negative signs.

Step 2: Simplify the Expression

Notice that subtracting a negative number is the same as adding its positive value. So, instead of \(-\frac{1}{2} - \left(-\frac{1}{4}\right)\), we can rewrite the expression as \(-\frac{1}{2} + \frac{1}{4}\).

Step 3: Find a Common Denominator

To add the fractions \(-\frac{1}{2}\) and \(\frac{1}{4}\), we need a common denominator. The denominators are 2 and 4, and the least common denominator is 4.

Step 4: Convert to Common Denominator

Convert \(-\frac{1}{2}\) to a fraction with a denominator of 4: \(-\frac{1}{2} = -\frac{2}{4}\). Now we have \(-\frac{2}{4} + \frac{1}{4}\).

Step 5: Perform the Addition

Add the fractions \(-\frac{2}{4} + \frac{1}{4}\). The result is \(\frac{-2 + 1}{4} = \frac{-1}{4}\).

Step 6: Conclusion

The result of the operation \(-\frac{1}{2} - \left(-\frac{1}{4}\right)\) is \(-\frac{1}{4}\).

Key Concepts

Negative Numbers

Negative numbers can be a bit tricky to handle, but learning how they work makes everything easier. A negative number is simply a number that is less than zero. In mathematics, it's represented with a minus sign (–) in front of a number. For example, -2 is a negative number.
When dealing with operations, remember:

• Subtracting a negative number is like adding a positive number. For instance, if we have \(-2 - (-3)\), it becomes \(-2 + 3\).
• When two negative numbers are multiplied or divided, the result is positive. But if a negative number is multiplied or divided by a positive number, the result remains negative.

In our exercise, we have to subtract \(-\frac{1}{4}\) from \(-\frac{1}{2}\). Since subtracting a negative is equivalent to adding a positive, this becomes a handy trick to remember when dealing with negatives!

Common Denominator

Finding a common denominator is crucial when adding or subtracting fractions because it ensures the fractions are compatible for the operation. The denominator is the bottom part of a fraction, representing how many parts the whole is divided into.
To find a common denominator:

• Identify the denominators of each fraction. In our exercise, these are 2 and 4.
• Find the least common denominator (LCD), which is the smallest number that both denominators divide evenly into. For 2 and 4, the LCD is 4.

By converting fractions to have the same denominator, it simplifies the process of addition or subtraction. When the denominators align, you add or subtract the numerators directly.

Fraction Addition

Once fractions have a common denominator, the next step is straightforward. It's all about the numerators, which are the top parts of the fractions. Here's how you proceed with fraction addition:

• Ensure the fractions have the same denominator. For our problem, we converted\(-\frac{1}{2}\) to\(-\frac{2}{4}\) so it has the same denominator as\(\frac{1}{4}\).
• Add the numerators while keeping the denominator constant. With our fractions\(-\frac{2}{4} + \frac{1}{4}\), you simply add the numerators: \(-2 + 1\), which equals\(-1\).
• The result is\(-\frac{1}{4}\), as the denominator remains as 4.

Adding fractions might seem complex at first, but once you have a common denominator, the numerators take center stage, making the operation quite simple!