Simplify 5 2/3 ÷ 2 7/5 / 6 1/3 ÷ 2 7/6

For the numerator:

$$5\frac{2}{3} = \frac{(5 \times 3) + 2}{3} = \frac{15 + 2}{3} = \frac{17}{3}$$ $$2\frac{7}{5} = \frac{(2 \times 5) + 7}{5} = \frac{10 + 7}{5} = \frac{17}{5}$$

For the denominator:

$$6\frac{1}{3} = \frac{(6 \times 3) + 1}{3} = \frac{18 + 1}{3} = \frac{19}{3}$$ $$2\frac{7}{6} = \frac{(2 \times 6) + 7}{6} = \frac{12 + 7}{6} = \frac{19}{6}$$

$$\frac{17}{3} \div \frac{17}{5} = \frac{17}{3} \times \frac{5}{17}$$

Cancel out the common factor of 17:

$$= \frac{5}{3}$$

$$\frac{19}{3} \div \frac{19}{6} = \frac{19}{3} \times \frac{6}{19}$$

Cancel out the common factor of 19:

$$= \frac{6}{3} = 2$$

The expression now becomes:

$$\frac{\frac{5}{3}}{2}$$

This is equivalent to $\frac{5}{3} \div 2$.

$$\frac{5}{3} \div 2 = \frac{5}{3} \times \frac{1}{2} = \frac{5 \times 1}{3 \times 2} = \frac{5}{6}$$

The simplified value is $\mathbf{\frac{5}{6}}$.