Question

select the correct answer. what is the solution for w in the equation? $w + \frac{1}{2} = \frac{1}{4}w + 2$ $\bigcirc$ $w = \frac{10}{3}$ $\bigcirc$ $w = \frac{9}{8}$ $\bigcirc$ $w = 2$ $\bigcirc$ $w = 4$

Explanation:

Step1: Subtract $\frac{1}{4}w$ from both sides

$w - \frac{1}{4}w + \frac{1}{2} = \frac{1}{4}w - \frac{1}{4}w + 2$
$\frac{3}{4}w + \frac{1}{2} = 2$

Step2: Subtract $\frac{1}{2}$ from both sides

$\frac{3}{4}w + \frac{1}{2} - \frac{1}{2} = 2 - \frac{1}{2}$
$\frac{3}{4}w = \frac{3}{2}$

Step3: Multiply both sides by $\frac{4}{3}$

$\frac{4}{3} \times \frac{3}{4}w = \frac{3}{2} \times \frac{4}{3}$
$w = 2$ (Wait, no, let's recalculate step3. $\frac{3}{2} \times \frac{4}{3} = \frac{12}{6} = 2$? Wait, no, wait: $\frac{3}{2} \times \frac{4}{3}$: the 3s cancel, 4/2=2. Wait, but let's check again. Wait, original equation: $w + \frac{1}{2} = \frac{1}{4}w + 2$. Let's plug w=2: left side 2 + 0.5=2.5, right side 0.5 + 2=2.5. Wait, but wait, when we did step1: $w - \frac{1}{4}w = \frac{3}{4}w$, correct. Step2: 2 - 0.5=1.5, which is $\frac{3}{2}$. Then step3: $\frac{3}{4}w = \frac{3}{2}$, multiply both sides by $\frac{4}{3}$: $w = \frac{3}{2} \times \frac{4}{3} = \frac{12}{6} = 2$. Wait, but let's check the options. One of the options is w=2. Wait, but let's check again. Wait, maybe I made a mistake. Wait, let's solve again:

$w + \frac{1}{2} = \frac{1}{4}w + 2$

Subtract $\frac{1}{4}w$ from both sides:

$w - \frac{1}{4}w + \frac{1}{2} = 2$

$\frac{3}{4}w + \frac{1}{2} = 2$

Subtract $\frac{1}{2}$:

$\frac{3}{4}w = 2 - \frac{1}{2} = \frac{3}{2}$

Multiply both sides by $\frac{4}{3}$:

$w = \frac{3}{2} \times \frac{4}{3} = 2$

Yes, so w=2. Wait, but let's check the options. The third option is w=2. So that's correct. Wait, but wait, maybe I messed up earlier. Let's verify with w=2:

Left side: 2 + 1/2 = 5/2 = 2.5

Right side: (1/4)*2 + 2 = 0.5 + 2 = 2.5. Correct. So the answer is w=2.

Wait, but wait, the first option is 10/3 ≈3.333, second 9/8=1.125, third 2, fourth 4. So when we solved, we got w=2, which is option C (assuming the options are A:10/3, B:9/8, C:2, D:4). So the correct answer is w=2.

Wait, but let's redo the calculation carefully:

Given $w + \frac{1}{2} = \frac{1}{4}w + 2$

1. Subtract $\frac{1}{4}w$ from both sides:

$w - \frac{1}{4}w + \frac{1}{2} = \frac{1}{4}w - \frac{1}{4}w + 2$

$\frac{3}{4}w + \frac{1}{2} = 2$

1. Subtract $\frac{1}{2}$ from both sides:

$\frac{3}{4}w + \frac{1}{2} - \frac{1}{2} = 2 - \frac{1}{2}$

$\frac{3}{4}w = \frac{3}{2}$

1. Multiply both sides by the reciprocal of $\frac{3}{4}$, which is $\frac{4}{3}$:

$w = \frac{3}{2} \times \frac{4}{3}$

Simplify: the 3s cancel, 4/2=2, so $w = 2$

Answer:

w = 2 (corresponding to the option "w = 2")