Problem 34 - Algebra A Combined Function

Question

Divide. \(\frac{36}{-9}\)

Step-by-Step Solution

Answer

The quotient of \(\frac{36}{-9}\) is -4.

Step 1: Understanding the Problem

The question asks us to divide 36 by -9. This means we need to find out how many times -9 fits into 36, or we need to calculate the result of dividing these two numbers.

Step 2: Identify the Sign

Division involves not only division of magnitude but also ensuring the correct sign of the result. In this case, we are dividing a positive number (36) by a negative number (-9). When dividing a positive and a negative number, the result will be negative.

Step 3: Perform the Division

Now, let's perform the division of the absolute values of the numbers. We divide 36 by 9. Calculate: \[36 \div 9 = 4\]So, the absolute value of our result is 4.

Step 4: Apply the Sign to the Quotient

Since we determined in step 2 that the result should be negative, we apply the negative sign to our quotient. Therefore, the division of 36 by -9 is -4.

Key Concepts

Positive and Negative Numbers

Understanding positive and negative numbers is crucial when solving problems involving integer division. These numbers are defined based on their position relative to zero on the number line:

• Positive numbers are greater than zero (e.g., 1, 2, 36).
• Negative numbers are less than zero (e.g., -1, -9, -36).

When you divide a positive number by a positive number, the result is positive. Similarly, dividing a negative number by a negative number also yields a positive result. However, when you divide numbers of different signs—one positive and one negative—the result is negative. Understanding this concept helps in predicting the sign of the result in division problems.

Division Sign Rules

In mathematics, division sign rules help determine the sign of the quotient. When dividing two numbers, it's important to consider these rules to get the correct sign:

• If both numbers (dividend and divisor) have the same sign (both positive or both negative), the result is positive.
• If the numbers have different signs (one positive and one negative), the result is negative.

These rules are consistent for any pair of numbers, ensuring we can always determine the sign of the answer simply by comparing the signs of the numbers involved. For instance, in our original problem where 36 is divided by -9, the dividend is positive and the divisor is negative, so the result is negative. This consistency helps provide predictability in calculations and ensures accuracy when solving division exercises.

Absolute Value

The concept of absolute value is fundamental in handling division problems involving negative numbers. The absolute value of a number is its distance from zero on a number line, ignoring the direction or sign of the number. This is an important tool, especially in division, because it allows us to focus on magnitude first:

• The absolute value of a positive number is the number itself (e.g., the absolute value of 36 is 36).
• The absolute value of a negative number is the opposite of that number (e.g., the absolute value of -9 is 9).

When performing division like \(36 \/ -9\), calculate the division using the absolute values of both numbers first. Divide 36 by 9 to get 4. Then, apply the appropriate sign based on the division rules to ensure the correct result, which in this case is -4. By understanding and applying absolute value, you can quickly find the magnitude of the quotient before addressing the sign.