Problem 31 - Algebra 1: Concepts and Skills

Question

Find the quotient. $$ \frac{-6}{-\frac{1}{2}} $$

Step-by-Step Solution

Answer

The quotient is 12.

Step 1: Identify the Fractions

First, identify the fractions involved in the division operation. Here, there are two fractions, namely \(-6 \) and \(- \frac{1}{2} \)

Step 2: Understand Division by a Fraction

Understand that dividing by a fraction is the same as multiplying by its multiplicative inverse or reciprocal. Hence, you need to multiply -6 with the reciprocal of -\frac{1}{2}, which is -2.

Step 3: Multiply

Now, perform the multiplication operation: \(-6 \times -2\)

Key Concepts

Multiplicative Inverse

The concept of a multiplicative inverse is integral when dealing with division of fractions. The multiplicative inverse, sometimes referred to as the reciprocal, is a number that, when multiplied by the original number, yields 1.
For a non-zero number, let's say \( a \), its multiplicative inverse is \( \frac{1}{a} \).
Multiplicative inverses help simplify division problems involving fractions. When dividing by a fraction, you multiply by its multiplicative inverse to achieve the same result.
For example, if you have \( \frac{2}{3} \), its multiplicative inverse is \( \frac{3}{2} \). Multiplying \( \frac{2}{3} \) by \( \frac{3}{2} \) would result in 1, demonstrating the concept of multiplicative inverse.

Reciprocal

Reciprocals play a pivotal role in fraction division, helping transform division into simpler multiplication. To find the reciprocal of a fraction, simply flip the numerator and denominator.
So, the reciprocal of \( \frac{a}{b} \) is \( \frac{b}{a} \).
This flip is key because it allows us to multiply the first fraction by the reciprocal of the second fraction instead of performing a division.
In the exercise given, the reciprocal of \(-\frac{1}{2}\) is \(-2\). This means you'll multiply by \(-2\) instead of dividing by \(-\frac{1}{2}\). It's an efficient trick that can feel intuitive with practice.

Negative Numbers

Negative numbers introduce some interesting changes to basic arithmetic operations like addition, subtraction, multiplication, and particularly division. When working with negative numbers, it’s important to remember a few key rules:

• Multiplying or dividing two negative numbers results in a positive number.
• Multiplying or dividing a positive number by a negative number results in a negative number.

In the exercise, both the dividend and the divisor are negative. Thus, when dividing, you are effectively multiplying two negative numbers, yielding a positive result.
For example, \(-6 \div -\frac{1}{2}\) turns into \(-6 \times -2\), resulting in 12, which is positive.

Fraction Multiplication

Fraction multiplication is straightforward once you understand the rules: multiply the numerators together and the denominators together. It simplifies many fraction operations, especially when division is transformed into multiplication using reciprocals.
For instance, in the given problem, after finding the reciprocal of the divisor \(-\frac{1}{2}\) and getting \(-2\), the multiplication step involves:

1. Multiply the numerators: \(-6 \times -2 = 12\).
2. Multiply the denominators, which are essentially just \(1\times1\), leaving 12 as the result.

Because multiplying two negative numbers results in a positive product, \(-6\) and \(-2\) combine to a positive 12. Fraction multiplication can thus be an efficient tool once one becomes comfortable with these processes.