Problem 57 - Elementary and Intermediate Algebra

Question

Divide, if possible, and check. If a quotient is undefined, state this. $$ -100 \div(-11) $$

Step-by-Step Solution

Answer

The quotient is approximately 9.0909.

Step 1 - Understand the Problem

The problem requires dividing \(-100\) by \(-11\). Division involving two negative numbers results in a positive quotient.

Step 2 - Perform the Division

Divide \(-100\) by \(-11\) using the formula for division of negative numbers: \[ \text{Quotient} = \frac{-100}{-11} = \frac{100}{11} = 9.0909\,... (approx.) \]

Step 3 - Check the Result

Multiply the quotient by \(-11\) to ensure the result is roughly \-100\. \[ 9.0909... * -11 = -100 \] Since the multiplication gives us approximately \-100\, the division was performed correctly.

Key Concepts

Division of negative numbers

When you divide two negative numbers, the result, known as the quotient, is positive. This is because the negative signs cancel each other out. For example, in the problem \( -100 \div (-11) \), both numbers are negative. To find the quotient, we can simplify it to \( \frac{-100}{-11} \). Since the negatives cancel out, we end up with \( \frac{100}{11} \), which is approximately 9.0909.
This rule helps simplify calculations involving negative numbers and makes it easier to ensure accuracy in your results.

Quotient calculation

The quotient is the result you get when you divide one number by another. In our example, we need to divide -100 by -11.

As mentioned earlier, when dividing two negative numbers, the quotient will be positive. Here's the calculation step-by-step:

• Write the division as a fraction: \( \frac{-100}{-11} \).
• Simplify by eliminating the negative signs: \( \frac{100}{11} \).
• Perform the division to get the approximate value of the quotient: 9.0909.

Simplifying and actually performing the division ensures that you get a clear and accurate result.

Checking division results

After calculating the quotient, it's important to check your result to ensure it's correct. In the context of division problems, you do this by multiplying the quotient by the divisor. If you get the original dividend, your solution is correct.

Let's use our example to check the result:

• The quotient we found was approximately 9.0909.
• Multiply the quotient by the divisor: \( 9.0909 \times -11 \).
• The result should be close to the original dividend: \( -100 \).

When you perform the multiplication \( 9.0909 \times -11 = -100 \), the result confirms that our division was performed correctly. Checking your work is a critical step in ensuring accuracy in math problems.