Problem 78 - Prealgebra and Introductory Algebra

Question

Divide. $$\frac{72}{-3}$$

Step-by-Step Solution

Answer

-24

Step 1: Identify dividend and divisor

Here, the dividend is 72, and the divisor is -3. There being a negative sign makes the result negative as well.

Step 2: Perform the Division

Now we need to divide. Divide 72 by 3. The result is 24.

Step 3: Apply the Sign Rule

Now, take into account the signs. Since there was a negative sign on the divisor, our final answer will also be negative. Thus, the result is -24.

Key Concepts

Dividend and Divisor

In the world of arithmetic, when we talk about dividing two numbers, we mention the terms "dividend" and "divisor". These two terms are crucial for understanding integer division.

• The **dividend** is the number that you want to divide. In our example, the dividend is 72. It is essentially the number that you start with.
• The **divisor**, on the other hand, is the number by which the dividend is divided. For instance, in \(\frac{72}{-3}\), the divisor is -3.

Using these terms helps to specify the operation at hand. In everyday language, you might hear it as asking "how many times does the divisor fit into the dividend?"

This helps in visualizing the division process more clearly. Knowing which number is the dividend and which is the divisor first is key before performing any calculations.

Negative Numbers

Negative numbers can sometimes confuse students, especially when performing arithmetic operations with them. A negative number is simply any number less than zero. These numbers lie to the left of zero on a number line.

In division, negative numbers can impact whether the result is positive or negative. When dividing 72, a positive number, by -3, a negative number, it's important to understand how negative signs affect your operation. Remember that negative numbers can intuitively seem like something you owe or a deficit.

• When a positive dividend is divided by a negative divisor, as in \(\frac{72}{-3}\), we must consider the fact that the negative divisor will change the result's sign.
• If both numbers had been negative, our calculations would change, due to the different sign rules that apply (which we will discuss in the next section).

Understanding the nature of negative numbers is part of understanding the broader topic of integers and operations involving them.

Sign Rules in Division

The rules of signs are vital when performing division with integers, especially when negative signs are involved. These rules help determine the sign of the answer when a division operation is performed. Here are the basic guidelines:

• If the **dividend and divisor** both have the same sign (either both are positive or both are negative), the quotient is positive.
• If the **dividend and divisor** have different signs, with one being negative and the other positive, the quotient is negative.

In our example, 72 (a positive number) divided by -3 (a negative number) results in a negative quotient, which is -24. Understanding these rules helps ensure you arrive at the correct sign of your answer.

By consistently applying these rules, you can solve division problems confidently, knowing exactly how to handle positive and negative numbers appropriately.