Problem 16 - Algebra for College Students

Question

Perform the following operations with real numbers. $$\frac{-75}{5}$$

Step-by-Step Solution

Answer

The result of \(\frac{-75}{5}\) is \(-15\).

Step 1: Understand the Problem

We need to simplify the expression \(\frac{-75}{5}\). This means we should divide \(-75\) by \(5\).

Step 2: Perform the Division

Divide \(-75\) by \(5\). To do this, divide the absolute values \(75\div 5\), which equals \(15\), and then apply the sign rule for division of real numbers: a negative divided by a positive is negative.

Step 3: Apply the Sign

Since \(-75\) is negative and \(5\) is positive, the result will be negative. So the division \(-75 \div 5 = -15\).

Key Concepts

Division of Real Numbers

Division of real numbers is a fundamental operation in arithmetic and algebra. It involves distributing a number (the dividend) by another number (the divisor). The result is called the quotient. In our example, dividing \(-75\) by \(5\), we focus on how to handle both positive and negative numbers.

When dividing real numbers, first focus on the absolute values. Ignore any negative signs initially and carry out a straightforward division as you would with positive numbers. For instance:

• Identify the absolute values, which ignore any negative signs. With \(-75\) and \(5\), this boils down to dividing 75 by 5.
• Compute the division: \(75 \div 5 = 15\).

With the calculated absolute quotient, you then decide on the sign of the final answer based on the sign rule for division, which we'll describe in the next sections.

Negative Numbers

Negative numbers are numbers with a value less than zero. These numbers are usually written with a minus sign in front. Understanding negative numbers is key when performing operations like division.

Negative numbers appear frequently in mathematics, especially in real number operations like subtraction and division. They represent values less than zero and have specific rules when combined with other numbers:

• Adding a negative number is the same as subtraction.
• Subtracting a negative number is akin to addition.
• Multiplying or dividing two negative numbers alone results in a positive number.
• Multiplying or dividing a negative and a positive number results in a negative number.

In our division example, the dividend is a negative number. It is crucial to apply these rules to determine the correct sign and magnitude of the result.

Sign Rule for Division

The sign rule for division helps determine the sign of the quotient when dividing two real numbers. In division, just like in multiplication, this rule is simple yet essential. Here's a quick breakdown for clarity:

• Negative \( \div \) Positive = Negative: When you divide a negative number by a positive number, the result is negative. As seen with \(-75 \div 5 = -15\).
• Positive \( \div \) Negative = Negative: Similarly, if a positive number is divided by a negative, the quotient is negative.
• Negative \( \div \) Negative = Positive: Dividing two negative numbers results in a positive quotient. For instance, \(-15 \div -3 = 5\).
• Positive \( \div \) Positive = Positive: This is straightforward – two positive numbers divide to give a positive result.

Appreciating this rule makes it easier to solve problems involving real numbers, ensuring the correct sign is used in the quotient. Understanding and correctly applying the sign rule is not only useful but necessary for accurate mathematical calculations.