Problem 39 - Algebra for College Students

Question

Perform the following operations with real numbers. $$\frac{-1.2}{-6}$$

Step-by-Step Solution

Answer

The result is 0.2.

Step 1: Understand the Problem

The problem requires division of two negative numbers, specifically \(-1.2\) divided by \(-6\). When we divide two negative numbers, the result is a positive number.

Step 2: Divide the Absolute Values

First, divide the absolute values of the numbers. The absolute value of \(-1.2\) is \(1.2\) and the absolute value of \(-6\) is \(6\). Divide \(1.2\) by \(6\).

Step 3: Perform the Division

Now, calculate \(1.2 \div 6\). Change \(1.2\) to a fraction or decimal to make division easier, \(1.2 = \frac{12}{10} = 0.12\). Then \(0.12 \div 6 = 0.2\).

Step 4: Determine the Final Answer

Since dividing two negative numbers results in a positive number, the result of \(-1.2 \div -6\) is \(0.2\).

Key Concepts

Division of Negative Numbers

Division is a basic arithmetic operation that might initially seem straightforward. However, when negative numbers come into play, it can become slightly confusing. To divide negative numbers and correctly determine the sign, remember these simple rules:

• If both numbers are negative, the result is positive. This is because the negatives effectively "cancel each other out." Hence, \(-1.2 \div -6\) results in a positive value.
• If one number is negative and the other is positive, the result is negative. This happens because there is only one negative sign involved.

Practicing with these straightforward rules will build your confidence when dealing with negative number division in any scenario.

Absolute Values

Understanding absolute values is fundamental in mathematics, especially when working with negative numbers. The absolute value of a number is its distance from zero on the number line, regardless of direction. It is always a positive number.
For example, the absolute value of \(-1.2\) is \(1.2\), and the absolute value of \(-6\) is \(6\). The absolute value essentially focuses on the magnitude while ignoring the sign.
When dividing \(-1.2\) by \(-6\), we calculate the division of their absolute values: \(1.2 \div 6\). Once computed, the division value shows the size or "size without sign" of the initial division problem.

Positive Result from Negative Division

It might seem counterintuitive at first, but dividing two negative numbers results in a positive number. Let's delve into why this happens.

• By definition, multiplying or dividing two numbers with the same sign (both positive or both negative) produces a positive result. This is an established rule in mathematics.
• This rule stems from the property of multiplication that states: the product of two negative integers equals a positive integer. Similarly, division follows this property since it’s the inverse operation of multiplication.

Applying this to the problem \(-1.2 \div -6\), the division outcome is positive because both numbers involved are negative, providing \(0.2\) as the final result. Understanding this principle can help ease the complexity when working with negative numbers in diverse math problems.