Using appropriate properties, find (i) (\frac{2}{3} \times \frac{3}{5} + \frac{5}{2} \times \frac{1}{6}) (ii) (\frac{2}{5} \times (\frac{3}{7}) - \frac{1}{6} \times \frac{3}{2} + \frac{1}{14} \times \frac{2}{5})

(i) (\frac{2}{3} \times \frac{3}{5} + \frac{5}{2} \times \frac{1}{6})
Using the commutative property, we can write the given expression as:
(\frac{2}{3} \times \frac{3}{5} + \frac{5}{2} \times \frac{1}{6})
Now, using the distributive property of multiplication over addition,
((\frac{2}{3} \times \frac{3}{5}) + (\frac{5}{2} \times \frac{1}{6}) = \frac{6}{15} + \frac{5}{12} = \frac{2}{5} + \frac{5}{12} = \frac{24 + 25}{60} = \frac{49}{60}
Hence, using appropriate properties, (\frac{2}{3} \times \frac{3}{5} + \frac{5}{2} \times \frac{1}{6} = \frac{49}{60})

(ii) (\frac{2}{5} \times (\frac{3}{7}) - \frac{1}{6} \times \frac{3}{2} + \frac{1}{14} \times \frac{2}{5})
Using the commutative property, we can write the same expression as:
(\frac{2}{5} \times \frac{3}{7} - \frac{1}{6} \times \frac{3}{2} + \frac{1}{14} \times \frac{2}{5})
Now, using the distributive property of multiplication over addition, we get:
(\frac{2}{5} \times (\frac{3}{7} + \frac{1}{14}) - \frac{1}{6} \times \frac{3}{2} = \frac{2}{5} \times (\frac{6 + 1}{14}) - \frac{3}{12} = \frac{2}{5} \times \frac{7}{14} - \frac{1}{4} = \frac{2}{5} \times \frac{1}{2} - \frac{1}{4} = \frac{1}{5} - \frac{1}{4} = \frac{4 - 5}{20} = -\frac{1}{20}
Hence, using appropriate properties,
(\frac{2}{5} \times (\frac{3}{7}) - \frac{1}{6} \times \frac{3}{2} + \frac{1}{14} \times \frac{2}{5} = -\frac{1}{20})