Problem 116 - Introductory Algebra for College Students

Question

Write a numerical expression for each phrase. Then simplify the numerical expression by performing the given operations. The difference between \(-11\) and the quotient of 20 and \(-5\)

Step-by-Step Solution

Answer

The simplified numerical expression is \(-7\)

Step 1 - Translate the Phrase into a Mathematical Expression

The difference between -11 and the quotient of 20 and -5 can be translated into the following mathematical expression: \(-11 - (20/-5)\)

Step 2 - Perform the Division

Next, perform the division operation in the parenthesis first due to the rule of operators precedence. So the expression becomes: \(-11 - (-4)\)

Step 3 - Perform the Subtraction

Lastly, perform the subtraction operation. Subtracting a negative number is the same as adding its absolute value, which makes the final expression: \(-11 + 4\)

Step 4 - Simplify the Expression

The result from the addition is: \(-7\)

Key Concepts

Understanding the Difference

In math, the term "difference" refers to the result of subtracting one number from another. The phrase "the difference between A and B" translates to the numerical expression \(A - B\). Here, it's crucial to understand what each part represents:

• A: The first number or term in the expression - the minuend.
• B: The second number or term being subtracted - the subtrahend.

The expression indicates how much one number differs from another. In our exercise, the difference is between \(-11\) and another value (identified as the quotient of 20 and \(-5\)), symbolically written as \(-11 - (20 / -5)\). We start solving such problems by calculating the inner expressions first, as informed by operator precedence rules.
The concept of difference is integral to understanding subtraction in algebra and other areas of mathematics.

What is a Quotient

The quotient in mathematics is the result of dividing one quantity by another. When you hear "the quotient of C and D," it usually means dividing C by D, mathematically expressed as \(C / D\). Here's a breakdown to make this crystal clear:

• C: The divisor, or number being divided.
• D: The dividend, or number by which C is divided.

In our given expression, the quotient of 20 and \(-5\) is the calculation that simplifies to \(-4\) because \(20 / -5 = -4\). Understanding a quotient is crucial since division as one of the four basic arithmetic operations frequently occurs in numerous math problems, and it plays a vital role in the order of operations.

Division Operation

Division is one of the fundamental operations in arithmetic. It involves splitting a quantity into equal parts. Let's explore the basics further:

• Dividend: The number you're dividing up (e.g., 20 in our example).
• Divisor: The number you're dividing by (e.g., -5 in our example).
• Quotient: The result or outcome of the division (e.g., -4, here).

To solve the division operation \(20 / -5\), we divide the positive 20 by the negative 5, resulting in a quotient of \(-4\). Performing division with negative numbers follows specific rules: a positive number divided by a negative number yields a negative result, and vice versa.
Practicing division helps develop critical skills needed in solving complex mathematical problems.

Operator Precedence

Operator precedence is a core concept in mathematics that dictates the order in which operations should be performed to accurately solve expressions. Without this guideline, calculations could yield incorrect results. In the world of math:

• Parentheses: First calculations using items enclosed in parentheses ( \(( )\) ). This step sets the ground for accurate solving of equations.
• Exponents: Solve powers before other operations.
• Multiplication and Division: Handle these from left to right as they appear in the expression.
• Addition and Subtraction: Finally, perform these operations, also from left to right.

In our example, the phrase translated into
\(-11 - (20 / -5)\) reveals the importance of starting with the division within the parentheses first. By doing the dividing first, we get \(-4\), which simplifies our comprehensive expression to
\(-11 - (-4)\). This detail ensures that mathematical expressions are solved efficiently and without error.