[tex]\begin{array}{ccc}200\ buttons&-&50\ shirts\\\\x\ buttons&-&350\ shirts\end{array}\ \ |cross\ multiply\\\\\\50x=200\times350\ \ \ \ |divide\ both\ sides\ by\ 50\\\\x=\frac{200\times350}{50}\\\\x=4\times350\\\\\boxed{x=1400}\\\\\boxed{\boxed{Answer:1400\ buttons/350\ shirts}}[/tex]