Let's solve these questions one by one.

(i) $\frac{1}{4} \text{ of } 2 \frac{2}{7} - \frac{3}{5}$

• Convert $2 \frac{2}{7}$ to an improper fraction:
  $2 \frac{2}{7} = \frac{16}{7}$

• So the expression becomes:
  $\frac{1}{4} \times \frac{16}{7} - \frac{3}{5}$

• Multiply:
  $\frac{1}{4} \times \frac{16}{7} = \frac{16}{28} = \frac{4}{7}$

• Now subtract:
  $\frac{4}{7} - \frac{3}{5}$

• Find the common denominator (35):
  $\frac{4}{7} = \frac{20}{35}, \quad \frac{3}{5} = \frac{21}{35}$

• Subtract:
  $\frac{20}{35} - \frac{21}{35} = -\frac{1}{35}$

Answer for (i): $-\frac{1}{35}$

(ii) $\frac{1}{4} \times \frac{1}{2} + \frac{1}{3}$

• Multiply:
  $\frac{1}{4} \times \frac{1}{2} = \frac{1}{8}$

• Now add:
  $\frac{1}{8} + \frac{1}{3}$

• Find the common denominator (24):
  $\frac{1}{8} = \frac{3}{24}, \quad \frac{1}{3} = \frac{8}{24}$

• Add:
  $\frac{3}{24} + \frac{8}{24} = \frac{11}{24}$

Answer for (ii): $\frac{11}{24}$

(iii) $6 \times \frac{1}{7} \times 0 \times \frac{5}{8}$

• The presence of $0$ makes the entire product $0$.

Answer for (iii): $0$

(iv) $\frac{3}{4} \times \frac{1}{3} \div \frac{3}{7} \times \frac{2}{5} \div \frac{5}{8}$

• First, resolve the divisions:
  $\frac{1}{3} \div \frac{3}{7} = \frac{1}{3} \times \frac{7}{3} = \frac{7}{9}$

  $\frac{2}{5} \div \frac{5}{8} = \frac{2}{5} \times \frac{8}{5} = \frac{16}{25}$

• Now multiply:
  $\frac{3}{4} \times \frac{7}{9} \times \frac{16}{25} = \frac{3 \times 7 \times 16}{4 \times 9 \times 25} = \frac{336}{900} = \frac{28}{75}$

Answer for (iv): $\frac{28}{75}$

(v) $\frac{2}{4} + \frac{2}{7} \div 1 \frac{1}{3} \times \frac{2}{3}$

• First, simplify the mixed fraction $1 \frac{1}{3} = \frac{4}{3}$

• Now resolve the division:
  $\frac{2}{7} \div \frac{4}{3} = \frac{2}{7} \times \frac{3}{4} = \frac{6}{28} = \frac{3}{14}$

• Now the expression becomes:
  $\frac{2}{4} + \frac{3}{14} \times \frac{2}{3}$

• Multiply:
  $\frac{3}{14} \times \frac{2}{3} = \frac{6}{42} = \frac{1}{7}$

• Now add:
  $\frac{2}{4} + \frac{1}{7} = \frac{1}{2} + \frac{1}{7}$

• Find the common denominator (14):
  $\frac{1}{2} = \frac{7}{14}$

• Add:
  $\frac{7}{14} + \frac{1}{14} = \frac{8}{14} = \frac{4}{7}$

Answer for (v): $\frac{4}{7}$

(vi) $\left(\frac{3}{7} - \frac{1}{2}\right) \text{ of } \frac{1}{7}$

• Subtract:
  $\frac{3}{7} - \frac{1}{2}$

• Find the common denominator (14):
  $\frac{3}{7} = \frac{6}{14}, \quad \frac{1}{2} = \frac{7}{14}$

• Subtract:
  $\frac{6}{14} - \frac{7}{14} = -\frac{1}{14}$

• Now multiply by $\frac{1}{7}$:
  $-\frac{1}{14} \times \frac{1}{7} = -\frac{1}{98}$

Answer for