Problem 67 - Introductory Algebra for College Students

Question

Perform the indicated division or state that the expression is undefined. $$\frac{-12.9}{3}$$

Step-by-Step Solution

Answer

-4.3

Step 1: Identify the Division

Here you can see that the problem requires dividing -12.9 by 3. A division operation requires taking one value -12.9, and seeing how many times the second value 3 can 'fit' inside it.

Step 2: Perform the Division

Now it is time to carry out the division. Hence, \( \frac{-12.9}{3} \) is equal to -4.3.

Step 3: Verify the Result

Finally, you must ensure the calculated result is accurate by multiplying the quotient back to the divisor and check if the original dividend is obtained. The exercise states the result is undefined if the division cannot be performed, but in this case, as the division is possible and has been accurately completed, the division operation's solution is -4.3.

Key Concepts

Division of Real Numbers

Division is the process of determining how many times one number, the divisor, can fit into another number, the dividend. In this case, we divide \( -12.9 \) by \( 3 \). To solve this, we calculate how many times \( 3 \) goes into \( -12.9 \). Since both are real numbers, this operation can be performed directly.
When calculating \( \frac{-12.9}{3} \), you determine how the dividend is shared equally among the divisor's value. The result here is \(-4.3\), which is the quotient. Real numbers, in this context, include all the numbers you typically deal with: positive numbers, negative numbers, and zero. Any division of real numbers is defined unless you are faced with division by zero, which is undefined because you cannot divide any number into zero parts.

Negative Numbers

Negative numbers are used to represent values less than zero. In this exercise, the dividend \(-12.9\) is negative, while the divisor \(3\) is positive.
When dividing a negative number by a positive number, the result will also be negative. This rule follows the sign convention:

• Positive divided by Positive = Positive
• Negative divided by Negative = Positive
• Negative divided by Positive = Negative
• Positive divided by Negative = Negative

This understanding helps in predicting the sign of the result even before performing the arithmetic operation. Thus, the division of \(-12.9\) by \(3\) results in \(-4.3\). Understanding and working with negative numbers is crucial in algebra as they are commonly used in various calculations and equations.

Arithmetic Operations

Arithmetic operations are the basic calculations we perform with numbers: addition, subtraction, multiplication, and division. Each of these has specific rules and properties.
In the context of division, particularly with decimals or negative numbers, it's important to apply the rules carefully. Arithmetic operations help to transform expressions and find specific solutions.
For students learning to perform division, it's essential to:

• Identify the dividend and divisor correctly.
• Understand whether the numbers involved are positive or negative.
• Apply rules of division and check your work to ensure accuracy.

In our exercise, dividing \(-12.9\) by \(3\), we broke it down into smaller parts to ensure correct calculation. This operation helps reinforce arithmetic skills and offers a clearer understanding of how to handle decimals and negative numbers effectively.