$12$ is related to $40$ following a certain logic. Following the same logic, $9$ is related to $31$. To which of the following is $16$ related using the same logic?

Let's find the functional mapping rule connecting the initial numbers to the corresponding outputs:
* First set: $12\to 40$

$$12\times 3+4=36+4=40$$

* Second set: $9\to 31$

$$9\times 3+4=27+4=31$$

The common logical formulation is: $\text{Output}=3\times \text{Input}+4$.

Applying this calculation format to $16$:

$$16\times 3+4=48+4=52$$