Problem 11 - Prealgebra

Question

Find each of the following quotients. (Divide.) [Examples 1–5] $$-22 \div 11$$

Step-by-Step Solution

Answer

The quotient is \(-2\).

Step 1: Identify the Problem

We need to determine what \(-22 \div 11\) evaluates to by performing division.

Step 2: Perform the Division

Divide the absolute values first: \(22 \div 11 = 2\).

Step 3: Determine the Sign of the Result

Since the signs of the dividend \(-22\) and the divisor \(11\) are different (one is negative, the other positive), the resulting quotient will be negative.

Step 4: Combine the Results

Combine the results from Steps 2 and 3. The absolute quotient is \(2\), and since it should be negative, the final result is \(-2\).

Key Concepts

Negative Numbers

Negative numbers are numbers that are less than zero. They have some unique properties, especially when involved in math operations like division. When dividing or multiplying with negative numbers, one important rule to remember is:

• If one of the numbers in a division or multiplication operation is negative, the result will be negative.
• If both numbers involved are negative, the result will be positive.

In the problem \[-22 \div 11\], we are dividing a negative number by a positive one. Because of the rule mentioned above, the quotient here is negative.

Quotients

A quotient is the result we get when we divide one number by another. In simple terms, it answers the question, 'How many times does one number fit into another?'

• The number being divided is called the dividend, in our example it is \(-22\).
• The number we are dividing by is called the divisor, here it is \(11\).

When calculating quotients, it is critical to consider both the magnitude and the sign of the numbers involved.
Since dividing changes the way numbers relate to each other, always be attentive to the sign of your quotient, as it can significantly change your result's interpretation.

Math Operations

Math operations refer to the basic mathematical procedures: addition, subtraction, multiplication, and division. Each operation follows specific rules, particularly when negative numbers are involved.For division:

• Always divide the absolute values first. This helps you understand the basic part of the operation without signs getting in the way.
• Determine the sign of the result after calculation. A 'positive' divided by a 'positive' produces a 'positive', while a 'negative' divided by a 'positive' results in a 'negative', and vice versa.

Using the knowledge of these operations, we calculated \(-22 \div 11\) by first ignoring the signs:

• Performing \(22 \div 11\), giving us \(2\).

Then, applying the sign rule, we acknowledged that the quotient is negative.