Problem 61 - Elementary and Intermediate Algebra

Question

Divide, if possible, and check. If a quotient is undefined, state this. $$ \frac{400}{-50} $$

Step-by-Step Solution

Answer

-8

Step 1 - Identify the Division

Understand the problem at hand: \( \frac{400}{-50} \). This means dividing 400 by -50.

Step 2 - Divide the Numbers

Perform the division: 400 divided by -50. \[ 400 \, \text{divided by} \, -50 = -8 \]

Step 3 - Verify the Quotient

Multiply the quotient back by the divisor to ensure it equals the original numerator. \[ -8 \, \text{times} \, -50 = 400 \] Since 400 is the original numerator, the quotient is correct.

Key Concepts

Division of Integers

Let's start with understanding how to divide integers. Division is one of the four basic arithmetic operations. It involves determining how many times one number (the divisor) is contained within another (the dividend). When dividing two integers, the result is called the quotient.

For example, if you have 400 apples and you want to divide them equally into groups of 50, you will end up with 8 groups. Mathematically, this is represented as:
\[ \frac{400}{50} = 8 \]
Here, 400 is the dividend, 50 is the divisor, and 8 is the quotient. It's important to perform the division step by step and verify the result, which we'll cover in the next sections.

Negative Numbers in Division

Now let's tackle the scenario where you have a negative number involved in the division. With our exercise, we need to divide 400 by -50. When dealing with negative numbers in division, it's essential to remember the rules:

• Positive ÷ Positive = Positive
• Negative ÷ Positive = Negative
• Positive ÷ Negative = Negative
• Negative ÷ Negative = Positive

Given the exercise \[ \frac{400}{-50} = -8 \], the positive number (400) divided by the negative number (-50) results in a negative quotient (-8). Applying the rule ensures that our understanding of division with negative numbers is clear and accurate.

Verifying Division Results

After performing the division, it’s crucial to verify the quotient to ensure that the calculation is correct. This is done by reversing the division process: multiplying the quotient by the divisor. If your result equals the original dividend, your division is accurate.

For our example, the division gave us (-8) as the quotient. We verify it by multiplying the quotient by the divisor:
\[ -8 \times -50 = 400 \]
Since multiplying -8 by -50 indeed gives us the original number 400, our division is confirmed correct. This step helps reinforce understanding and ensures accuracy in mathematical operations.