Probability Venn Diagram: Class of 100 students; Violin and Piano In a class of $100$ students, $x$ students play the violin and $2x$ play the piano. If $\frac{x}{2}$ students play both the violin and the piano, and there are 3x non-piano players, find the probability of a student playing neither piano or violin. How would you solve this question using logic, and then also algebra?

Assuming that the non-piano probability is $\frac{3x}{100}$ instead(probabilities cannot be integers $>1$), we get that there are $3x$ non-piano players, so $2x + 3x = 5x$ people in total (piano and non-piano are complementary). So $x=20$.

In that case there are $10$ violin-only players, $10$ play both piano and violin while $30$ play piano only. So we have $50$ players of either instrument and $50$ that play neither. So the asked for probability is $\frac{1}{2}$.