Problem 21 - Elementary Algebra

Question

Determine the quotient of -36 and 3 .

Step-by-Step Solution

Answer

The quotient of -36 and 3 is -12.

Step 1: Understand the Problem

We are tasked with finding the quotient of -36 and 3. This means we need to divide -36 by 3.

Step 2: Set Up the Division

We set up the division expression as \(-36 \div 3\). This is the expression we need to solve to find the quotient.

Step 3: Perform the Division

Divide the numbers as you normally would, ignoring the sign for now: \(\frac{36}{3} = 12\).

Step 4: Apply the Sign Rule

Since we are dividing a negative number by a positive number, the quotient will be negative. So, the quotient is \(-12\).

Key Concepts

Negative Numbers

Negative numbers are numbers that are less than zero. They are an essential part of mathematics, often representing values below zero, like temperatures below freezing or a bank account in deficit.
Negative numbers are usually prefixed with a minus sign (-). For example, -36 means thirty-six units less than zero. Understanding negative numbers helps with many math operations and solving real-world problems.

• Negative numbers can be added or subtracted like positive numbers, but with attention to their direction on the number line.
• The multiplication or division of two negatives results in a positive number.
• A negative divided by a positive equals a negative, and vice versa.

Negative numbers can thus change the sign of the result when involved in arithmetic operations like division.

Quotient

The quotient is the result of a division operation. In our exercise, it is the solution to dividing \(-36 \div 3\), which results in -12. The quotient gives us an idea of how many times one number can "fit into" another.
It's important to understand how the sign of the numbers affects the quotient:

• If both numbers are positive, the quotient is positive.
• If one number is negative and the other is positive, the quotient is negative.
• If both numbers are negative, the quotient is positive.

In general, when finding a quotient, consider both the magnitude and the sign of the numbers involved.

Basic Arithmetic Operations

Basic arithmetic operations include addition, subtraction, multiplication, and division. Division, like in our exercise, is one of these core operations and is used to distribute numbers evenly into groups.
The operations have different rules when applied to negative numbers, especially for subtraction and division. In division, pay attention to the sign:

• When a negative number is divided by a positive number, the quotient is negative.
• When dividing positive numbers, the result is positive.

Understanding these operations and their rules, especially with negative numbers involved, is crucial for more complex math problems later on.