Lunasa, Merlin, and Lyrica each has an instrument. We know the following about the prices of their instruments: (a) If we raise the price of Lunasa's violin by 50% and decrease the price of Merlin's trumpet by 50%, the violin will be $50 more expensive than the trumpet;

(b) If we raise the price of Merlin's trumpet by 50% and decrease the price of Lyrica's piano by 50%, the trumpet will be $50 more expensive than the piano.

Given these conditions only, there exist integers m and n such that if we raise the price of Lunasa's violin by $m\%$ and decrease the price of Lyrica's piano by $m\%$, the violin must be exactly $\$n$ more expensive than the piano. Find $100m+n$.