Solve the Fraction Equation: 1/8(1/2-x) = 1/4(x-1/4)

To solve the equation $\frac{1}{8}(\frac{1}{2} - x) = \frac{1}{4}(x - \frac{1}{4})$, we will first distribute the fractions:

• Distribute $\frac{1}{8}$ through $(\frac{1}{2} - x)$:
• $\frac{1}{8} \cdot \frac{1}{2} - \frac{1}{8}x = \frac{1}{16} - \frac{1}{8}x$
• Distribute $\frac{1}{4}$ through $(x - \frac{1}{4})$:
• $\frac{1}{4}x - \frac{1}{4} \cdot \frac{1}{4} = \frac{1}{4}x - \frac{1}{16}$

With distributed terms, the equation becomes:

$\frac{1}{16} - \frac{1}{8}x = \frac{1}{4}x - \frac{1}{16}$.

We will eliminate the fractions by multiplying the entire equation by 16, the least common multiple of the denominators 8 and 4, to eliminate fractions:

• $16 \times (\frac{1}{16} - \frac{1}{8}x) = 16 \times (\frac{1}{4}x - \frac{1}{16})$
• This simplifies to: $1 - 2x = 4x - 1$.

Now, we'll solve the equation:

1. Add $2x$ to both sides to gather $x$ terms on one side:

$1 = 6x - 1$

2. Add $1$ to both sides:

$2 = 6x$

3. Divide both sides by $6$ to isolate $x$:

$x = \frac{2}{6} = \frac{1}{3}$

Thus, the solution to the equation is $x = \frac{1}{3}$.