Problem 8 - Prealgebra

Question

Find each of the following quotients. (Divide.) [Examples 1–5] $$\frac{-18}{-6}$$

Step-by-Step Solution

Answer

The quotient is 3.

Step 1: Identify the Division

The expression given is a division problem: \( \frac{-18}{-6} \). This means you need to divide -18 by -6.

Step 2: Simplify Absolute Values

Consider the absolute values of the numbers involved in the division. The absolute value of -18 is 18 and the absolute value of -6 is 6. Therefore, you perform the division using these numbers: \( \frac{18}{6} = 3 \).

Step 3: Determine the Sign of the Quotient

Since both the dividend and the divisor are negative, the rule for division of integers states that dividing two numbers of the same sign yields a positive result. Therefore, the quotient is positive 3.

Key Concepts

Understanding Absolute Value

Absolute value is a concept that helps us understand the magnitude of a number without considering its sign. In simple terms, the absolute value of a number is how far that number is from zero on the number line. This can be particularly useful in calculations such as division, where the magnitude of numbers is more significant than their direction (positive or negative).

When we look at the absolute value, we represent it with vertical bars, such as \(|x|\). Whether we are dealing with a positive or negative number, the absolute value is always non-negative.

• For instance, the absolute value of -5 is 5, represented as \(|-5| = 5\), because it is 5 units away from zero.
• Similarly, the absolute value of 8 is 8, so \(|8| = 8\).

Absolute value plays a critical role in division because it simplifies the process by ensuring we are focusing solely on the magnitudes of the numbers. This helps in obtaining the basic division result before any sign considerations.

Division Rules for Integers

When dividing integers, there are some specific rules we need to follow to determine the correct quotient, including its sign. These rules are crucial to perform accurate calculations:

• Positive divided by Positive: The result is always positive. For example, \(24 \/ 3 = 8\).
• Negative divided by Negative: The result is positive, similar to our initial exercise with \(-18 \/ -6 = 3\).
• Positive divided by Negative: The result is negative, like in \(12 \/ -4 = -3\).
• Negative divided by Positive: Also results in a negative quotient, such as \(-20 \/ 5 = -4\).

These division rules help us not only with understanding the magnitude but also ensuring that the quotient reflects the correct sign based on the original integers involved. Always consider both the rule and the concept of absolute value to make the calculation seamless.

Executing Integer Operations

Integer operations encompass addition, subtraction, multiplication, and division of whole numbers. Each operation follows its own set of rules that need understanding. Let’s focus on division, an essential integer operation that often causes confusion due to sign considerations.

When dealing with integer division, one of the most critical steps is to handle negative signs correctly. Unlike other operations, division of integers requires careful attention to both magnitude and sign. By first considering the absolute values, as seen in the previous example, we can simplify our calculations:

• Start the division as if both numbers are positive, providing a preliminary quotient.
• Apply integer division rules to ascertain the correct sign for the final quotient.

Handling integer operations consistently will make math problems less daunting and more intuitive, especially when dealing with more complex calculations that combine both positive and negative numbers.