Problem 42 - Beginning and Intermediate Algebra

Question

Find each quotient. \(\frac{38}{-19}\)

Step-by-Step Solution

Answer

-2

Step 1: Identify the Numbers

In this step, identify the numbers involved in the division. Here, the two numbers are 38 and -19.

Step 2: Consider the Signs

When dividing numbers with different signs, the quotient will be negative. In this case, you have positive 38 and negative -19, so the result will be negative.

Step 3: Divide the Absolute Values

Ignore the signs for a moment and divide the absolute values of the numbers. Calculate \(\frac{38}{19}\). The result is 2.

Step 4: Apply the Correct Sign

Since you are dividing a positive number by a negative number, the final quotient should be negative. Therefore, \(\frac{38}{-19} = -2\).

Key Concepts

Quotient

Let's start by understanding what a quotient is. A quotient is the result you get when you divide one number by another. In the exercise provided, you are asked to find the quotient of \(\frac{38}{-19}\). This simply means, you need to see how many times one number (the divisor) goes into another number (the dividend). In this case, the dividend is 38 and the divisor is -19.
The quotient can be a whole number, a fraction, or even a negative number, depending on the numbers you are dividing.

Absolute Values

Absolute value is the non-negative value of a number without considering its sign. It tells you how far a number is from zero. For instance, the absolute value of both 38 and -19 is 38 and 19, respectively.
When you need to divide numbers, considering absolute values can simplify the process. In our exercise, even though we are dealing with 38 and -19, we temporarily ignore the signs and consider only 38 and 19. This makes our division straightforward: \(\frac{38}{19} = 2\).
Only after dividing the absolute values do we reapply the sign based on the original numbers.

Sign Rules

When working with integer division, knowing the sign rules is essential. The sign of the quotient depends on the signs of the dividend and divisor.
Here are the rules to remember:

• Positive ÷ Positive = Positive
• Negative ÷ Negative = Positive
• Positive ÷ Negative = Negative
• Negative ÷ Positive = Negative

In the provided exercise, 38 is positive, and -19 is negative. According to our sign rules, a positive number divided by a negative number will result in a negative quotient. That is why \(\frac{38}{-19} = -2\).