A set $S$ consists of 30 different numbers, if n is in the set $S$ and $n$ is $6$ times the average of the other 29 numbers in $S$, then what is the ratio of $n$ to the sum of the $30$ numbers in the list.
I've been thinking for this questions for so long. Any explanation? Thanks in advance.
Suppose $A = a_1 + \cdots + a_{29} $, so we have:
$$n = 6\times \frac{a_1+\cdots + a_{29}}{29} = 6\times \frac{A}{29}$$ $$\frac{n}{a_1 + \cdots + a_{29} + n} = \frac{6\times \frac{A}{29}}{A + 6\times \frac{A}{29}} = \frac{6\times \frac{A}{29}}{35\times \frac{A}{29}} = \frac{6}{35}$$