Problem 39 - Introductory Algebra for College Students

Question

Write each English phrase as an algebraic expression. Let the variable \(x\) represent the number. the sum of 10 divided by a number and that number divided by 10

Step-by-Step Solution

Answer

The algebraic expression for the English phrase given is \(10/x + x/10.\)

Step 1: Translate English to Math - Part 1

The phrase 'the sum of' signifies addition. So, whatever comes next will be added together.

Step 2: Translate English to Math - Part 2

The phrase '10 divided by a number' translates to \(10/x\). 'Divided by' implies division, and since 'a number' is represented by the variable \(x\), this becomes \(10/x\).

Step 3: Translate English to Math - Part 3

Next, the phrase 'that number divided by 10' translates to \(x / 10\). Again 'divided by' stands for division, and 'that number' again refers to \(x\), hence making it \(x/10\).

Step 4: Final Translation

The final step involves putting these translations together: 'the sum of' \(10/x\) and \(x/10\). In algebraic expression, this becomes \(10/x + x/10.\)

Key Concepts

Algebraic Expressions

In the realm of mathematics, an algebraic expression is a combination of numbers, variables, and operations. These expressions are the foundation of algebra and are used to represent real-world situations in mathematical terms.

For instance, the exercise involves converting the English phrase 'the sum of 10 divided by a number and that number divided by 10' into an algebraic expression. The process requires a good understanding of the keywords in English that correspond to mathematical operations. 'The sum of' suggests addition, and 'divided by' indicates division. When applied, these operations help us structure the expression with the use of variables to accurately depict the relationship described in words.

The resulting algebraic expression from the provided exercise is \(10/x + x/10\), which succinctly combines the operations of addition and division within one mathematical sentence.

Variables in Algebra

In algebra, variables act as placeholders for numbers that may vary or that we do not yet know. They are often represented by letters such as \(x\), \(y\), or \(z\). Understanding how to use variables is crucial when translating English phrases into algebraic expressions.

Consider our exercise, where 'a number' is unknown. By representing it with a variable, \(x\), we convey flexibility in that expression can represent any number. This abstraction is essential in algebra, as it allows mathematicians and students alike to work with not just specific numbers but entire classes of numbers at once.

Learning to identify when a word or phrase represents a variable and what operation it entails is key. It's also vital to understand the convention of using lowercase letters for variables, while uppercase letters often denote constants or specific coefficients.

Basic Algebra Concepts

Beginning with basic algebra concepts is fundamental for anyone diving into the study of algebra. It involves understanding the role of operations (addition, subtraction, multiplication, and division) and how they interact with variables to form expressions and equations.

In our exercise, the English phrases needed to be deciphered into mathematical operations and represented with variables. The translation process hinges on a few basic algebra principles. Firstly, identifying keywords that signal specific mathematical operations, such as 'the sum of' for addition, and translating them correctly. Secondly, recognizing that algebraic expressions follow specific rules of order—meaning you should always respect the mathematical order of operations when interpreting phrases. Lastly, the recognition that a variable can represent an unknown number, allowing the expression to remain valid for all possible values of that number.

Students should consistently practice these fundamentals to sharpen their problem-solving skills and gain confidence in translating verbal descriptions into algebraic language accurately.