If $$A \# B = \frac{A \times B}{A + B}$$, the value of $$\frac{12 \# 8}{8 \# 4} + \frac{10 \# 6}{6 \# 2}$$ is

Computing each part, $$12 \# 8 = \frac{96}{20} = \frac{24}{5}$$ and $$8 \# 4 = \frac{32}{12} = \frac{8}{3}$$, so $$\frac{12 \# 8}{8 \# 4} = \frac{24}{5} \times \frac{3}{8} = \frac{9}{5}$$. Similarly $$10 \# 6 = \frac{60}{16} = \frac{15}{4}$$ and $$6 \# 2 = \frac{12}{8} = \frac{3}{2}$$, so $$\frac{10 \# 6}{6 \# 2} = \frac{15}{4} \times \frac{2}{3} = \frac{5}{2}$$. Adding gives $$\frac{9}{5} + \frac{5}{2} = \frac{43}{10} = 4.3$$.