Solution:
$$\eqalign{ {\text{Given expression,}} \cr \frac{{4 \times {3^3} + {3^2} + 3 + 1}}{{4 \times {3^3}}} \cr = \frac{{108 + 9 + 3 + 1}}{{108}} \cr = \frac{{121}}{{108}} \cr} $$

Solution:
$$\eqalign{ {\text{According to question,}} \cr {\text{3}}\frac{1}{2} - \left[ {2\frac{1}{4} \div \left\{ {1\frac{1}{4} - \frac{1}{2}\left( {1\frac{1}{2} - \frac{1}{3} - \frac{1}{6}} \right)} \right\}} \right] \cr = \frac{7}{2} - \left[ {\frac{9}{4} \div \left\{ {\frac{5}{4} - \frac{1}{2}\left( {\frac{3}{2} - \frac{1}{3} - \frac{1}{6}} \right)} \right\}} \right] \cr = \frac{7}{2} - \left[ {\frac{9}{4} \div \left\{ {\frac{5}{4} - \frac{1}{2} \times 1} \right\}} \right] \cr = \frac{7}{2} - \left[ {\frac{9}{4} \div \frac{3}{4}} \right] \cr = \frac{7}{2} - \left[ {\frac{9}{4} \times \frac{4}{3}} \right] \cr = \frac{7}{2} - 3 \cr = \frac{{7 - 6}}{2} \cr = \frac{1}{2} \cr} $$

Solution:
$$\eqalign{ \frac{1}{{4.263}} = 0.2346 \cr \frac{{1000}}{{4263}} = 0.2346 \cr \frac{1}{{0.0004263}} \cr = \frac{{10000000}}{{4263}} \cr = 10000 \times 0.2346 \cr = 2346 \cr} $$

Solution:
$$\eqalign{ \frac{m}{n}{\text{ = }}\frac{4}{3}{\text{ and }}\frac{r}{t}{\text{ = }}\frac{9}{{14}} \cr \Rightarrow \frac{{mr}}{{nt}} = \frac{4}{3} \times \frac{9}{{14}} \cr \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{6}{7} \cr \therefore \frac{{3mr - nt}}{{4nt - 7mr}}{\text{ }} \cr = \frac{{3\frac{{mr}}{{nt}} - 1}}{{4 - 7\frac{{mr}}{{nt}}}} \cr = \frac{{3 \times \frac{6}{7} - 1}}{{4 - 7 \times \frac{6}{7}}} \cr = \frac{{\frac{{18}}{7} - 1}}{{4 - 6}} \cr = \frac{{11}}{7} \times \left( { - \frac{1}{2}} \right) \cr = - \frac{{11}}{{14}} \cr} $$

Solution:
$$\eqalign{ \frac{{16 - 0.002}}{{8 + 0.4 + 0.02}} \cr = \frac{{16000 - 2}}{{8000 + 400 + 20}} \cr = \frac{{15098}}{{8420}} \cr = 1.9{\text{ Answer}} \cr} $$

Solution:
$$\eqalign{ \frac{{\left( {187 \div 17 \times a - 3 \times 3} \right)}}{{\left( {{8^2} - 9 \times 7 + {a^2}} \right)}} = 1 \cr 11a - 9 = 64 - 63 + {a^2} \cr 11a - 9 = 1 + {a^2} \cr {a^2} - 11a + 10 = 0 \cr {a^2} - \left( {10 + 1} \right)a + 10 = 0 \cr {a^2} - 10a - a + 10 = 0 \cr a\left( {a - 10} \right) - 1\left( {a - 10} \right) = 0 \cr \left( {a - 10} \right)\left( {a - 1} \right) = 0 \cr a = 1,\,10{\text{ Answer}} \cr} $$

Solution:
2.8 + (5.2 ÷ 1.3 × 2) - 6 × 3 ÷ 8 + 2
= 2.8 + (4 × 2) - $$\frac{{6 \times 3}}{8}$$ + 2
= 2.8 + 8 - 2.25 + 2
= 10.55