Problem 3 - Prealgebra

Question

Find the quotient in each case by replacing the divisor by its reciprocal and multiplying. $$-\frac{2}{3} \div \frac{1}{2}$$

Step-by-Step Solution

Answer

The quotient is \\(-\frac{4}{3}\\)."

Step 1: Understand the Problem

We are asked to divide \(-\frac{2}{3}\) by \(\frac{1}{2}\). In division of fractions, instead of dividing, we multiply by the reciprocal of the divisor.

Step 2: Identify the Reciprocal

The reciprocal of a fraction is obtained by switching its numerator and denominator. The divisor here is \(\frac{1}{2}\), so its reciprocal is \(\frac{2}{1}\) or simply 2.

Step 3: Convert Division to Multiplication

Replace the division operation with multiplication by the reciprocal. That transforms the problem into \(-\frac{2}{3} \times 2\).

Step 4: Multiply Fractions

Multiply the fractions by multiplying the numerators and the denominators separately. Here, \(-\frac{2}{3} \times 2 = -\frac{2 \times 2}{3 \times 1} = -\frac{4}{3}\).

Step 5: Simplify if Necessary

The fraction \(-\frac{4}{3}\) is already in its simplest form. Thus, the quotient does not need further simplification.

Key Concepts

Reciprocal: An Essential Part of Fractions

When we're dealing with fractions, the concept of the "reciprocal" is crucial, especially in division. So, what exactly is a reciprocal? In simple terms, the reciprocal of a fraction is obtained by flipping its numerator and denominator.

Let's break this down: if you have a fraction like \(\frac{a}{b}\), its reciprocal would be \(\frac{b}{a}\).
To find the reciprocal of a fraction, just exchange the top (numerator) and the bottom (denominator) numbers.

For example, the reciprocal of \(\frac{1}{2}\) is \(\frac{2}{1}\), or simply 2. This simple flip helps transform division problems into multiplication problems, making calculations a lot easier. Always remember: reciprocals can turn the tricky act of dividing fractions into a much simpler task!

Multiplying Fractions: Simplifying the Process

Multiplying fractions is quite straightforward once you grasp the basics. Here's how it works: when multiplying fractions, you multiply the numerators together and the denominators together.

For instance, if you have two fractions like \(\frac{a}{b}\) and \(\frac{c}{d}\), the product is achieved by \(\frac{a \times c}{b \times d}\).

• Numerator times numerator
• Denominator times denominator

When dealing with a whole number, like 2, as in the division problem example, treat it as \(\frac{2}{1}\) to maintain consistency.

This approach turns division problems into manageable multiplication ones. Remember: once you replace the divisor with its reciprocal, you're essentially set to multiply, leading you directly to the result.

Simplifying Fractions: The Final Touch

Simplifying fractions is the step that ensures your solution is as simple as possible. To simplify a fraction, you divide both the numerator and the denominator by their greatest common divisor (GCD).

In some cases, though, your fraction may already be in its simplest form, like in our example \(-\frac{4}{3}\).
No further action is needed here, but understanding the process is beneficial for other instances where simplification might be required.

• Check if both numerator and denominator have a common factor
• Divide them by the greatest common factor to simplify

Simplifying makes fractions neater and, often, easier to interpret. It's like polishing up your final answer for clarity and precision. Keep these principles in mind as they will enhance your math-solving skills across various problems.