Problem 39 - Intermediate Algebra

Question

Perform the operations. See Examples 4 and 5 . $$ \frac{-8}{4} $$

Step-by-Step Solution

Answer

The result of \( \frac{-8}{4} \) is -2.

Step 1: Identify the Operation

We are given the expression \( \frac{-8}{4} \). This expression represents the division of two numbers: -8 (the numerator) and 4 (the denominator). Our task is to divide the numerator by the denominator.

Step 2: Divide the Numerator by the Denominator

Divide the numerator -8 by the denominator 4. Carry out the division as follows: \( \frac{-8}{4} = -2 \). The quotient of -8 divided by 4 is -2.

Step 3: Simplify the Result

Verify that the result from the division cannot be simplified further. Since -2 is already in its simplest form, no further simplification is needed.

Key Concepts

Numerator

In a fraction, the numerator is the number that sits above the division line. It represents the number of parts we have. In the expression \( \frac{-8}{4} \), the numerator is -8. This part of the fraction is crucial as it tells us about the quantity or value that we are dividing into smaller sections. Here, -8 is being divided. We can think of the numerator as the subject of division, indicating what is being divided.

Denominator

The denominator is the number found below the fraction line. It tells you into how many parts the whole is divided. In our fraction, \( \frac{-8}{4} \), the denominator is 4. This means that -8 is being divided into 4 equal parts.

Understanding denominators is key in division as they determine the size of each part when performing division. If the denominator changes, the size of each portion also changes, altering the result of the division.

Simplification

Simplification in mathematics often refers to reducing fractions or expressions to their simplest form. This means that the fraction has been reduced in such a way that both the numerator and the denominator have no common factors other than 1.

In the case of \( \frac{-8}{4} \), the solution gives us -2. Here, the fraction is simplified by performing the division. Since -2 is a whole number, it indicates that the fraction has been reduced to its simplest form, confirming that -8 divided by 4 gives us -2 directly. Simplification makes expressions easier to understand and work with.

Quotient

The quotient is the result you get after you divide the numerator by the denominator. In the expression \( \frac{-8}{4} \), when we divide, we get -2 as the quotient.

• This result represents the number of times the denominator can fit into the numerator.
• It also tells you how many equal parts you end up with after the division.

In arithmetic, understanding the quotient helps you see the end result of a division procedure, giving you a clear answer to a division problem.