Problem 12 - Algebra 1

Question

Find the quotient. $$-12 \div 3$$

Step-by-Step Solution

Answer

-4

Step 1: Identify the dividend and divisor

In this problem, the dividend, or the number to be divided, is -12. The divisor, or the number by which the dividend is to be divided, is 3.

Step 2: Perform the division

Divide the dividend, -12, by the divisor, 3. The sign of the division will be negative, as one number is negative and the other one is positive.

Step 3: Write down the result

The result of -12 divided by 3 is -4. This is the quotient.

Key Concepts

Quotient

In division, the **quotient** is the result you get when you divide one number by another. Imagine having 12 apples and distributing them equally to your three friends. Each friend would get 4 apples; thus, 4 is the quotient. However, when we deal with negative numbers, the concept remains the same but becomes slightly more complex. For instance, dividing \(-12\) by 3, we use the same division process as with positive numbers, but with an awareness of the role negative numbers play. When one number is negative, like in this division, the process influences the sign of the quotient.

Dividend

The **dividend** in the division is the number you want to divide. Think of it as the total amount you are starting with. In our exercise, the dividend is \(-12\). This number can be positive or negative.

• If it's positive, like 12, you just distribute that amount by the divisor.
• If it's negative, like \(-12\) in our exercise, it usually signals a decrease or a debt.

When you divide a negative dividend, it means you're distributing a shortage, resulting in a negative quotient when the divisor is positive.

Divisor

The **divisor** is the number by which you divide the dividend. It determines how that dividend will be split or grouped. In the exercise, our divisor is 3, a positive number.

• When the divisor is a positive number, it typically signals fairness or equal distribution in real-life scenarios.
• The sign of the divisor directly affects whether the quotient is positive or negative. If the divisor is negative, the roles reverse, leading to different outcomes for the quotient.

Understanding the divisor's role helps manage how we approach division, especially with negative values. It shifts the result's sign when divided with negative numbers.

Negative Numbers

**Negative numbers** often introduce an extra layer into mathematical operations. They represent values less than zero, such as debts or temperatures below freezing. When dividing with negative numbers, a few simple rules apply:

• If you divide a negative number by a positive number, like \(-12 \div 3\), the quotient is negative.
• Conversely, if you divide a positive number by a negative one, the result is also negative.
• However, two negatives make a positive when dividing: dividing \(-12\) by \(-3\) results in a positive 4.

These rules help maintain consistency across operations, ensuring understanding when dealing with negative values in division.