Problem 88 - Algebra for College Students

Question

Simplify each numerical expression. $$-\frac{1}{2}-\left(\frac{3}{8}+\frac{1}{4}\right)$$

Step-by-Step Solution

Answer

The simplified expression is \(-\frac{9}{8}\).

Step 1: Understand the Problem

We need to simplify the expression \(-\frac{1}{2}-\left(\frac{3}{8}+\frac{1}{4}\right)\). This involves combining fractions by finding a common denominator and simplifying the expression.

Step 2: Simplify Inside the Parentheses

The expression inside the parentheses is \(\frac{3}{8} + \frac{1}{4}\). To add these two fractions, we need a common denominator. The least common denominator of 8 and 4 is 8. Thus, we convert \(\frac{1}{4}\) into \(\frac{2}{8}\). Now the expression inside the parentheses becomes \(\frac{3}{8} + \frac{2}{8}\).

Step 3: Add Fractions

Now that the fractions inside the parentheses have the same denominator, we can add them. \(\frac{3}{8} + \frac{2}{8} = \frac{5}{8}\).

Step 4: Subtract from the First Term

The expression now is \(-\frac{1}{2} - \frac{5}{8}\). To subtract these, we first need a common denominator. The least common denominator of 2 and 8 is 8. Convert \(-\frac{1}{2}\) into \(-\frac{4}{8}\). Now the expression becomes \(-\frac{4}{8} - \frac{5}{8}\).

Step 5: Subtract the Fractions

Both fractions now have a common denominator, so \(-\frac{4}{8} - \frac{5}{8} = \frac{-4 - 5}{8} = \frac{-9}{8}\). Thus, the simplified expression is \(-\frac{9}{8}\).

Key Concepts

Common Denominator

When adding or subtracting fractions, it is crucial to have the same denominator. This is called the "common denominator."
It allows us to easily perform the arithmetic operations on fractions, just like with whole numbers.

• A common denominator is like a shared "base" that makes fractions compatible for addition or subtraction.
• To find a common denominator, you typically look for the least common multiple (LCM) of the denominators in question.

In our exercise, we are dealing with fractions that have denominators of 8 and 4. The LCM of 8 and 4 is 8. Therefore, 8 is our common denominator.
When we converted \(\frac{1}{4}\) into \(\frac{2}{8}\), we were actually just making it have the same denominator as \(\frac{3}{8}\). The common denominator simplifies the addition of these fractions.

Fraction Addition

Adding fractions may seem tricky at first, but it's simply about aligning the denominators and then dealing with the numerators.
Here's a straightforward approach:

• First, ensure all fractions involved have the same denominator.
• Add or subtract the numerators while keeping the common denominator unchanged.
• This process combines the numerators over a shared denominator, much like adding pieces that fit neatly into the same puzzle.

In our exercise, once we have a common denominator of 8, we can easily add \(\frac{3}{8}\) and \(\frac{2}{8}\), resulting in \(\frac{5}{8}\).
This sum is then used in further calculations.

Negative Fractions

Handling negative fractions is a regular part of fraction arithmetic and is used to indicate the direction of a value or reduce its quantity.
The minus sign can either be in front of the fraction (like \(-\frac{1}{2}\)) or with the numerator (like \(\frac{-1}{2}\)). Both formats mean the same thing.

• When subtracting negative fractions, having a common denominator helps keep calculations organized.
• Negative signs simply demonstrate direction: subtracting more means moving further in the negative direction.

In our example, we went from \(-\frac{1}{2} - \frac{5}{8}\) to \(-\frac{4}{8} - \frac{5}{8}\).
After rewriting \(-\frac{1}{2}\) as \(-\frac{4}{8}\), we could easily see that combining these negative fractions resulted in \(-\frac{9}{8}\).
Managing negative fractions efficiently is about maintaining clarity with signs and values involved.