Problem 50 - Algebra A Combined Function

Question

Perform the indicated operation. \(\frac{-45}{-9}\)

Step-by-Step Solution

Answer

The result of \(\frac{-45}{-9}\) is 5.

Step 1: Identify the operation

The exercise asks us to divide two numbers: -45 divided by -9. This is an example of division with negative numbers.

Step 2: Divide the absolute values

Start by ignoring the negative signs and dividing the absolute values of the numbers. So, divide 45 by 9. \[ 45 \div 9 = 5 \]

Step 3: Determine the sign of the result

Since you're dividing two negative numbers, according to the rules of division, the result will be positive. Therefore, the quotient from the division is positive.

Step 4: Final result

Combine the result from Step 2 and Step 3. The final answer to \(\frac{-45}{-9}\) is 5.

Key Concepts

Absolute Values

When you encounter negative numbers, it's often helpful to focus on their absolute values first. An absolute value measures how far a number is from zero, ignoring whether it's positive or negative. You can think of it as the number's size without considering direction. For instance:

• The absolute value of -45 is 45.
• The absolute value of -9 is 9.

This approach helps simplify division by focusing on basic math without worrying about signs just yet. Once you have divided the absolute values, you can deal with the signs separately.

Division Rules

Division rules guide us on the direction of the result's sign. When dividing, it's essential to understand how the sign of the numbers affects the quotient. Here’s a quick guide for division with signs:

• Positive divided by positive equals a positive result.
• Positive divided by negative gives a negative result.
• Negative divided by positive results in a negative.
• Negative divided by negative yields a positive outcome.

These rules are fundamental in math and help determine the final sign of any calculation involving negative numbers.

Quotient

In division, the quotient is the result of dividing one number by another. It tells you how many times the divisor fits into the dividend. In the problem \( \frac{-45}{-9} \), the absolute values \( 45 \) divided by \( 9 \) give a quotient of \( 5 \). This means -9 fits into -45 a total of 5 times.
Remember, this conclusion is based purely on the mathematical sizes of the numbers through absolute values. To finalize our result, we also need to incorporate the division rules for signs. For this exercise, that means applying the understanding that dividing two negative numbers results in a positive quotient.

Negative Signs

Negative signs can feel tricky, but their behavior is systematic. Whether multiplying or dividing, you can apply specific rules to handle them correctly. In our exercise, we had:
- a negative dividend (-45)
- a negative divisor (-9)

When you divide a negative by another negative, the negatives cancel each other out. This is why the negative signs in \( \frac{-45}{-9} \) result in a positive outcome.
The visual imagery is that two negatives make a positive, much like how two 'wrongs' in some contexts make a 'right'. Understanding this principle helps simplify and correctly evaluate similar exercises.