Problem 20 - Elementary Algebra Within Reach

Question

In Exercises \(11-22\), translate the verbal phrase into an algebraic expression. $$ d \text { divided by } 100 $$

Step-by-Step Solution

Answer

The algebraic expression for the given verbal phrase is \(\frac{d}{100}\)

Step 1: Identify the Variables

We recognize 'd' as the variable from the verbal phrase.

Step 2: Identify the Operation

The phrase 'divided by' is an indication that we are dealing with the division operation.

Step 3: Form the Algebraic Expression

Given the variable 'd' and the operation 'divided by', followed by the number '100', the algebraic expression is \(\frac{d}{100}\).

Key Concepts

Verbal Phrases

Verbal phrases in mathematics are groups of words that translate into mathematical operations. Identifying and understanding these phrases is key to translating everyday language into algebraic expressions. In this context, the verbal phrase "divided by" points to a division operation in mathematical terms. Here, the expression " \( d \text{ divided by } 100 \) " is a clear verbal phrase. It indicates that the variable \( d \) should be divided by the number \( 100 \). This translation from words to symbols helps in forming algebraic expressions used in problem-solving. Verbal phrases often include keywords like “sum,” “difference,” “product,” and “quotient,” each representing a different mathematical operation.

Variables in Algebra

A variable in algebra is a symbol, often a letter, that represents an unknown value or can represent multiple values. In the exercise given, the letter \( d \) is the variable. It stands for a value that can change or that we might need to find. Variables allow us to write expressions and equations that refer to numbers in a general way. Using variables can help in solving problems flexibly, enabling us to describe mathematical relationships efficiently.

• Variables are not fixed but placeholders for data values.
• They enable you to create expressions applicable in various scenarios.
• Commonly used letters in algebra are \( x, y, z, \) and \( d, \) as seen in this example.

Variables are fundamental in forming equations and expressing relationships between different quantities.

Division Operation in Algebra

The division operation in algebra takes the form of dividing one quantity by another. This operation is represented by the division symbol \( \div \) or a slash \( / \), but in algebraic expressions, it often appears as a fraction. In this exercise, the expression \( \frac{d}{100} \) translates the verbal phrase " \( d \text{ divided by } 100 \) ". It means you are taking the value of \( d \) and splitting it equally into \( 100 \) parts. Understanding division in algebra involves recognizing that:

• It indicates how many times one number is contained within another.
• Fractions are often a more compact representation of division in algebra.
• Dividing by \( 100 \) implies finding a part per hundred or a percentage.

This concept is crucial for solving problems involving rates, proportions, and other scenarios where division is applied to real-world situations.