If the degree measures of the angles of a triangle are in the ratio $3:3:4$, what is the degree measure of the largest angle of the triangle?
The sum of the ratios is $10$.  Since the sum of the angles of a triangle is $180^{\circ}$, the ratio can be scaled up to $54:54:72$ $(3\cdot 18:3\cdot 18:4\cdot 18).$  The numbers in the ratio $54:54:72$ represent the angles of the triangle.  The question asks for the largest, so the answer is $\fbox{72}$.