Problem 53 - Intermediate Algebra

Question

Perform the following operations with real numbers. $$ \frac{3}{4} \div\left(-\frac{1}{2}\right) $$

Step-by-Step Solution

Answer

The result is \( -\frac{3}{2} \).

Step 1: Understand the Divisional Operation

The problem requires us to divide fractions \( \frac{3}{4} \) by \( -\frac{1}{2} \). Dividing by a fraction is equivalent to multiplying by its reciprocal.

Step 2: Find the Reciprocal

The reciprocal of a fraction \( \frac{a}{b} \) is \( \frac{b}{a} \). So, the reciprocal of \( -\frac{1}{2} \) is \( -2 \).

Step 3: Apply Multiplication

Rewrite the expression using multiplication by the reciprocal: \( \frac{3}{4} \times (-2) \).

Step 4: Multiply the Numerators

Multiply the numerators of the fractions: \( 3 \times (-2) = -6 \).

Step 5: Multiply the Denominators

Multiply the denominators of the fractions: \( 4 \times 1 = 4 \).

Step 6: Simplify the Fraction

Form the new fraction with the results: \( \frac{-6}{4} \). Simplify \( \frac{-6}{4} \) by dividing the numerator and the denominator by their greatest common divisor (GCD), which is 2. So, \( \frac{-6}{4} = \frac{-3}{2} \).

Key Concepts

Fractions Division

Fractions division might seem challenging, but it becomes easy with a simple rule: when you divide by a fraction, you are essentially multiplying by its reciprocal. Think of dividing fractions as a two-step process: first, find the reciprocal of the divisor (the fraction you are dividing by), then multiply it with the dividend (the fraction you are dividing).

For instance, let's apply this to our example:

• Start with the initial fractions: \( \frac{3}{4} \div \left(-\frac{1}{2}\right) \).
• Find the reciprocal of the divisor: \(-\frac{1}{2}\), which is \(-2\).
• Rewrite the problem as multiplication: \( \frac{3}{4} \times (-2) \).

Thus, division of fractions changes the operation to multiplication, making it straightforward to solve.

Reciprocal Operation

Understanding the reciprocal operation is crucial in math, especially in fraction division. The reciprocal of a number is an equivalent fraction that, when multiplied with the original, results in 1. This concept is used to transform division into multiplication by flipping the numerator and the denominator of the divisor.

For the fraction \(-\frac{1}{2}\):

• The reciprocal is found by swapping the numerator and the denominator.
• The reciprocal of \(-\frac{1}{2}\) is \( -2 \) because dividing \(-1\) by \( \frac{1}{2} \) equals \(-2\).

By using this operation, division becomes multiplication, simplifying the calculation process.

Simplifying Fractions

After performing operations on fractions, we often need to simplify the result. This means reducing the fraction to its lowest terms by dividing both the numerator and the denominator by their greatest common divisor (GCD). Simplifying makes it easier to understand and communicate the result.

In our example:

• The result of multiplying the numerators is \(-6\) and the denominators is \(4\), forming the fraction \( \frac{-6}{4} \).
• Identify the GCD of \(-6\) and \(4\), which is \(2\).
• Divide both the numerator and the denominator by \(2\) to obtain \( \frac{-3}{2} \).

Simplification is a key step in ensuring final answers are clear and meaningful, reflecting the smallest possible fraction that still represents the same value.