Find the sum of all [positive](https://artofproblemsolving.com/wiki/index.php/Positive_number) [rational numbers](https://artofproblemsolving.com/wiki/index.php/Rational_number) that are less than 10 and that have [denominator](https://artofproblemsolving.com/wiki/index.php/Denominator) 30 when written in [ lowest terms](https://artofproblemsolving.com/wiki/index.php/Reduced_fraction).
Solution 1
There are 8 [fractions](https://artofproblemsolving.com/wiki/index.php/Fraction) 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 $4(10)+8(1+2+3+\cdots+9)=\boxed{400}.$

Solution 2
By [Euler's Totient Function](https://artofproblemsolving.com/wiki/index.php/Euler%27s_Totient_Function), there are $8$ numbers that are relatively prime to $30$, less than $30$. Note that they come in pairs $(m,30-m)$ which result in sums of $1$; thus the sum of the smallest $8$ rational numbers satisfying this is $\frac12\cdot8\cdot1=4$. Now refer to solution 1.

Solution 3
Note that if $x$ is a solution, then $(300-x)$ is a solution. We know that $\phi(300) = 80.$ Therefore the answer is $\frac{80}{2} \cdot\frac{300}{30} = \boxed{400}.$



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