Problem 106 - 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 \(\frac{1}{3}\) of itself

Step-by-Step Solution

Answer

The algebraic expression is \(x - \frac{1}{3}x\).

Step 1: Represent the unknown number

Firstly, the problem states 'A number' which is unknown. We'll represent this number by the variable 'x'.

Step 2: Identify the operation

Next, understand that 'decreased by' implies subtraction in mathematics. It means we have to subtract something from 'x'.

Step 3: Translate 'a third of itself'

'A third of itself' refers to dividing 'x' by 3, or \(\frac{1}{3}\) of 'x'. This is what we'll subtract from 'x'.

Step 4: Construct the algebraic expression

We then form the complete algebraic expression by combining steps 1, 2, and 3. That is, 'x' (the number) decreased by \(\frac{1}{3}\) of 'x' (itself). So, this translates into an algebraic expression as \(x - \frac{1}{3}x\).

Key Concepts

Mathematical Translation

Mathematical translation involves turning a word problem into a mathematical expression or equation. This is a crucial skill in algebra that helps us understand and solve problems more systematically.
Let's break down the process:

• Identify key phrases: Look for phrases like "a number," "decreased by," or "a third of," as they indicate mathematical operations or variables.
• Match phrases with operations: For example, "decreased by" usually means subtraction.
• Use variables: Assign a letter, like 'x,' to represent unknown quantities.

When you translate words into math, you turn everyday language into something you can solve with numbers, making complex problems simpler.

Subtraction in Algebra

Subtraction in algebra is similar to subtraction in arithmetic, but it often involves variables. It's used to find the difference between numbers or expressions.
Here's how it works in the context of algebra:

• Identifying the minuend and subtrahend: In an algebraic expression like \(x - \frac{1}{3}x\), 'x' is the minuend (the starting value) and \(\frac{1}{3}x\) is what we subtract, or the subtrahend.
• Performing the operation: Subtract the subtrahend from the minuend using basic arithmetic rules applied to algebraic terms.

Being comfortable with subtraction in algebra helps you simplify expressions and solve equations more effectively.

Variables in Algebra

In algebra, variables are symbols or letters that represent unknown numbers or values. They allow us to generalize problems and find solutions that apply to numerous situations.
Here’s why variables are essential:

• Representation of unknowns: Use variables like 'x' to stand in for values we don't yet know.
• Flexibility: Variables keep expressions flexible, allowing the same formula to solve different problems.
• Calculations: They make it possible to perform calculations and solve equations systematically.

Variables are the backbone of algebra, providing structure and clarity to mathematical expressions and equations.