Simplify: \(\left\{\frac{15}{42} \div \frac{15}{3}\right\} \div \left(\frac{3}{10} \times \frac{20}{9} + \frac{5}{14}\right) + \frac{3}{6} \div \frac{43}{20} \text{ of } \frac{20}{6}\)
\(\frac{6}{43}\)
First bracket: \(\frac{15}{42} \div \frac{15}{3} = \frac{15}{42} \times \frac{3}{15} = \frac{3}{42} = \frac{1}{14}\).
Second bracket: \(\frac{3}{10} \times \frac{20}{9} = \frac{2}{3}\), then \(\frac{2}{3} + \frac{5}{14} = \frac{28 + 15}{42} = \frac{43}{42}\).
First term: \(\frac{1}{14} \div \frac{43}{42} = \frac{1}{14} \times \frac{42}{43} = \frac{3}{43}\).
Last term, resolving 'of' first: \(\frac{43}{20} \text{ of } \frac{20}{6} = \frac{43}{20} \times \frac{20}{6} = \frac{43}{6}\); then \(\frac{3}{6} \div \frac{43}{6} = \frac{1}{2} \times \frac{6}{43} = \frac{3}{43}\).
Adding both parts: \(\frac{3}{43} + \frac{3}{43} = \frac{6}{43}\).
Hence, the simplified value is \(\frac{6}{43}\).
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