jmc

We notice that 6! is a common factor of the numerator and denominator and then simplify: $$\begin{array}{rl}\frac{7!+8!}{6!+7!}& =\frac{7\cdot 6!+8\cdot 7\cdot 6!}{6!+7\cdot 6!}\\ & =\frac{6!(7+8\cdot 7)}{6!(1+7)}\\ & =\frac{7+56}{1+7}\\ & =\boxed{\frac{63}{8}}\end{array}$$