Mathematicians and Musicians

In a given nation, every $20^{th}$ mathematician is also a musician, while every $30^{th}$ musician is also a mathematician. Are there more mathematicians or musicians in the nation? How many times more?

Solution

Reference

1. Sergey Dorichenko, A Moscow Math Circle: Week-by-Week Problem Sets (MSRI Mathematical Circles Library, 2012), #4.3

In a given nation, every $20^{th}$ mathematician is also a musician, while every $30^{th}$ musician is also a mathematician. Are there more mathematicians or musicians in the nation? How many times more?

Solution 1

Assume there are $M$ mathematicians and $N$ musicians in the nation. Then we can say that in the very least M/20,$M/20$,$N/20$,$M/30$,$N/30$ musicians. But more can be said. These $M/20$ mathematicians that are also musicians, are musicians that are also mathematicians. The number of such musicians is at least N/30,$M/20$,$N/20$,$M/30$,$N/30$, implying that $M/20\ge N/30.$ In a similar manner, we could obtain $N/30\ge M/20,$ implying that the numbers are equal: $M/20= N/30,$ i.e., $N=1.5M.$ There are 50\%,$30\%$,$40\%$,$50\%$,$60\%$ more musicians than mathematicians.

The fact that we got an equality $M/20= N/30$ suggests a shortcut

Solution 2

Let $X$ be the number of people in the nation who are both mathematicians and musicians. We then see that the number of mathematicians in the nation equals 20X,$20X$,$30X$, while the number of musicians is 30X,$20X$,$30X$. It follows that there are $50\%$ more musicians than mathematicians.