Problem 27 - Algebra 1: Concepts and Skills

Question

Find the quotient. $$ 6 \div\left(-\frac{1}{2}\right) $$

Step-by-Step Solution

Answer

-12

Step 1: Recognizing Division as Multiplication by the Reciprocal

Dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of a number is just 1 divided by that number. Therefore, 6 ÷ (-1/2) is the same as 6 * -2/1.

Step 2: Perform the Multiplication

Now that we have converted the division into a multiplication problem, we can proceed to solve. Multiply 6 by -2/1 to get -12.

Key Concepts

Understanding Multiplication and Division

In mathematics, multiplication and division are inverse operations. This means that they undo each other. When you multiply a number by another, you are essentially scaling that number. Division, on the other hand, breaks down or partitions a number into smaller equal parts.
To illustrate, consider multiplication as repeated addition. If you multiply 4 by 3, you add 4, three times: 4 + 4 + 4 = 12. Division works in reverse; if you have 12 and divide by 3, you ask how many times 3 can be added to reach 12, which is 4.

• Multiplication scales up numbers.
• Division scales down numbers.
• They are opposite processes.

In problems involving fractions, like our original exercise, knowing how to switch between multiplication and division by using the reciprocal is essential.

Reciprocal Explained

The reciprocal of a number is what you multiply by the original number to get 1. Think of it as flipping the number. For a fraction, you simply swap the numerator and the denominator. Thus, the reciprocal of \( \frac{1}{2} \) is \( \frac{2}{1} \).
In our exercise, dividing by \( -\frac{1}{2} \) is equivalent to multiplying by its reciprocal \( -\frac{2}{1} \). This is a crucial skill when dividing fractions, as it allows you to change a division problem into multiplication, which is generally easier to handle.

• Find a reciprocal by flipping the fraction upside down.
• Use the reciprocal to turn division into multiplication.
• Helps simplify complex calculations with fractions.

Mastering the reciprocal helps with quick calculations in various mathematical situations.

Working with Negative Numbers

Negative numbers represent values less than zero. They follow specific rules, especially in multiplication and division, that differ from their positive counterparts. When you multiply or divide two numbers and one is negative, the result is negative.
In our mathematical exercise, the conversion of division into multiplication involved a negative number, \(-\frac{1}{2}\). When multiplied, one positive number and one negative number give a negative product, so \(6 \times -\frac{2}{1}\) results in \(-12\).

• Negative × Positive = Negative
• Positive × Negative = Negative
• Negative × Negative = Positive

Understanding these rules is vital and helps prevent errors in calculations involving negative numbers. It is especially important in algebra and other advanced mathematical topics.