The amount of carbon-14 present in a paint after $t$ years is given by $y=y_{0} e^{-0.00012 t}$. The paint contains $21 \%$ of its carbon-14. How old are the paintings?

Step 1 ：The amount of carbon-14 present in a paint after \(t\) years is given by \(y=y_{0} e^{-0.00012 t}\).

Step 2 ：The paint contains \(21 \%\) of its carbon-14, which means that \(y = 0.21y_0\).

Step 3 ：We can substitute this into the equation and solve for \(t\).

Step 4 ：Let \(y_0 = 1\) and \(y = 0.21\).

Step 5 ：Solving the equation gives \(t = 13005.39790220557\).

Step 6 ：Rounding to the nearest whole number, we find that the paintings are approximately \(\boxed{13005}\) years old.