If all angles are measured in degrees, the ratio of three times the measure of $\angle A$ to four times the measure of the complement of $\angle A$ to half the measure of the supplement of $\angle A$ is $3:14:4$. What is the number of degrees in the measure of the complement of $\angle A$?
Let $x$ be the number of degrees in the measure of $\angle A$.  Then we have  \[
\frac{3x}{4(90-x)}=\frac{3}{14},
\]from the information "the ratio of three times the measure of $\angle A$ to four times the measure of the complement of $\angle A$ is $3:14$." Multiplying both sides by $\frac{2}{3}$ and clearing denominators, we find $$7x=180-2x\implies 9x=180\implies x=20.$$The measure of the complement of 20 degrees is $\boxed{70}$ degrees.

Note: The hypothesis "if all angles are measured in degrees'' is not necessary.  The angle is determined uniquely by the given information regardless of the units used.