1) $$1 \frac{1}{2} \cdot \frac{2}{3} = \frac{3}{2} \cdot \frac{2}{3} = \frac{3 \cdot 2}{2 \cdot 3} = \frac{6}{6} = 1$$
2) $$\frac{2}{3} \cdot \frac{2}{5} = \frac{2 \cdot 2}{3 \cdot 5} = \frac{4}{15}$$
4) $$\frac{3}{8} \cdot \frac{2}{3} = \frac{3 \cdot 2}{8 \cdot 3} = \frac{6}{24} = \frac{1}{4}$$
5) $$\frac{3}{7} \cdot \frac{5}{8} = \frac{3 \cdot 5}{7 \cdot 8} = \frac{15}{56}$$
6) $$\frac{4}{7} : \frac{4}{9} = \frac{4}{7} \cdot \frac{9}{4} = \frac{4 \cdot 9}{7 \cdot 4} = \frac{36}{28} = \frac{9}{7} = 1 \frac{2}{7}$$