Adding tricky fractions

Answer: 1

Change the order of the fractions in the sum so that fractions whose denominators share more factors are added first: $$\begin{array}{rl}& \frac{1}{7}+\frac{1}{8}+\frac{1}{9}+\frac{1}{10}+\frac{1}{11}+\frac{1}{12}+\frac{1}{14}+\frac{1}{15}+\frac{1}{18}+\frac{1}{22}+\frac{1}{24}+\frac{1}{28}+\frac{1}{33}\\ & =\frac{1}{7}+\frac{1}{14}+\frac{1}{28}+\frac{1}{8}+\frac{1}{12}+\frac{1}{24}+\frac{1}{9}+\frac{1}{18}+\frac{1}{10}+\frac{1}{15}+\frac{1}{11}+\frac{1}{22}+\frac{1}{33}\\ & =\frac{4}{28}+\frac{2}{28}+\frac{1}{28}+\frac{3}{24}+\frac{2}{24}+\frac{1}{24}+\frac{2}{18}+\frac{1}{18}+\frac{3}{30}+\frac{2}{30}+\frac{6}{66}+\frac{3}{66}+\frac{2}{66}\\ & =\frac{7}{28}+\frac{6}{24}+\frac{3}{18}+\frac{5}{30}+\frac{11}{66}\\ & =\frac{1}{4}+\frac{1}{4}+\frac{1}{6}+\frac{1}{6}+\frac{1}{6}\\ & =1\end{array}$$