Problem 19 - Elementary Algebra Within Reach

Question

In Exercises \(11-22\), translate the verbal phrase into an algebraic expression. $$ w \text { divided by } 3 $$

Step-by-Step Solution

Answer

The algebraic expression for the verbal phrase 'w divided by 3' is \(w / 3\).

Step 1: Identify the Variable and Number

The verbal phrase provides us with the variable 'w' and the number '3'.

Step 2: Understand the Meaning of 'Divided By'

In algebraic expressions, 'divided by' is represented by the symbol '/'.

Step 3: Form the Algebraic Expression

Combine the variable, the 'divided by' symbol, and the number to form the algebraic expression. The variable 'w' is divided by '3', which can be represented as \(w / 3\).

Key Concepts

Translation of Verbal Phrases

Translating verbal phrases into algebraic expressions is a foundational skill in algebra that allows us to convert words into mathematical equations or expressions. This process involves turning descriptive phrases into symbols and numbers.
Consider the phrase "w divided by 3":

• The word "divided by" indicates that we need to perform division.
• The letter 'w' represents the variable involved in the division.
• The number '3' is the divisor, which is the number by which the variable is divided.

By identifying keywords and numbers, we can transform verbal phrases into a more concise and mathematical form that is easier to analyze and solve.

Variables in Algebra

Variables are symbols used in algebra to represent unknown values or quantities that can vary. They are typically denoted by letters such as x, y, or in this case, w.
An important aspect of working with variables is understanding that:

• They act as placeholders for numbers.
• They allow for generalization in mathematical formulas and expressions.
• They enable us to solve equations and explore relationships between different quantities.

In algebraic expressions like "w divided by 3", the variable 'w' can represent a wide range of possible values, making it a powerful tool for modeling real-world situations.

Division in Algebra

Division in algebra involves splitting a quantity into equal parts and is represented using the division symbol '/'. Understanding how to perform division with variables is crucial for solving various types of algebraic problems.
Here is how you can approach division with variables:

• Identify the dividend (the number or variable being divided) and the divisor (the number by which you divide).
• Write the algebraic expression in a clear, simplified form that typically involves the dividend followed by the division symbol and then the divisor.

For example, when the phrase "w divided by 3" is translated into an algebraic expression, it becomes \( w / 3 \), indicating that the variable w is split into three equal parts. This reasoning helps in maintaining clarity and precision in mathematical communication.