Problem 33 - Prealgebra

Question

Use any of the rules developed in this chapter and the rule for order of operations to simplify each of the following expressions as much as possible. [Examples 6–9] $$\frac{4-8}{8-4}$$

Step-by-Step Solution

Answer

The expression simplifies to -1.

Step 1: Perform Subtraction in the Numerator

Subtract the numbers in the numerator: \(4 - 8\). This equals \(-4\).

Step 2: Perform Subtraction in the Denominator

Subtract the numbers in the denominator: \(8 - 4\). This equals \(4\).

Step 3: Division of the Results

Divide the result of the numerator by the result of the denominator: \(-4 \div 4\). This equals \(-1\).

Key Concepts

Simplification

Simplification is all about making complex expressions easier to work with by breaking them down into simpler forms. It is akin to tidying up a complicated math problem to see it clearly.
When you simplify a fraction or expression, the goal is to find an equivalent that is easier to handle. This often involves reducing, factoring, or performing arithmetic operations.
For example, consider the expression \(\frac{4-8}{8-4}\). Simplifying involves:

• Performing the subtraction in both the numerator and denominator.
• Then dividing the simplified numerator by the denominator.

The expression reduces to \[\frac{-4}{4} = -1\]

• We arrived at a simpler result by executing the basic arithmetic operations.
• This makes the conceptually complex expression much easier to interpret and utilize.

Numerator and Denominator

The terms *numerator* and *denominator* are integral to understanding fractions. A fraction consists of two main parts: the numerator and the denominator.
The **numerator** is the top part of the fraction and represents the number of parts considered. In our provided exercise, initially, this was expressed as \(4-8\).
The **denominator** is the bottom part of the fraction and indicates how many parts make up a whole. For the exercise, it was \(8-4\) initially.
The process:

• First, simplify the numerator, \(4-8\), to get \(-4\).
• Then, simplify the denominator, \(8-4\), to reach \(4\).
• Ultimately, the fraction \(\frac{-4}{4}\) tells us one of \(-4\) divided by \(4\).

Understanding this relationship aids in performing more complex operations like division.

Division

Division is a fundamental operation in mathematics where you determine how many times one number is contained in another. In our exercise, the final step required dividing the simplified numerator by the denominator.
Here's how to accurately interpret this:

• You take the result from your numerator, \(-4\), and divide it by \(4\).
• The division \(-4 \div 4\) equals \(-1\).

**Key Points to Remember:**

• When dividing two numbers with the same absolute value but opposite signs, the result is negative.
• This concept can greatly simplify expressions, bringing convoluted fractions into a single, understandable number.
• It's crucial to execute the division only after simplifying the numerator and denominator separately.

By learning to break down and execute such operations, division becomes a straightforward, manageable task, essential in math.