One-third of a number is $12$ and then the number is increased by $6$. What is one-fourth of the new number?

Let the original number be $x$.
According to the problem:

$$\frac{1}{3}\times x=12$$

$$x=12\times 3=36$$

Now, the number is increased by $6$ to get the new number:

$$\text{New Number}=36+6=42$$

We need to find one-fourth of this new number:

$$\text{One-fourth of new number}=\frac{1}{4}\times 42=10.5$$