Difference between revisions of "1992 AIME Problems/Problem 1"

Problem

Find the sum of all positive rational numbers that are less than 10 and that have denominator 30 when written in lowest terms.

Solution

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

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