Problem 51 - Introductory and Intermediate Algebra for College Students 4th

Question

In Exercises \(47-76\), perform the indicated division or state that the expression is undefined. $$\frac{-90}{-3}$$

Step-by-Step Solution

Answer

The solution to the division operation \(\frac{-90}{-3}\) is \(30\).

Step 1: Recognize the Division of Negative Numbers

The equation can be rewritten as \(-90 ÷ -3\). Both numbers are negative. According to the rules in mathematics, the division of two negative numbers will give a positive result.

Step 2: Perform the division

Perform the division of 90 by 3 to yield 30.

Step 3: Apply the rule for division of negative numbers

Because both numbers in this scenario are negative, the outcome of their division is positive, yielding a final answer of 30.

Key Concepts

Negative Numbers

Understanding negative numbers is crucial for solving various math problems, including division. Negative numbers are simply numbers with a minus sign (−) in front of them. They represent values less than zero and are often used to describe debts, temperatures below freezing, or elevations below sea level. When you divide negative numbers, it is important to keep the rules of mathematics in mind.

• A negative number divided by another negative number results in a positive number.
• If you divide a negative number by a positive number, the result is negative.
• Conversely, a positive number divided by a negative number also results in a negative number.

Negative numbers behave differently in various operations, so it’s essential to recognize when and how to apply these rules correctly. This understanding serves as a strong foundation for more complex mathematical concepts.

Mathematical Rules

In mathematics, rules regarding how to operate with numbers are central to finding the correct solutions. One such fundamental rule is how to divide numbers, particularly negatives. To divide any numbers, you need to follow a sequence of operations. With negative numbers, one of the primary rules is that the division of two negative numbers results in a positive outcome. This might seem counterintuitive, especially if you're just starting with algebra. However, these rules are consistent and logical when understanding the number line and how negatives mirror positives. The mathematical rules are:

• A negative divided by a negative equals a positive.
• A positive divided by a negative equals a negative.
• A negative divided by a positive is negative.

By consistently applying these rules for dividing negative numbers, you can solve numerous problems accurately! Remember, practice makes perfect, so keep these rules handy as you work on division exercises.

Positive Result

Understanding when a positive result occurs is a critical part of mastering division, especially involving negative numbers. As demonstrated in the problem \(\frac{-90}{-3}\), where both the dividend and divisor are negative, the result is positive due to the rule that two negatives make a positive. This is analogous to saying that a debt (negative influence) can be 'canceled out' by another debt, leading to a neutral or positive scenario.The outcome of such divisions should always be checked to ensure accuracy:

• After dividing the absolute values of the numbers, check the signs.
• If both signs are negative, the result is positive.

So, in this exercise, dividing -90 by -3 gives a positive 30 because of these rules. Recognizing the simplicity in these outcomes aids in building a strong foundation in number operations. Thus, understanding the result of negative divisions not only simplifies calculations but also strengthens your overall math skills.