Problem 44 - Introductory and Intermediate Algebra for College Students 4th

Question

In Exercises \(43-46\) a. Rewrite the division as multiplication involving a multiplicative inverse. b. Use the multiplication from part ( \(a\) ) to find the given quotient. \(-18 \div 6\)

Step-by-Step Solution

Answer

The quotient of the division \(-18 \div 6\) is \(-3\).

Step 1: Identify the division operation

Here, we have the division operation \(-18 \div 6\).

Step 2: Rewrite the division using the multiplicative inverse

Multiplicative inverse of 6 is \(1/6\). So, \(-18 \div 6\) is equivalent to \(-18 \times (1/6)\).

Step 3: Calculate the quotient

Now, execute the multiplication to get the quotient. \(-18 \times (1/6)\) equals \(-3\).

Key Concepts

Division Operation

When we talk about a division operation, we're discussing how to split something into equal parts. For example:

• Division helps us understand how many times one number can fit into another.
• In the division \(-18 \div 6\), we're asking how many times 6 can evenly "fit into" -18.

It's like distributing -18 apples to 6 friends, where each friend gets the same number of apples. To solve this, we utilize division, turning large problems into manageable pieces. Keep in mind, division can also result in a remainder, but for this problem, numbers divide evenly!

Multiplication

Multiplication is essentially repeated addition. If we multiply a number, we're adding it to itself a certain number of times. Consider the role of multiplication in solving division problems:

• When we rewrite division as multiplication by using a multiplicative inverse, we're seeking the same end result through a different method.
• Think of \(-18 \times (1/6)\) as "how many 1/6 fractions fit intoa whole number -18?"

Each step in multiplication, just like an addition, takes us closer to the answer, piece by piece. Understanding multiplication will make arithmetic less intimidating and more systematic.

Quotient

The name given to the result of a division operation is the quotient. Simply put:

• It tells us the number of times one number can be divided by another.
• In our problem, \(-18 \div 6\), the quotient is \(-3\).

Think of the quotient as the answer to your division question. It's what you're left with after dividing the dividend by the divisor. Grasping the concept of a quotient can make complex problems seem more approachable and less daunting.

Introductory Algebra

Introductory algebra introduces us to the concept of using letters and symbols to represent numbers and operations in order to solve problems. This foundational skill helps in understanding equations such as our exercise here:

• With algebra, we manipulate expressions to unveil the unknowns.
• In our exercise: rewriting \(-18 \div 6\) as \(-18 \times (1/6)\) provides a different perspective using algebraic concepts.

Algebra makes it possible to understand how components within a problem interact and lead to a solution, laying groundwork for more advanced mathematics.