jmc

If Anthony makes $2/3$ of his next $24$ attempts, he will make another $16$ free throws. Then he will have $5+16=21$ successful throws in $12+24=36$ attempts. That's a success rate of $21/36=7/12$, which is $58.3\%$. His success rate before was $5/12$, which is $41.6\%$. The increase is $58.3-41.6=16.7$, or $\boxed{17\%}$ to the nearest whole number.