Solve the following equations :
`(3x)/(4)-(1)/(4)(x-20)=(x)/(4)+32`

Answer

To solve the equation \(\frac{3x}{4} - \frac{1}{4}(x - 20) = \frac{x}{4} + 32\), we will follow these steps:

### Step 1: Eliminate the fractions
To eliminate the fractions, we can multiply the entire equation by 4 (the least common multiple of the denominators).

\[ 4 \left(\frac{3x}{4}\right) - 4 \left(\frac{1}{4}(x - 20)\right) = 4 \left(\frac{x}{4}\right) + 4(32) \]

This simplifies to:

\[ 3x - (x - 20) = x + 128 \]

### Step 2: Simplify the equation
Now, distribute the negative sign on the left side:

\[ 3x - x + 20 = x + 128 \]

Combine like terms:

\[ 2x + 20 = x + 128 \]

### Step 3: Isolate the variable
Next, we will isolate \(x\) by moving \(x\) to one side and constants to the other side. Subtract \(x\) from both sides:

\[ 2x - x + 20 = 128 \]

This simplifies to:

\[ x + 20 = 128 \]

### Step 4: Solve for \(x\)
Now, subtract 20 from both sides to solve for \(x\):

\[ x = 128 - 20 \]

This gives us:

\[ x = 108 \]

### Final Answer
Thus, the solution to the equation is:

\[ \boxed{108} \]

---