Problem 15 - Prealgebra

Question

Find each of the following quotients. (Divide.) [Examples 1–5] $$125 \div(-25)$$

Step-by-Step Solution

Answer

The quotient is -5.

Step 1: Understand the Division Problem

We need to divide 125 by -25. This means we have 125 being divided by negative 25.

Step 2: Determine the Sign

When dividing a positive number by a negative number, the quotient will be negative. This is because the rules of division state that a positive divided by a negative equals a negative.

Step 3: Divide the Absolute Values

Ignore the signs momentarily and divide the absolute values: \( 125 \div 25 = 5 \).

Step 4: Assign the Sign to the Answer

Since we determined the result should be negative, the final quotient is \(-5\).

Key Concepts

Quotients

In division, when we refer to "quotients," we're talking about the result you get after dividing one number by another. Quotients help us understand how many times one number is contained within another. For example, when we say "find the quotient of 125 divided by -25," we want to know how many times -25 goes into 125.

To compute a quotient, you simply divide the dividend by the divisor. The formula for this is:

• Dividend ÷ Divisor = Quotient

For our example, 125 is the dividend, and -25 is the divisor. A successful division process helps you quickly find the quotient and understand the relationship between the numbers involved. By keeping these terms in mind, the division becomes a clearer and more structured process.

Sign Rules in Division

Understanding the sign rules in division is crucial. This tells us whether the answer will be positive or negative. These rules are consistent and follow simple guidelines:

• A positive number divided by a positive number results in a positive quotient.
• A negative number divided by a negative number also results in a positive quotient, since the negatives cancel each other out.
• A positive number divided by a negative number results in a negative quotient.
• Likewise, a negative number divided by a positive number results in a negative quotient.

In the example, dividing 125 (a positive number) by -25 (a negative number) results in a negative quotient. Therefore, the resulting quotient is extbf{-5}.

Absolute Values in Division

Absolute values are crucial when performing division with positive and negative numbers. In mathematics, the absolute value of a number is its distance from zero on the number line, ignoring the sign. It helps simplify division by focusing solely on the magnitude of the numbers involved.

When dividing integers, first consider only their absolute values:

• Absolute value of 125 is 125.
• Absolute value of -25 is 25.

By using the absolute values, you perform division as if both numbers are positive. So, dividing 125 by 25 gives you 5. After calculating this, determine the sign of your answer using the sign rules. Absolute values allow you to separate the division process into manageable parts, which makes solving problems like these less daunting.