Algebra Word Problem Breakdown

This problem asks us to find a specific unknown number. We are given clues about operations performed on this number: first, subtracting $\frac{1}{2}$, and second, multiplying the result of that subtraction by $\frac{1}{2}$. The final outcome of these operations is stated as $\frac{1}{8}$.

Equation Setup from Word Problem

Let's represent the unknown number with a variable, say '$x$'. We can translate the problem description into a mathematical equation step-by-step:

• "subtract $\frac{1}{2}$ from a number": This translates to $x - \frac{1}{2}$.
• "multiply the result by $\frac{1}{2}$": This means we take the previous expression and multiply it by $\frac{1}{2}$, giving us $\frac{1}{2} \left( x - \frac{1}{2} \right)$.
• "you get $\frac{1}{8}$": This sets the expression equal to $\frac{1}{8}$.

So, the equation we need to solve is:

Solving the Fraction Equation

Now, we solve the equation for '$x$' using algebraic methods:

1. Distribute or Isolate: We can first multiply both sides of the equation by 2 to eliminate the fraction on the left side. $ 2 \times \frac{1}{2} \left( x - \frac{1}{2} \right) = 2 \times \frac{1}{8} $ $ x - \frac{1}{2} = \frac{2}{8} $
2. Simplify the Result: Simplify the fraction on the right side. $ x - \frac{1}{2} = \frac{1}{4} $
3. Isolate the Variable: To find '$x$', we need to add $\frac{1}{2}$ to both sides of the equation. $ x = \frac{1}{4} + \frac{1}{2} $
4. Add Fractions: To add the fractions, find a common denominator, which is 4. Convert $\frac{1}{2}$ to $\frac{2}{4}$. $ x = \frac{1}{4} + \frac{2}{4} $ $ x = \frac{1+2}{4} $ $ x = \frac{3}{4} $

Identifying the Number

The value we found for '$x$' is $\frac{3}{4}$. This is the number that satisfies the conditions given in the problem.

Let's check: If we take $\frac{3}{4}$, subtract $\frac{1}{2}$ (which is $\frac{2}{4}$), we get $\frac{3}{4} - \frac{2}{4} = \frac{1}{4}$. If we multiply this result ($\frac{1}{4}$) by $\frac{1}{2}$, we get $\frac{1}{2} \times \frac{1}{4} = \frac{1}{8}$. This matches the condition given in the problem.

Therefore, the number is $\frac{3}{4}$.