Solve Complex Fraction Expression: (a²/2)÷(a/3)-(3a+12÷(4a·3))

To solve this problem, we'll follow these steps:

• Step 1: Simplify $\frac{a^2}{2}:\frac{a}{3}$.
• Step 2: Simplify $3a + 12:(4a \cdot 3)$.
• Step 3: Perform the subtraction from Step 1 and Step 2 results.

Now, let's work through each step:

Step 1:
The expression $\frac{a^2}{2}:\frac{a}{3}$ translates to $\frac{a^2}{2} \div \frac{a}{3}$.
Dividing by a fraction is equivalent to multiplying by its reciprocal, thus:
$\frac{a^2}{2} \times \frac{3}{a} = \frac{a^2 \cdot 3}{2 \cdot a} = \frac{3a}{2}$.

Step 2:
Handle $3a + 12:(4a \cdot 3)$.
First, calculate $12:(4a \cdot 3)$, which translates to $\frac{12}{12a} = \frac{1}{a}$.
Thus, $3a + \frac{1}{a}$ remains unchanged as $3a + \frac{1}{a}$.

Step 3:
Subtract the result from Step 2 from the result in Step 1:
$\frac{3a}{2} - (3a + \frac{1}{a})$.
This simplifies to:
$\frac{3a}{2} - 3a - \frac{1}{a}$.
To combine $\frac{3a}{2}$ and $-3a$, find a common denominator:
Convert $-3a$ to $-\frac{6a}{2}$, then:
$\frac{3a}{2} - \frac{6a}{2} = -\frac{3a}{2}$.
Therefore, the expression simplifies to:
$-\frac{3a}{2} - \frac{1}{a}$.

Therefore, the solution to the problem is $-1.5a - \frac{1}{a}$.