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$6 \frac{3}{8}+(-8 \frac{1}{2})+(-5 \frac{1}{6})+4 \frac{2}{3}=-2,625$

$6 \frac{3}{8} = \frac{6*8+3}{8} = \frac{51}{8}$

$-8 \frac{1}{2}=- \frac{8*2+1}{2} =-\frac{17}{2}$

$-5 \frac{1}{6} =- \frac{5*6+1}{6} =- \frac{31}{6}$

$4 \frac{2}{3}= \frac{4*3+2}{3}= \frac{14}{3}$

$\frac{51}{8} +(- \frac{17}{2})+(- \frac{31}{6} )+ \frac{14}{3} =-2,625$

1) $\frac{51}{8} +(- \frac{17}{2}) = \frac{51}{8}-\frac{17}{2} = \frac{1*51-4*17}{8}= \frac{51-68}{8}=- \frac{17}{8}$

2) $- \frac{17}{8} +(- \frac{31}{6} )= -\frac{17}{8}-\frac{31}{6}= -\frac{3*17 -4*31 }{24}=- \frac{51-124}{24}=- \frac{175}{24}$

3) $- \frac{175}{24} + \frac{14}{3}= -\frac{1*175+8*14}{24}= \frac{-175+112}{24}= -\frac{63}{24} = -2,625$