Sum of Rationals

If a number fits this property than it has a numerator, n , where $1 \le n \le 299$ and $gcd(n,30) = 1$ . If $1 \le n \le 30$ there are 8 valid values of n , namely $1,7,11,13,17,19,23,29$ . Now, if $gcd(n,30)=1$ then $gcd(n+30k,30)=1$ and if $gcd(n,30) \ne 1$ then $gcd(n+30k,30) \ne 1$ . Therefore all possible values of n are just these first 8 values plus multiples of 30. The sum of these first 8 number is $1+7+11+13+17+19+23+29 = 120$ . Therefore the sum of all possible numbers (including division by 30) is $\dfrac{(120)+(120 + 30 \times 8) + (120 + 60 \times 8) + ... + (120 + 270 \times 8)}{30}$ $= \dfrac{120 \times 10 + 240(1+2+...+9)}{30}$ $= 40 + 8*45 = \fbox{400}$

A sweet solution to a sweet problem

There are 8 fractions which fit the conditions between 0 and 1: $\frac{1}{30},\frac{7}{30},\frac{11}{30},\frac{13}{30},\frac{17}{30},\frac{19}{30},\frac{23}{30},\frac{29}{30}$

Their sum is 4. Note that there are also 8 terms between 1 and 2 which we can obtain by adding 1 to each of our first 8 terms. For example, $1+\frac{19}{30}=\frac{49}{30}$ . Following this pattern, our answer is $4(10)+8(1+2+3+\cdots+9)=\boxed{400}.$