Tính bằng cách thuận tiện:

a) \[\frac{5}{9} + \frac{5}{{11}} + \frac{1}{9} + \frac{4}{{11}} + \frac{1}{3} + \frac{2}{{11}}\]        b) \[\frac{6}{7} \times \frac{2}{7} + \frac{6}{7} \times \frac{{11}}{{14}} + \frac{6}{7}\]

c) \[\frac{2}{5} \times \frac{3}{8} + \frac{2}{5} \times \frac{1}{4} + \frac{2}{5} \times \frac{5}{6} + \frac{2}{5} \times \frac{1}{3}\]                            d) \[\frac{1}{7}:\frac{2}{5} + \frac{1}{7}:\frac{4}{3} + \frac{1}{7}:\frac{{12}}{7}\]

a) \[\frac{5}{9} + \frac{5}{{11}} + \frac{1}{9} + \frac{4}{{11}} + \frac{1}{3} + \frac{2}{{11}}\]

\[ = \left( {\frac{5}{9} + \frac{1}{9} + \frac{1}{3}} \right) + \left( {\frac{5}{{11}} + \frac{4}{{11}} + \frac{2}{{11}}} \right)\]

\[ = \left( {\frac{5}{9} + \frac{1}{9} + \frac{3}{9}} \right) + \frac{{11}}{{11}}\]

\[ = \frac{9}{9} + \frac{{11}}{{11}}\]

= 1 + 1 = 2

b) \[\frac{6}{7} \times \frac{2}{7} + \frac{6}{7} \times \frac{{11}}{{14}} + \frac{6}{7}\]

\[\begin{array}{l} = \frac{6}{7} \times \left( {\frac{2}{7} + \frac{{11}}{{14}} + 1} \right)\\ = \frac{6}{7} \times \left( {\frac{4}{{14}} + \frac{{11}}{{14}} + \frac{{14}}{{14}}} \right)\\ = \frac{6}{7} \times \frac{{29}}{{14}}\\ = \frac{{174}}{{98}} = \frac{{87}}{{49}}\end{array}\]

c) \[\frac{2}{5} \times \frac{3}{8} + \frac{2}{5} \times \frac{1}{4} + \frac{2}{5} \times \frac{5}{6} + \frac{2}{5} \times \frac{1}{3}\]

\[ = \frac{2}{5} \times \left( {\frac{3}{8} + \frac{1}{4} + \frac{5}{6} + \frac{1}{3}} \right)\]

\[ = \frac{2}{5} \times \left( {\frac{9}{{24}} + \frac{6}{{24}} + \frac{{20}}{{24}} + \frac{8}{{24}}} \right)\]

\[ = \frac{2}{5} \times \frac{{43}}{{24}}\]

\[ = \frac{{43}}{{60}}\]

d) \[\frac{1}{7}:\frac{2}{5} + \frac{1}{7}:\frac{4}{3} + \frac{1}{7}:\frac{{12}}{7}\]

\[ = \frac{1}{7} \times \frac{5}{2} + \frac{1}{7} \times \frac{3}{4} + \frac{1}{7} \times \frac{7}{{12}}\]

\[ = \frac{1}{7} \times \left( {\frac{5}{2} + \frac{3}{4} + \frac{7}{{12}}} \right)\]

\[ = \frac{1}{7} \times \left( {\frac{{30}}{{12}} + \frac{9}{{12}} + \frac{7}{{12}}} \right)\]

\[ = \frac{1}{7} \times \frac{{46}}{{12}}\]

\[ = \frac{{46}}{{84}} = \frac{{23}}{{42}}\]