-3 3/5+(-2 1/3)+4 8/15-1 5/6-(-6 7/10)+1/2

$-3\frac{3}{5}+(-2\frac{1}{3})+4\frac{8}{15}-1\frac{5}{6}-(-6\frac{7}{10})+\frac{1}{2}=4\frac{1}{6}$

1) $-3\frac{3}{5}+(-2\frac{1}{3})=-3\frac{3}{5}-2\frac{1}{3}=-(3\frac{3}{5}+2\frac{1}{3})=-(\frac{18}{5}+\frac{7}{3})=-(\frac{51+35}{15})=-\frac{86}{15}=-5\frac{11}{15}$

2) $-5\frac{11}{15}+4\frac{8}{15}=-\frac{86}{15}+\frac{68}{15}=\frac{68}{15}-\frac{86}{15}=\frac{68-86}{15}=-\frac{18}{15}=-\frac{6}{5}=-1\frac{1}{5}$

3) $-1\frac{1}{5}-1\frac{5}{6}=-\frac{6}{5}-\frac{11}{6}=-(\frac{6}{5}+\frac{11}{6})=-(\frac{36+55}{30})=-\frac{91}{30}=-3\frac{1}{30}$

4) $-3\frac{1}{30}-(-6\frac{7}{10})=-3\frac{1}{30}+6\frac{7}{10}=-\frac{91}{30}+\frac{67}{10}= \frac{67}{10}-\frac{91}{30}=\frac{201-91}{30}=\frac{110}{30}=\frac{11}{3}=3\frac{2}{3}$

5) $3\frac{2}{3}+\frac{1}{2}=\frac{11}{3}+\frac{1}{2}=\frac{22+3}{6}=\frac{25}{6}=4\frac{1}{6}$