$1\frac{1}{3}-\frac{7}{12}+\frac{9}{20}-\frac{11}{30}+\frac{13}{42}-\frac{15}{56}$
$= (1+\frac{1}{3})- (\frac{1}{3}+\frac{1}{4})+ (\frac{1}{4}+\frac{1}{5})- (\frac{1}{5}+\frac{1}{6}) +(\frac{1}{6}+\frac{1}{7})-(\frac{1}{7}+\frac{1}{8})$
$=1+\frac{1}{3}-\frac{1}{3}-\frac{1}{4}+\frac{1}{4}+\frac{1}{5}-\frac{1}{5}-\frac{1}{6}+\frac{1}{6}+\frac{1}{7}-\frac{1}{7}-\frac{1}{8}$
$=1-\frac{1}{8}$
$=\frac{7}{8}$