Problem 25 - Prealgebra

Question

Find each quotient. Use an area model if necessary. $$-\frac{1}{2} \div \frac{5}{6}$$

Step-by-Step Solution

Answer

The quotient is \(-\frac{3}{5}\).

Step 1: Invert the Divisor

To divide fractions, start by taking the reciprocal of the divisor (the second fraction). The divisor is \( \frac{5}{6} \), and its reciprocal is \( \frac{6}{5} \).

Step 2: Multiply

Change the division of fractions to multiplication by using the reciprocal. The problem \( -\frac{1}{2} \div \frac{5}{6} \) becomes \( -\frac{1}{2} \times \frac{6}{5} \).

Step 3: Multiply the Numerators

Multiply the numerators of the fractions. So, \(-1 \times 6 = -6\).

Step 4: Multiply the Denominators

Multiply the denominators of the fractions. Thus, \(2 \times 5 = 10\).

Step 5: Simplify the Result

Combine the products from Steps 3 and 4 to get \(-\frac{6}{10}\). Simplify \(-\frac{6}{10}\) by dividing the numerator and the denominator by their greatest common divisor, 2, resulting in \(-\frac{3}{5}\).

Key Concepts

Reciprocal

Understanding the concept of a reciprocal is integral when dividing fractions. The reciprocal of a fraction is formed by flipping or inverting the positions of the numerator and denominator. For example, the reciprocal of \( \frac{5}{6} \) is \( \frac{6}{5} \). This means swapping the top and bottom parts of the fraction. The reciprocal is powerful because it transforms division into multiplication, making calculations easier. When you take the reciprocal of a fraction, you're essentially finding a value that, when multiplied by the original fraction, equals 1. This is because anything divided by itself equals 1, and multiplying a fraction by its reciprocal will have this effect.

Simplifying division to multiplication allows you to manage fractions using straightforward arithmetic. Remember, when dividing by a fraction, always use its reciprocal. It clears up any complexity and helps you tackle the operation with clarity.

Multiplication of Fractions

Once you've taken the reciprocal of a fraction for division, the problem shifts into a multiplication problem. For example, \( -\frac{1}{2} \div \frac{5}{6} \) becomes \( -\frac{1}{2} \times \frac{6}{5} \). So how does multiplication of fractions actually work?

To multiply fractions, follow a simple method:

• Multiply the numerators (the top numbers) together.
• Multiply the denominators (the bottom numbers) together.

For our example:

• Numerators: \(-1 \times 6 = -6\)
• Denominators: \(2 \times 5 = 10\)

The result is \(-\frac{6}{10}\). This method is straightforward and ensures clarity when multiplying fractions. There's no need to find a common denominator as you do with addition or subtraction of fractions. Keeping this in mind, you can confidently tackle any fraction multiplication!

Simplifying Fractions

Simplifying fractions is the process of making a fraction as simple as possible, by ensuring the numerator and the denominator have no common factors (other than 1). After multiplying the fractions as in our earlier example — resulting in \(-\frac{6}{10}\) — the final step is simplification.

Here's how you simplify:

• Find the greatest common divisor (GCD) of the numerator and denominator.
• Divide both the numerator and the denominator by their GCD.

For \(-\frac{6}{10}\), the GCD is 2.

• Divide both the numerator and the denominator by 2: \(-\frac{6}{2} \) and \( \frac{10}{2} \) which gives \(-3\) and \(5\) respectively.

Thus, the fraction simplifies to \(-\frac{3}{5}\). This means \(-\frac{3}{5}\) is the simplest form of the fraction created in the previous steps. Simplifying helps clarify results, making fractions easier to understand and compare. Always check the factors of numbers involved to simplify fractions effectively.