Problem 8 - Algebra 1: Concepts and Skills

Question

Find the quotient. \begin{equation} -7 \div-\frac{1}{2} \end{equation}

Step-by-Step Solution

Answer

The answer is 14.

Step 1: Understand Division by Fraction

When dividing by a fraction, the principle of 'invert and multiply' is applied. In other words, dividing by a fraction is the same as multiplying by its reciprocal.

Step 2: Identify the Reciprocal

First, we need to find the reciprocal of \(-\frac{1}{2}\). The reciprocal of a fraction is obtained by swapping its numerator and denominator. Hence, the reciprocal of \(-\frac{1}{2}\) is \(-2\).

Step 3: Apply Invert and Multiply

Now, replace the division operation by a multiplication operation with the reciprocal: -7 multiplied by -2.

Step 4: Perform the Multiplication

Perform the multiplication operation. Two negative numbers multiplied give a positive number. Hence, the answer is 14.

Key Concepts

Reciprocal of a Fraction

When dealing with fractions, understanding the concept of the reciprocal is essential. A reciprocal is simply what you get when you flip the numerator and the denominator of a fraction.

For example, the reciprocal of \( \frac{a}{b} \) is \( \frac{b}{a} \). This means for the fraction \( -\frac{1}{2} \), the reciprocal would be \( -2 \). Here's why it's important: reciprocals play a key role in fraction division.

• Flip the fraction: To find the reciprocal of \(-\frac{1}{2}\), switch the positions of 1 and 2. Thus, it becomes \(-2\).
• Keep the sign: The negative sign remains as it is a part of the fraction.

This is a crucial step whenever you are dividing fractions because it forms the basis of the "invert and multiply rule".

Multiplying Negative Numbers

Multiplying negative numbers might seem a bit tricky at first, but it's straightforward once you get the hang of it. The key rule to remember is:

The product of two negative numbers is a positive number.

• For instance, if you multiply two negatives, like \(-7 \times -2\), you'll end up with a positive 14.
• This happens because the negatives cancel each other out, similar to subtracting a negative value, which essentially adds a positive value.

Think about it like this: when you remove a negative, you are actually creating a positive. So, every time you multiply two negative numbers, you can confidently state that the result will be a positive number.

Invert and Multiply Rule

The "invert and multiply rule" is a key technique used in division of fractions like in our original exercise with \(-7 \div -\frac{1}{2}\). Let's break it down:

• Invert the divisor: First, find the reciprocal of the fraction you are dividing by. In our example, \(-\frac{1}{2}\) becomes \(-2\).
• Change the operation: Replace the division sign \( \div \) with a multiplication sign \( \times \).
• Multiply: Use the new reciprocal to multiply with the original number, \(-7 \times -2\).

When you multiply \(-7\) by \(-2\), the product is \(14\), a positive number. This method simplifies division problems, turning them into multiplication problems, which are often easier to solve.