Problem 77 - Beginning and Intermediate Algebra

Question

Divide. $$ \frac{30}{-2} $$

Step-by-Step Solution

Answer

The result of dividing 30 by -2 is -15.

Step 1: Identify the operation

The operation required is division, where 30 needs to be divided by -2.

Step 2: Understand the division process

In division, you are finding out how many times the divisor fits into the dividend. In this case, you need to find out how many times -2 fits into 30.

Step 3: Divide the absolute values

First, ignore the signs and divide the absolute values: \( 30 \div 2 = 15 \).

Step 4: Apply the sign rule

According to the rules of division of integers, when you divide a positive number by a negative number, the result is negative. Therefore, \( +30 \div -2 = -15 \).

Key Concepts

Integer Arithmetic

When performing integer arithmetic, you are dealing with whole numbers. This includes operations like addition, subtraction, multiplication, and division, which are some of the basic math operations. Each operation has its own set of rules when integers are involved.

Integer arithmetic is special because it follows rules that take into account the signs (positive or negative) of the numbers. Whole numbers are integers, and can be either positive (like 3, 4, 5) or negative (like -1, -2, -3), as well as zero.

• For addition or subtraction, the sign of the numbers affects whether their magnitudes add or cancel each other out.
• Multiplication and division also have sign-specific rules, which dictate the sign of the result based on the signs of the operands.
• These fundamental rules ensure consistency across math operations.

Understanding integer arithmetic helps in setting the stage for properly performing division, especially when one or both of the numbers involved are negative.

Operations with Positive and Negative Numbers

Operations with positive and negative numbers follow specific rules that determine the result's sign. This becomes especially important in multiplication and division.

With division, the sign of the result follows this simple rule:

• When you divide two numbers with the same sign (both positive or both negative), the result is positive.
• When you divide numbers with different signs (one positive and one negative), the result is negative.

For example, when dividing 30 by -2, we see that the signs are different: 30 is positive, and -2 is negative. Hence, the result of the division is negative. This same logic applies universally, making it crucial to recognize the numbers' signs to correctly determine the division outcome.

Basic Division Rules

Understanding basic division rules involves knowing how to handle the numbers involved and their respective signs. Division is essentially finding out how many times one number, called the divisor, fits into another number, called the dividend.

Here is what you need to understand when dividing integers:

• First, perform the division on the absolute values of the numbers involved. The absolute value is the non-negative value of a number without considering its sign.
• After dividing the absolute values, determine the result's sign using the rules of signs - same signs give a positive result, different signs give a negative result.
• If you have two positive numbers, the result remains positive. Example: \( \frac{6}{2} = 3 \).
• If you divide a positive number by a negative number, like in \( \frac{30}{-2} = -15 \), the result is negative.

Always follow these rules, and you'll navigate through division problems with ease.