Problem 55 - Elementary and Intermediate Algebra

Question

Divide, if possible, and check. If a quotient is undefined, state this. $$ \frac{-50}{5} $$

Step-by-Step Solution

Answer

The division results in \(-10\).

Step 1 - Identify the Dividend and Divisor

In the given problem, \(\frac{-50}{5}\), \(-50\) is the dividend and \(5\) is the divisor.

Step 2 - Check for Undefined Quotient

A quotient is undefined if the divisor is zero, because division by zero is not possible. In this case, the divisor is \(5\), which is not zero. Therefore, the quotient is defined.

Step 3 - Perform the Division

Now, divide the dividend by the divisor: \(-50 \div 5 = -10\).

Step 4 - Check the Result

To verify the result, multiply the quotient by the divisor: \(-10 \times 5 = -50\). Since the product is equal to the original dividend, the division was performed correctly.

Key Concepts

dividing negative numbers

When dividing negative numbers, there are essential rules to follow. One common situation is dividing a negative number by a positive number, as seen in \(\frac{-50}{5}\). The result of such a division is always negative.
Here's why:
\(-50\) (negative) \div 5 (positive) yields -10 (negative)\. To understand this rule, imagine removing groups of 5 from -50. Each group represents a negative increment, thus, the result is negative.
This is different from:

• \(50 \div -5\) = -10
• \(-50 \div -5\) = 10
• Division where both numbers are negative yields a positive result.

checking division results

Verifying your division results is crucial to ensure accuracy. After calculating \(\frac{-50}{5} = -10\), you should multiply the quotient by the divisor:
10 x 5 = 50
When the original numbers include negatives, this becomes:

• -10 x 5 = -50

If the product equals the original dividend, the calculation is correct. This method confirms the reliability of the division outcome, ensuring no errors occurred.

undefined quotient

An undefined quotient occurs when the divisor is zero. Division by zero is mathematically impossible and undefined because it doesn't follow the rules of arithmetic.
Let's check our exercise \(\frac{-50}{5}\):

• Dividend: -50
• Divisor: 5 (non-zero)

Since the divisor is not zero, the quotient is defined. However, be cautious with scenarios such as: \(\frac{-50}{0} \). This is undefined because you can't distribute any number into zero parts.