Problem 68 - Fundamentals of Mathematics

Question

Perform each operation. $$ \frac{-15}{-3} $$

Step-by-Step Solution

Answer

5

Step 1: Understand the Operation

The given problem involves division. You need to divide the numerator \(-15\) by the denominator \(-3\).

Step 2: Divide the Absolute Values

First, divide the absolute values of the numbers: \(15 \div 3 = 5\).

Step 3: Determine the Sign of the Result

When dividing two numbers with the same sign (both negative in this case), the result is positive. Hence, the quotient \(5\) is positive.

Step 4: Write the Final Answer

The result of dividing \(-15\) by \(-3\) is \(+5\). The complete expression evaluates to \(5\).

Key Concepts

Quotient

In division, the term "quotient" refers to the result obtained when one number is divided by another. It is the answer to a division problem. For example, in the expression \( \frac{-15}{-3} \), the quotient is the number you get after dividing \(-15\) by \(-3\). The idea of a quotient is central to understanding division:

• It illustrates how many times the divisor fits into the dividend.
• In this example, \(-3\) fits into \(-15\) a total of 5 times.

Finding the quotient involves several steps. The absolute values of the numbers are divided first, and the sign of the quotient is determined by the signs of the numbers involved in the division, as discussed in the steps below.

Negative Numbers

Negative numbers are numbers less than zero, symbolized by a minus sign (\(-\)). They can be found on the left side of the number line. When performing division with negative numbers, understanding how their signs affect the outcome is crucial. Here's how to handle negative numbers in division:

• Two negative numbers divide to yield a positive quotient.
• If one number is negative and the other is positive, the quotient is negative.

In our example, both \(-15\) and \(-3\) are negative. Thus, dividing them results in a positive quotient of \(5\). This important concept ensures that the rules of division remain consistent with mathematical principles involving signs.

Absolute Values

The absolute value of a number is its distance from zero on the number line, regardless of direction. It is always non-negative. When calculating division, working with absolute values can simplify the process.Here are key points about absolute values in division:

• Convert each number to its absolute value (ignore the sign).
• Perform the division with these absolute values as if they were positive.
• For the division \( \frac{-15}{-3} \), use the absolute values (15 and 3).
• After finding the quotient (5), apply the correct sign using the rules for negative numbers.

Understanding absolute values is vital for accurately determining the magnitude of division results, independent of their original signs. This step removes any confusion about signs until the final result is assessed.