Problem 59 - Introductory Algebra for College Students

Question

Perform the indicated division or state that the expression is undefined. $$120 \div(-10)$$

Step-by-Step Solution

Answer

-12

Step 1: Identify the numbers

Identify the two numbers involved in the division operation. Here, number 120 is to be divided by -10.

Step 2: Perform the Division

Perform the division operation as usual. But pay attention to the signs. The division of a positive number by a negative number will give a negative result. So calculate \( 120 \div -10 = -12 \)

Step 3: Write the Result

Finalize the result of division. In this case, after performing the division operation, we get -12, therefore \(120 \div(-10) = -12\)

Key Concepts

Understanding Positive and Negative Numbers

Positive and negative numbers are crucial components of arithmetic, particularly in operations like division. Positive numbers are greater than zero, including numbers like 1, 2, and 10.
They are often used to represent quantities, increases, or gains. Negative numbers, on the other hand, are less than zero, such as -1, -2, and -10.
These numbers typically represent decreases, losses, or debts. When working with positive and negative numbers, it's important to know that they dictate the direction in which the number line moves.

• Positive numbers move to the right on the number line.
• Negative numbers move to the left on the number line.

The difference between positive and negative numbers greatly impacts the result of operations, such as division, as will be explained.

The Division Operation

The division operation is one of the four fundamental arithmetic operations, alongside addition, subtraction, and multiplication.
It involves splitting a number, called the dividend, into a specified number of equal parts as indicated by another number, called the divisor. To perform a division operation, you divide the dividend by the divisor. For example, in the original problem, we divided 120 (dividend) by -10 (divisor). Key concepts to understand in division include:

• The result of division is called the quotient.
• Division by zero is undefined because you cannot distribute a number into zero parts.

A division symbol, which can be represented by the characters "/" or the sign "÷", distinguishes this operation.

Results of Division with Different Signs

The results of a division operation depend significantly on the signs of the numbers involved. When dividing numbers:

• A positive number divided by another positive number gives a positive quotient.
• A negative number divided by another negative number also yields a positive quotient.
• A positive number divided by a negative number results in a negative quotient, as was seen with 120 ÷ -10 = -12.
• A negative number divided by a positive number also results in a negative quotient.

Understanding these outcomes helps avoid confusion and errors in arithmetic. The same rules apply for multiplication and division operations involving positive and negative numbers.
Always remember to keep track of the signs to determine the correct result.