Solution:
Converting each of the given fractions in to decimal form, we get
$$\eqalign{ \frac{1}{3} = 0.33 \cr \frac{2}{5} = 0.4 \cr \frac{3}{7} = 0.42 \cr \frac{4}{5} = 0.8 \cr \frac{5}{6} = 0.83 \cr \frac{6}{7} = 0.85 \cr} $$
Clearly, 0.85 > 0.83 > 0.8 > 0.42 > 0.4 > 0.33
So, $$\frac{6}{7}$$ > $$\frac{5}{6}$$ > $$\frac{4}{5}$$ > $$\frac{3}{7}$$ > $$\frac{2}{5}$$ > $$\frac{1}{3}$$

Solution:
Converting each of the given fractions into decimal form, we get :
$$\frac{2}{3}$$ = 0.66
$$\frac{3}{4}$$ = 0.75
$$\frac{4}{5}$$ = 0.8
$$\frac{5}{6}$$ = 0.833
Since 0.833 > 0.8 > 0.75 > 0.66
So, $$\frac{5}{6}$$ > $$\frac{4}{5}$$ > $$\frac{3}{4}$$ > $$\frac{2}{3}$$
∴ Required difference = $$\left( {\frac{5}{6} - \frac{2}{3}} \right) = \frac{1}{6}$$

Solution:
Given expression :
$$\eqalign{ \frac{{0.125 + 0.027}}{{0.5 \times 0.5 + 0.09 - 0.15}} \cr = \frac{{{{\left( {0.5} \right)}^3} + {{\left( {0.3} \right)}^3}}}{{{{\left( {0.5} \right)}^2} + {{\left( {0.3} \right)}^2} - \left( {0.5 \times 0.3} \right)}} \cr = \left( {\frac{{{a^3} + {b^3}}}{{{a^2} + {b^2} - ab}}} \right) \cr = \frac{{\left( {a + b} \right)\left( {{a^2} - ab + {b^2}} \right)}}{{\left( {{a^2} - ab + {b^2}} \right)}} \cr = (a + b) \cr = \left( {0.5 + 0.3} \right) \cr = 0.8 \cr} $$

Solution:
Given expression :
(833.25 - 384.45) ÷ 24
= 18.7

Solution:
$$\eqalign{ 0.\overline {142857} \div 0.\overline {285714} \cr = \frac{{142857}}{{999999}} \div \frac{{285714}}{{999999}} \cr = \left( {\frac{{142857}}{{999999}} \times \frac{{999999}}{{285714}}} \right) \cr = \frac{1}{2} \cr} $$

Solution:
Given expression :
= $$\frac{0.2 (0.2 + 0.02)}{0.044}$$
= $$\frac{0.2 ×0.22}{0.044}$$
= $$\frac{0.044}{0.044}$$
= 1

Solution:
= 3899 ÷ 11.99 - 2379 ÷ 13.97
= 3899 ÷ 12 - 2380 ÷ 14
≈ 325 - 170
= 155