Problem 103 - Introductory and Intermediate Algebra for College Students 4th

Question

Translate from English to an algebraic expression or equation, whichever is appropriate. Let the variable \(x\) represent the number. \(\frac{1}{5}\) of a number

Step-by-Step Solution

Answer

\(\frac{1}{5}x\). This is the algebraic expression that represents \(\frac{1}{5}\) of a number

Step 1: Identify the Expression's Components

The statement \(\frac{1}{5}\) of a number mentions a fraction and a number. The word 'of' in mathematics often implies multiplication, especially when involving fractions and whole numbers.

Step 2: Assign the Variable

Let's denote the number mentioned in the phrase by the variable \(x\). The problem statement tells us to use \(x\) to represent this number.

Step 3: Translate to Algebra

The statement \(\frac{1}{5}\) of a number translated to algebraic form becomes \(\frac{1}{5} * x\), or simply \(\frac{1}{5}x\).

Key Concepts

Algebraic Expressions

Understanding algebraic expressions is critical for solving various types of mathematical problems. An algebraic expression is a combination of numbers, variables, and mathematical operations that represents a particular value or set of values. In algebra, we frequently encounter expressions like \(\frac{1}{5}x\), which was derived from the phrase '\frac{1}{5}' of a number'. This expression shows us how algebra can be used to translate words into a mathematical language.

To create an algebraic expression from an English statement, we must be on the lookout for keywords that indicate mathematical operations. For instance, the word 'of' often suggests multiplication, particularly when it's related to fractions or percentages. In the given exercise, '\frac{1}{5}' of a number translates to multiplying the fractional value \(\frac{1}{5}\) by the variable that represents the number. Hence, the algebraic expression \(\frac{1}{5}x\) correctly captures the mathematical essence of the original English phrase.

Mathematical Operations in Algebra

Identifying Mathematical Operations

When translating English to algebraic expressions, recognizing the mathematical operations involved is essential. The primary operations include addition, subtraction, multiplication, division, and exponentiation. In our example, we encountered multiplication, which we denoted using the asterisk (*) or sometimes implied as seen between \(\frac{1}{5}\) and \(x\).

Algebra allows for flexibility in writing mathematical operations. For instance, \(\frac{1}{5}*x\) can also be written as \(\frac{1}{5}x\) without the multiplication sign when there is no ambiguity. Students must remember that this convenience can result in different, yet equivalent, forms of the same expression, understanding that regardless of how it is written, it represents the same mathematical relationship.

Variables in Algebra

Variables play a pivotal role in algebra and are represented by letters such as \(x\), \(y\), or \(z\) that stand in for unknown numbers. In the exercise, \(x\) was assigned to be the number in question. Variables allow us to write general expressions that can be applied to many different situations. For example, the expression \(\frac{1}{5}x\) can apply to any value of \(x\); whether \(x\) is 2, 10, or 100, the expression provides a way to calculate \(\frac{1}{5}\) of that number.

When dealing with variables, it's essential to understand they are placeholders that can change or vary—which is why they are called 'variables.' It's also important for students to remember to define what variable represents in the context of a problem, as it ensures clarity and helps prevent mistakes while solving for or manipulating expressions.