2014 AMC 12B Problem 7

Write $$\dfrac{n}{30-n} = \dfrac{30}{30-n} - 1.$$ For this to be a positive integer, $30-n$ must be a positive divisor of $30$ with $\dfrac{30}{30-n} \ge 2,$ i.e. $30-n \le 15.$ The divisors of $30$ that are at most $15$ are $1, 2, 3, 5, 6, 10, 15,$ giving $7$ values of $n$ (namely $15, 20, 24, 25, 27, 28, 29$). Thus, the correct answer is D.