Problem 10 - Algebra 1

Question

Use the subtraction rule to rewrite the subtraction expression as an equivalent addition expression. Then evaluate the expression. $$ \frac{1}{2}-\frac{1}{4} $$

Step-by-Step Solution

Answer

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

Step 1: Rewrite the subtraction as an equivalent addition

The subtraction rule states subtraction is the same as adding the opposite. In this case, change \(\frac{1}{2} - \frac{1}{4}\) to \(\frac{1}{2} + -\frac{1}{4}\), evaluating to the same in order to apply this rule.

Step 2: Simplify the expression

Next, add the fractions according to the normal rules of arithmetic fractions: \(\frac{1}{2} + -\frac{1}{4} = \frac{1}{2} - \frac{1}{4}\) is equivalent to \(\frac{2}{4} - \frac{1}{4}\). This expression simplifies further to \(\frac{1}{4}\).

Key Concepts

Subtraction Rule

Understanding the subtraction rule can help simplify complex expressions. In mathematics, subtraction can be seen as an addition of an additive inverse.
This means that when you subtract a number, it is the same as adding its opposite. This rule is handy when working with complex operations or even basic arithmetic.
Taking the exercise into account, where we have \(\frac{1}{2} - \frac{1}{4}\), the subtraction rule allows us to transform this into an equivalent addition expression: \(\frac{1}{2} + -\frac{1}{4}\).
Here are some key points to remember about the subtraction rule:

• Subtraction is equivalent to adding the opposite.
• Converts subtraction problems into potentially simpler addition problems.
• Simplifies operations, especially useful within equations.

Utilizing this rule helps streamline problem-solving and makes it easier to comprehend and manage large or complex sets of operations.

Equivalent Addition

Equivalent addition is a concept closely tied to the subtraction rule. This principle allows you to restate any subtraction expression into addition, which frequently simplifies calculations and enhances understanding.
The key to equivalent addition lies in converting the subtracted term into its negative counterpart, making it possible to carry out normal addition operations.
In our exercise: \(\frac{1}{2} - \frac{1}{4}\) becomes \(\frac{1}{2} + -\frac{1}{4}\). This transformation didn't change the value or the relationship between numbers.
Benefits of using equivalent addition include:

• Maintaining the same value while simplifying expression manipulation.
• Easier to work with commutative properties of addition.
• Enhancing visualization of negative and positive influences within equations.

Employing equivalent addition creates cleaner and often more intuitive equations, facilitating easier problem solving.

Simplifying Fractions

Simplifying fractions is a critical skill, aimed at reducing fractions to their simplest form, which aids in clearer interpretation and calculation.
When simplifying, the goal is to express a fraction with the smallest possible numerator and denominator, maintaining the original value.
In the exercise, we transformed \(\frac{1}{2} + -\frac{1}{4}\) to \(\frac{2}{4} + -\frac{1}{4}\). This simplifies to \(\frac{1}{4}\), representing the same value more simply.
Steps involved in simplifying:

• Find the greatest common divisor (GCD) for numerator and denominator.
• Divide both the numerator and the denominator by the GCD.
• Re-write the fraction using these new values.

Mastering this technique can make arithmetic operations less cumbersome and allow for quicker, more confident solutions during examinations and real-life applications.