Problem 64 - Elementary and Intermediate Algebra

Question

Divide. See Example 5. $$ \frac{-13.5}{-1.5} $$

Step-by-Step Solution

Answer

The result is 9.

Step 1: Identify the division operation

The exercise asks us to divide \(-13.5\) by \(-1.5\). It's important to recognize that both numbers are negative, which will affect the sign of the result.

Step 2: Simplify the division of absolute values

Ignore the negative signs for a moment and divide the absolute values of the numbers. Compute \(\frac{13.5}{1.5}\). To do this, you can treat the division problem as a question of how many times 1.5 fits into 13.5.

Step 3: Perform the division

Calculate \(13.5 \div 1.5\). Start by interpreting this as \(135 \div 15\). Recognize that 15 goes into 135 exactly 9 times, since 15 times 9 gives 135.

Step 4: Consider the sign of the result

Recall that dividing two negative numbers gives a positive result. Therefore, the division \(\frac{-13.5}{-1.5}\) results in a positive number.

Step 5: Write down the final result

Based on our calculations and sign considerations, \(\frac{-13.5}{-1.5} = 9\).

Key Concepts

Understanding Absolute Value

The absolute value of a number is its distance from zero on the number line, without considering its direction. This means, it is always a positive number or zero. For example, the absolute value of both -3 and 3 is 3, because both are three units away from zero.

Important points to remember about absolute values:

• Always consider absolute values to be non-negative.
• They help in simplifying calculations, as seen in our exercise where \(-13.5\) and \(-1.5\) become 13.5 and 1.5.
• The absolute value is denoted by two vertical bars, like this: \(|x|\).

Sign Rules for Division

When dividing numbers, the sign of the result depends on the signs of the numbers you are dividing. It is crucial to understand this rule to avoid confusing your results. The sign rules for division are straightforward:

• If you divide two numbers with the same sign (both positive or both negative), the result is positive.
• If you divide two numbers with opposite signs (one positive and one negative), the result is negative.

In this exercise, since both -13.5 and -1.5 are negative, their division results in a positive number, which is why the final answer is 9.

Navigating Negative Numbers

Negative numbers are numbers with a minus sign, indicating they are less than zero. They are typically found on the left of zero on a number line.

Some important properties of negative numbers include:

• Adding a negative number is like subtracting its positive counterpart.
• Subtracting a negative number is like adding its positive counterpart.
• Multiplying or dividing two negative numbers results in a positive number.
• Multiplying or dividing a positive number by a negative number results in a negative number.

In dividing \(-13.5\) by \(-1.5\), the negative signs cancel each other, leaving us with a positive result.

Simplifying Fractions

Simplifying fractions involves reducing them to their simplest form such that the numerator and the denominator are as small as possible while still accurate in ratio. The process includes:

• Finding the greatest common factor (GCF) of the numerator and the denominator.
• Dividing both the numerator and the denominator by their GCF.

In terms of dividing \(-13.5\) by \(-1.5\), we considered the fraction of 13.5 over 1.5 without their signs initially to simplify our work using ratios easily, ultimately finding how 15 goes into 135 nine times. This concept helps in solving rational number operations swiftly by breaking down complex fractions into more manageable pieces.