Problem 16 - Prealgebra

Question

Find each of the following quotients. (Divide.) [Examples 1–5] $$-144 \div(-9)$$

Step-by-Step Solution

Answer

The quotient is 16.

Step 1: Identify the Problem

The exercise asks us to divide \(-144\) by \(-9\). This operation involves dividing two negative numbers.

Step 2: Apply the Division Rule for Negatives

Recall that when you divide a negative number by another negative number, the quotient is positive. In this case, dividing \(-144\) by \(-9\) results in a positive quotient.

Step 3: Perform the Division

Calculate the division of the absolute values of the numbers. Divide 144 by 9: \(144 \div 9 = 16\).

Step 4: Finalize the Answer

Since both numbers in the original division were negative, and the rule states that a negative divided by a negative is positive, the final answer is positive \(16\).

Key Concepts

Negative Numbers

Negative numbers are integers less than zero, marked with a minus sign (-). These numbers can be found to the left of zero on a number line.
They are used to represent values less than nothing, often in contexts like temperatures below freezing or financial debt.
Understanding negative numbers is crucial when performing operations like addition, subtraction, multiplication, and division, especially since they follow distinct rules different from positive numbers.
Negative numbers also participate in unique rules when it comes to division and multiplication. For instance, when a negative number is divided by another negative number, the result is a positive number.

Positive Quotient

A quotient is the result obtained by dividing one number by another.
When involving two negative numbers, the final quotient turns out positive. Why, you might ask? Well, it's all about the rules of multiplication and division.
When we multiply or divide two negative numbers, they "negate" each other, resulting in a positive product or quotient.
This is an important rule to remember as it helps prevent mistakes when dealing with negative values in mathematical operations.
In the exercise \[ -144 \div (-9) \], dividing two negative numbers results in a positive quotient, which is \(16\).

Absolute Values

The absolute value of a number is its distance from zero on a number line, without considering direction.
It's always a non-negative number. For instance, both 7 and -7 have an absolute value of 7 because they are seven units away from zero.
Absolute values are used in operations to simplify the handling of negative numbers.
In division, you first find the absolute values of the numbers involved to perform the division easily.
By overlooking the signs initially, then addressing them with division rules afterward, you simplify the process, as seen in \(144 \div 9 = 16\), before applying the sign rules.

Division Rules

Division of integers involves following specific rules concerning their signs:

• Dividing two positive numbers yields a positive quotient.
• Dividing a positive number by a negative number results in a negative quotient.
• Dividing a negative number by a positive number also gives a negative quotient.
• Lastly, dividing two negative numbers results in a positive quotient, as they cancel each other out.

These rules are essential in ensuring accuracy when performing arithmetic operations.
Understanding and applying these rules, like when dividing \(-144 \div (-9)\), make complex mathematical problems easier to solve, ensuring you obtain the correct positivity or negativity of the result.