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 $100m+n$.

[i]Proposed by Yannick Yao[/i]