[tex]\displaystyle\\
4 \frac{1}{2}~kg - 1 \frac{5}{6}~kg =\\\\
= \frac{4\times 2+1}{2}~kg - \frac{1\times6+ 5}{6}~kg = \\\\
= \frac{9}{2}~kg - \frac{11}{6}~kg = \\\\
= \frac{9\times3}{2\times3}~kg - \frac{11}{6}~kg = \\\\
= \frac{27}{6}~kg - \frac{11}{6}~kg =\frac{27-11}{6}~kg =\frac{16}{6}~kg = \frac{8}{3}~kg=\boxed{\bf2\frac{2}{3}~kg}
[/tex]