Problem 105 - 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. A number decreased by \(\frac{1}{4}\) of itself

Step-by-Step Solution

Answer

\(\frac{3}{4}*x\)

Step 1: Identify the Number

The number being referred to in the problem is \(x\).

Step 2: Identify the Operation

The problem mentions 'a number decreased by', which implies subtraction. Further, '\(\frac{1}{4}\) of itself' refers to \(\frac{1}{4}*x\).

Step 3: Formulate the Algebraic Expression

Subtract \(\frac{1}{4}\) of the number \(x\) from the number itself which is \(x - \frac{1}{4}*x\).

Step 4: Simplify the Algebraic Expression

By finding the common factor of \(x\), we can simplify to \(x*(1 - \frac{1}{4}) = \frac{3}{4}*x\).

Key Concepts

Variable Representation

When dealing with algebraic expressions, understanding variable representation is crucial. In a problem, certain phrases or descriptions are used to describe unknown quantities. Here, the unknown number we are referring to is represented by the variable \( x \). Variables are typically letters used to stand in for values that can change or that are not yet known. By selecting \( x \) as our variable, we assign this letter the role of representing the number mentioned in the problem.
Consider variables as placeholders, just like how we use boxes to store items. Variables allow us to perform mathematical operations and transformations just as we might manipulate those items in physical boxes. This flexibility makes it important to carefully choose and consistently use variable names as we solve problems.

Translation from Words to Expressions

The process of translating words into algebraic expressions involves careful reading of the given problem and understanding which mathematical operations are implied. In our example problem, the phrase "a number decreased by" suggests subtraction. Here, we are told a number is decreased by "\( \frac{1}{4} \) of itself".
The expression "\( \frac{1}{4} \) of itself" means we take a quarter of the unknown number \( x \). Mathematically, this is written as \( \frac{1}{4} \times x \). Therefore, when translating the sentence into an algebraic expression, we represent the entire phrase as \( x - \frac{1}{4} \times x \). This step involves understanding the language of mathematics and how verbal descriptions convert to numbers and operations, making algebra both a language and a tool for solving problems.

Simplifying Expressions

Simplifying expressions is about making them easier to understand and work with. It involves reducing expressions to their simplest form by carrying out operations and combining like terms.
In the example, we have the expression \( x - \frac{1}{4} \times x \). Here, both terms involve the variable \( x \), which allows us to factor it out. Factoring is like finding a common element in all terms; in this case, \( x \) is the common factor:

• First, rewrite the expression as a product: \( x \times (1 - \frac{1}{4}) \).
• Next, compute \( 1 - \frac{1}{4} \) to simplify further. \( 1 \) can be represented as \( \frac{4}{4} \), so \( 1 - \frac{1}{4} = \frac{4}{4} - \frac{1}{4} = \frac{3}{4} \).

This gives us the simplified expression \( \frac{3}{4} \times x \). Simplification helps us not only solve problems more easily but also understand deeper connections and structure within algebraic expressions.