To find out how much of the can of paint Andy would need to cover the whole wall, we start by determining how much paint he used per square unit of the wall.
Andy used \(\frac{16}{12}\) of a can of paint for 12 units of the wall. We need to simplify that to find out how much paint is used per unit of the wall:
\[ \frac{16}{12} = \frac{4}{3} \]
This means Andy uses \(\frac{4}{3}\) of a can of paint to cover 12 units of the wall.
Now, we need to find the total paint needed to cover the entire wall, which can be considered as 1 full wall unit.
Let \( x \) be the amount of paint needed for 1 wall unit. We establish the following proportion:
\[ \frac{4}{3} \text{ cans of paint} \text{ for } 12 \text{ wall units} \]
Thus, the amount of paint needed for 1 wall unit would be:
\[ x = \frac{4}{3} \div 12 = \frac{4}{3} \times \frac{1}{12} = \frac{4}{36} = \frac{1}{9} \]
To find out how much paint Andy used to paint the whole wall, we set up the proportion for a total wall which is 1 unit:
- He used \(\frac{4}{3}\) cans for 12 units, then for 1 unit, it is \(\frac{4}{3} \times \frac{1}{12} = \frac{1}{9}\).
Now, since he covered a total area needing the entire wall, we would multiply by what remains and calculate accordingly.
However, a misunderstanding must be corrected: if he covered the whole wall unevenly and depending on the need, we would have calculated \(\frac{1}{9}\) accordingly.
The original question posed the complete consumption rather than ratio and proportion here needs a reading of the area directly.
Hence;
If he uses \(16\) of the can for \(12\) clearly the answer which represents the ratio in context of a complete wall equivalent here in answer modeling style should correctly look at totaling \(x = total \space of 1 full wall = 12 \space of \space paint used.\)
So through sheer ratio modelling, one looks correctly towards representing \( 12 \) again.
Thus the answer is:
\[ \boxed{12} \]