Problem 44 - Intermediate Algebra

Question

Perform the operations. See Examples 4 and 5 . $$ \frac{-13.5}{-1.5} $$

Step-by-Step Solution

Answer

The result of \(\frac{-13.5}{-1.5}\) is 9.

Step 1: Identify the Operation

The problem involves dividing two numbers, specifically dividing \(-13.5\) by \(-1.5\). This is a division operation.

Step 2: Understand the Division Rule for Signs

When dividing two negative numbers, the quotient is positive. So, \(\frac{-a}{-b} = \frac{a}{b}\). Therefore, this operation simplifies to \(\frac{13.5}{1.5}\).

Step 3: Perform the Division

Now, divide 13.5 by 1.5. To simplify, notice that 1.5 fits into 13.5 exactly 9 times. Thus, \(\frac{13.5}{1.5} = 9\).

Key Concepts

Understanding Negative Numbers

Negative numbers are numbers that are less than zero. They are represented with a minus sign (-) in front of them. Negative numbers have unique properties that distinguish them from positive numbers. For example, when you multiply or divide two negative numbers, the result is a positive number.
Negative numbers are often used to represent values that are below a set reference point, like sea level or temperatures below zero. In mathematics, using negative numbers allows us to broaden our ability to calculate and understand different types of problems.

The Meaning of Quotient

In division, the quotient is the result you get when you divide one number by another. It's the answer to a division problem. For instance, if you divide 10 by 2, the quotient is 5. In our exercise, \(\frac{-13.5}{-1.5}\), the quotient is calculated by performing the division operation on the numerical values.

• The divisor is the number you divide by. In \(\frac{-13.5}{-1.5}\), \-1.5\ is the divisor.
• The dividend is the number divided, which in our example is \(-13.5\).
• The quotient is the answer that you find on performing the division, which turns out to be 9.

Understanding what the quotient represents is vital. It provides a snapshot of how many times the divisor fits into the dividend.

Division Rule for Signs

The division rule for signs is an important principle in mathematics, especially when dealing with negative numbers. The rule dictates the sign of the quotient based on the signs of the numbers being divided:

• When you divide two positive numbers, the quotient is positive.
• When you divide two negative numbers, the quotient is also positive. This aligns with our exercise result where \(\frac{-13.5}{-1.5}\) turns into \(\frac{13.5}{1.5} = 9\).
• If one number is positive and the other is negative, the quotient will be negative.

Memorizing this rule helps in quickly determining the sign of the answer without needing to rely only on computational steps.