Problem 6 - Prealgebra

Question

Find each of the following quotients. (Divide.) [Examples 1–5] $$-50 \div(-25)$$

Step-by-Step Solution

Answer

The quotient of \(-50\) divided by \(-25\) is 2.

Step 1: Understand the division problem

The problem asks us to find the quotient of \(-50\) divided by \(-25\). This means we need to determine how many times \(-25\) goes into \(-50\).

Step 2: Simplify the signs

When dividing two numbers with the same sign (both are negative in this case), the result will be positive. Therefore, we ignore the negative signs temporarily and focus on dividing the absolute values of the numbers.

Step 3: Divide the absolute values

The absolute value of \(-50\) is 50 and the absolute value of \(-25\) is 25. We divide the absolute values: \(50 \div 25\), which equals 2.

Step 4: Determine the sign of the quotient

As determined in Step 2, dividing two numbers with the same sign results in a positive quotient. Therefore, the quotient \(2\) is positive.

Key Concepts

Understanding Absolute Value

When we talk about absolute value, we're looking at the size or magnitude of a number, regardless of its sign. It's like looking at the distance from zero on a number line, where both positive and negative numbers have a positive distance.
The absolute value of a number is always non-negative. In math notation, the absolute value of a number \(x\) is written as \(|x|\). For example:

• The absolute value of \( -50 \) is \( 50 \).
• The absolute value of \( 25 \) is \( 25 \).

Understanding absolute value helps simplify division problems by removing negative signs temporarily, making it easier to focus on the division itself. Once you divide the absolute values, don't forget to adjust for sign if necessary.

Handling Negative Numbers in Division

Negative numbers can be tricky, but they follow predictable rules. A negative number, simply, is a number less than zero.
When dividing numbers, the sign of the result depends on the signs of the numbers involved. Here are the basic rules:

• Dividing two positive numbers results in a positive quotient.
• Dividing two negative numbers also results in a positive quotient.
• Dividing a positive number by a negative number, or vice versa, results in a negative quotient.

In our exercise, \(-50\) and \(-25\) are both negative. According to the rules, the quotient will be positive, since a negative divided by a negative equals a positive. After calculating the division using their absolute values, this rule tells us what sign the final answer should have.

What is a Quotient?

The term "quotient" is used in mathematics to describe the result of a division problem. It is what you get when you divide one number by another.
Let's explore this through an example:

• If you divide 10 by 2, the quotient is 5.
• If you divide \(-50\) by \(-25\), the quotient is 2.

In equations, the division sign ( \(\div\) ) signifies this operation, pointing out that one number is being divided by another. Understanding what the quotient represents allows you to interpret division results clearly, especially when negative numbers and absolute values are in play.