Problem 2 - Prealgebra

Question

Find each of the following quotients. (Divide.) [Examples 1–5] $$15 \div(-3)$$

Step-by-Step Solution

Answer

The quotient is -5.

Step 1: Identify the Division Problem

The problem given is to divide 15 by -3. It can be expressed mathematically as \(15 \div (-3)\).

Step 2: Understand the Sign Rule for Division

When dividing numbers, if the signs of the two numbers are different, the quotient will be negative. Here, 15 is positive and -3 is negative, so the result will be negative.

Step 3: Perform the Division

Divide the absolute values of the numbers: 15 divided by 3 equals 5. Since one number is positive and the other is negative, the quotient is -5.

Key Concepts

Integer Division

In mathematics, dividing integers is a fundamental operation that often requires special attention compared to dividing regular numbers. Integer division is the process of dividing one integer (a whole number) by another. This operation finds out how many times one number is contained within another, in terms of whole numbers.
For example, in the problem stated as "15 divided by -3," it involves dividing a positive integer by a negative integer. The operation can be thought of as repeated subtraction of the divisor from the dividend.

• If you take 15 as a pile of apples and you want to divide them into groups of -3, theoretically, you would be creating groups in the opposite direction (e.g., removing apples).
• The essence of integer division is not the actual removal, but how it conceptually fits as a whole number.

Sign Rules for Division

Understanding sign rules is crucial when performing division involving integers. These rules help you determine the sign of the quotient.
Here are the basic sign rules you should remember when dealing with division involving positive and negative integers:

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

To apply this to our example of dividing 15 by -3, you notice that 15 is positive and -3 is negative. According to our sign rule, their quotient will be negative. This simple rule assists in quickly identifying the sign of the result without needing to calculate the division fully.

Negative Quotient

The term "negative quotient" refers to the result of a division operation when one integer is positive and the other is negative. In such cases, the outcome will always be a negative number.
Consider the division problem 15 divided by -3. By applying the sign rules, you know the quotient must be negative. The quotient can be found by calculating the division of their absolute values (i.e., ignoring the negative sign) and then applying the negative sign afterward.

• First, divide 15 (a positive number) by 3 (ignoring the negative sign of -3), which gives you 5.
• Then, because you divided a positive by a negative, your final answer is -5.

Being aware of the concept of a negative quotient allows you to tackle more complex problems involving both positive and negative integers with confidence.