jmc

It might be easier to find the integers less than or equal to 30 which are NOT relatively prime to 30. They include 2, 4, 6, 8, 10, $\dots$, 28, 30, or 15 even integers. They also include 3, 9, 15, 21, 27, or the odd multiples of 3. And also, 5, 25, the multiples of 5 relatively prime to 2 and 3. So we have a total of $15+5+2=22$ numbers sharing a factor with 30. So there are 8 relatively prime integers, giving us a ratio of $\frac{8}{30}=\boxed{\frac{4}{15}}$.

Notice that the prime divisors of 30 are 2, 3, and 5, and we have $$30\left(1-\frac{1}{2}\right)\left(1-\frac{1}{3}\right)\left(1-\frac{1}{5}\right)=30\cdot \frac{1}{2}\cdot \frac{2}{3}\cdot \frac{4}{5}=8,$$ which equals the number of positive integers less than 30 that are relatively prime to 30. Is this a coincidence?