Problem 108 - Introductory Algebra for College Students

Question

Translate from English to an algebraic expression or equation, whichever is appropriate. Let the variable \(x\) represent the number. A number decreased by \(\frac{1}{3}\) is half of that number.

Step-by-Step Solution

Answer

The algebraic equation that represents the given English statement is \(x - \frac{1}{3} = \frac{x}{2}\).

Step 1: Identify the variable

In this problem, it is given to use \(x\) to represent 'the number'. Therefore, 'a number' is represented as \(x\).

Step 2: Translate the expression 'a number decreased by \(\frac{1}{3}\)'

The term 'a number decreased by \(\frac{1}{3}\)' can be translated to \(x-\frac{1}{3}\). In this expression, \(x\) is 'the number' and \(-\frac{1}{3}\) represents 'decreased by \(\frac{1}{3}\)'.

Step 3: Translate the expression 'half of that number'

'Half of that number' can be translated as \(\frac{1}{2}*x\) or simply \(\frac{x}{2}\). In this expression, \(x\) is 'the number' and '\(\frac{1}{2}\)' represents 'half'.

Step 4: Formulate the equation

Since the problem states that 'a number decreased by \(\frac{1}{3}\)' is 'half of that number', these two expressions are set equal to each other. Therefore, the algebraic equation is \(x - \frac{1}{3} = \frac{x}{2}\).