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?
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The total number of students is 100. All students come under the 2 categories of either piano players, or non-piano players. Therefore:
Total number of students = piano players + non-piano players
100 = 5x
x = 20
We know that: x students play the violin, 2 x play the piano, x/2 play both. Violinists = 20 Pianists = 40
Total musicians = (20+40) - 10 = 50
Therefore: Pr(Student playing NO instrument) = 50/100 = 0.5
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 $\frac{1}{2}$.