Problem 75 - Beginning and Intermediate Algebra

Question

Divide. $$ \frac{-12}{-4} $$

Step-by-Step Solution

Answer

The result of \( \frac{-12}{-4} \) is 3.

Step 1: Understand Division with Negative Numbers

When dividing two negative numbers, the negatives cancel out, resulting in a positive quotient. Thus, dividing \(-12\) by \(-4\) is the same as dividing 12 by 4.

Step 2: Simplify the Division

Consider the positive numbers only: \(12 \div 4\). Perform the division by calculating how many times 4 fits into 12.

Step 3: Perform the Division

Divide 12 by 4: \(12 \div 4 = 3\). Therefore, the result of the division \(-12 \div -4\) is equivalent to 3.

Key Concepts

Division Rules

When it comes to dividing numbers, understanding division rules is crucial. Division is essentially splitting a number into equal parts. In mathematics, certain rules help simplify calculations:

• If both numbers in the division problem are negative, the negatives cancel each other out, resulting in a positive quotient.
• If one number is negative and the other is positive, the quotient will be negative.

Understanding these rules makes it easier to tackle problems involving negative numbers. For instance, \(\frac{-12}{-4}\) becomes \(\frac{12}{4}\), yielding a positive result after canceling the negatives. These rules provide a simple way to navigate through seemingly complex arithmetic problems.

Integers

Integers are numbers that can be positive, negative, or zero, and do not include fractions or decimals. They form an essential part of the number system used in arithmetic operations like addition, subtraction, multiplication, and division. Integers follow unique rules when involved in operations:

• Adding two positive integers results in a positive sum.
• Adding two negative integers gives a negative sum.
• Adding a negative integer and a positive integer involves finding the difference between the numbers.

With division, you apply specific rules, such as when dividing \(\-12\) by \(\-4\), both being negative integers results in the negatives canceling each other and yielding a positive quotient. Understanding how integers work helps students perform all arithmetic operations more effectively.

Arithmetic Operations

Arithmetic operations consist of addition, subtraction, multiplication, and division. These operations follow specific mathematical principles and properties that simplify calculations.Each operation has its rules:

• Addition: Combine values. Positive numbers add to make a larger positive number, while negative numbers create a larger negative number.
• Subtraction: Taking away one number from another. An effective way to change the sign of numbers is to see subtraction as the addition of a negative.
• Multiplication: Repeated addition of the same number. Two negatives multiplied make a positive.
• Division: Splitting a number into equal parts. Two negatives divide to yield a positive result, as demonstrated in dividing \(\-12\) by \(\-4\).

Recognizing how these operations interact with negative numbers lets students confidently solve problems that appear daunting at first. Understanding these basics is the cornerstone of mastering more complex arithmetic.