Problem 113 - Introductory Algebra for College Students

Question

Write a numerical expression for each phrase. Then simplify the numerical expression by performing the given operations. The quotient of \(-18\) and the sum of \(-15\) and 12

Step-by-Step Solution

Answer

The simplified numerical expression is 6.

Step 1: Translate the phrase to a mathematical expression

The word 'quotient' means division, 'and' means plus in this context, and 'the sum of' means to take the result of adding two numbers. Thus, 'The quotient of -18 and the sum of -15 and 12' translates into the expression: -18 ÷ (-15 + 12).

Step 2: Apply the order of operations

The order of operations instructs us to perform any operations inside parentheses before doing division. Here inside the parentheses we have (-15 + 12). So, -15 + 12 = -3. Thus, the expression simplifies to: -18 ÷ -3.

Step 3: Perform the division

Finally, divide -18 by -3 to get 6. Hence, the simplified numerical expression is 6.

Key Concepts

Understanding the Concept of Quotient

When faced with the word "quotient" in a mathematical problem, it's essential to recognize it as a signal for division. But what exactly does quotient mean? In simplest terms, the quotient is the result you get when you divide one number by another. For example, in the expression \(-18 \div (-3)\), the number \(-18\) is the dividend, \(-3\) is the divisor, and the result of this division, which is \(6\), is the quotient.

Understanding how to identify these components in word problems is crucial for transforming a sentence into a mathematical expression that you can work with. In our given problem, the phrase "the quotient of \(-18\) and the sum of \(-15\) and \(12\)" points directly towards a division operation.

Mastering the Order of Operations

The order of operations is a fundamental concept you will often use when dealing with numerical expressions. It is sometimes remembered by the acronym PEMDAS: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).

In our problem, the first step involves dealing with the operations inside the parentheses. Inside the parentheses, \(-15 + 12\) leads to \(-3\). Only after you perform operations in brackets should you proceed with division as instructed by our expression, \(-18 \div (-3)\).

• Parentheses first: Simplify expressions within them.
• Follow up with division or multiplication next.
• Finish with addition or subtraction.

By applying these rules carefully, you'll always ensure that you do not make mistakes in simplifying complex expressions.

Simplifying Expressions Made Easy

Simplifying expressions is the process of making them easier to work with, by reducing all operations to their simplest form. This involves regulating steps like combining equal terms and performing operations like addition, subtraction, multiplication, and division.

In the example given, the expression \(-18 \div (-3)\) can be simplified by thinking of how the negative sign affects division. Since dividing two negative numbers results in a positive number, \(-18 \div (-3) = 6\).

• Combine like terms to make expressions simpler.
• Follow the order of operations for accuracy.
• Simplify progressively from the complex to the straightforward.

Learning to simplify expressions ensures that you can solve problems more efficiently and accurately, offering clearer insights into the results.