Problem 67 - Introductory and Intermediate Algebra for College Students 4th

Question

Use the order of operations to simplify each expression. $$\frac{\frac{1}{4}-\frac{1}{2}}{\frac{1}{3}}$$

Step-by-Step Solution

Answer

The simplified form of the expression is \(-\frac{3}{4}\)

Step 1: Simplify Brackets or Numerator

Subtract \(\frac{1}{2}\) from \(\frac{1}{4}\) to give: \(-\frac{1}{4}\)

Step 2: Simplify the Whole Fraction

The entire fraction now simplifies to: \(\frac{-1/4}{1/3}\). This looks like a complex fraction, but it's just division. We can simplify by multiplying the numerator (-1/4)) by the reciprocal of the denominator (3/1). This will simplify to: \(-\frac{1}{4} * \frac{3}{1} = -\frac{3}{4}\)

Step 3: Simplify the Result

After the multiplication, the result simplifies to \(-\frac{3}{4}\)

Key Concepts

Simplification

Simplification in mathematics involves reducing an expression to its simplest form. This means making it as concise and readable as possible.
One common way to simplify an expression is to perform operations in a specific order, often denoted by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). In our original exercise, we started by simplifying the expression inside the numerator.

• First, the subtraction is performed: \(\frac{1}{4} - \frac{1}{2}\), which simplifies to \(-\frac{1}{4}\).
• Next, division is addressed by rewriting it as multiplication using the reciprocal method.

Through these steps, the goal is always to make the algebraic or numerical expression easier to manage and understand.

Complex Fractions

Complex fractions have fractions in the numerator, denominator, or both. These can seem complicated, but with the right techniques, they can be simplified easily.
Imagine a fraction where each part is itself a fraction. It's important to work through these step by step to avoid confusion.

• First, look at the fraction in the numerator, such as \(\frac{1}{4} - \frac{1}{2}\). Simplify this part by performing any operations, like subtraction.
• Once simplified, consider the overall fraction: \(\frac{-1/4}{1/3}\). It's often helpful to think of this as a basic division problem.
• With practice, breaking down each component becomes intuitive, allowing for straightforward simplification of complex fractions.

Reciprocals

Reciprocals are key to simplifying fractions, especially within division.
A reciprocal is simply what you get when you flip the numerator and the denominator of a fraction. For example, the reciprocal of \(\frac{1}{3}\) is \(\frac{3}{1}\).

• In division of fractions, we multiply by the reciprocal of the divisor. Hence, \(\frac{-1/4}{1/3}\) becomes \(-\frac{1}{4} \times \frac{3}{1}\).
• This method changes the division into multiplication, which is often easier to handle.

By understanding reciprocals, the division of fractions transitions into a more manageable multiplication problem, yielding a simplified result.