Cho các biểu thức:

\(A = \dfrac{{34}}{{7.13}} + \dfrac{{51}}{{13.22}} + \dfrac{{85}}{{22.37}} + \dfrac{{68}}{{37.49}};\)

\(  B = \dfrac{{39}}{{7.16}} + \dfrac{{65}}{{16.31}} + \dfrac{{52}}{{31.43}} + \dfrac{{26}}{{43.49}}\)

Tỉ số \(\dfrac{A}{B}\) bằng:

\(A = \dfrac{{34}}{{7.13}} + \dfrac{{51}}{{13.22}} + \dfrac{{85}}{{22.37}} + \dfrac{{68}}{{37.49}}\)

\(= \dfrac{{34}}{6}.\dfrac{6}{{7.13}} + \dfrac{{51}}{9}.\dfrac{9}{{13.22}}\)\(\, + \dfrac{{85}}{{15}}.\dfrac{1}{{22.37}} + \dfrac{{68}}{{12}}.\dfrac{{12}}{{37.49}}\)

\(= \dfrac{{17}}{3}.\left( {\dfrac{1}{7} - \dfrac{1}{{13}}} \right) + \dfrac{{17}}{3}.\left( {\dfrac{1}{{13}} - \dfrac{1}{{22}}} \right) \)\(\,+ \dfrac{{17}}{3}.\left( {\dfrac{1}{{22}} - \dfrac{1}{{37}}} \right) \)\(\,+ \dfrac{{17}}{3}.\left( {\dfrac{1}{{37}} - \dfrac{1}{{49}}} \right)\)

\(= \dfrac{{17}}{3}.\dfrac{1}{7} - \dfrac{{17}}{3}.\dfrac{1}{{13}} + \dfrac{{17}}{3}.\dfrac{1}{{13}} \)\(\,- \dfrac{{17}}{3}.\dfrac{1}{{22}} + \dfrac{{17}}{3}.\dfrac{1}{{22}} - \dfrac{{17}}{3}.\dfrac{1}{{37}} \)\(\,+ \dfrac{{17}}{3}.\dfrac{1}{{37}} - \dfrac{{17}}{3}.\dfrac{1}{{49}}\)

\(= \dfrac{{17}}{3}.\dfrac{1}{7} - \dfrac{{17}}{3}.\dfrac{1}{{49}} = \dfrac{{17}}{3}\left( {\dfrac{1}{7} - \dfrac{1}{{49}}} \right)\)

\(= \dfrac{{17}}{3}.\left( {\dfrac{7}{{49}} - \dfrac{1}{{49}}} \right) = \dfrac{{17}}{3}.\dfrac{6}{{49}} = \dfrac{{34}}{{49}}\)

\(B = \dfrac{{39}}{{7.16}} + \dfrac{{65}}{{16.31}} + \dfrac{{52}}{{31.43}} + \dfrac{{26}}{{43.49}}\)

\(= \dfrac{{39}}{9}.\dfrac{9}{{7.16}} + \dfrac{{65}}{{15}}.\dfrac{{15}}{{16.31}} \)\(\,+ \dfrac{{52}}{{12}}.\dfrac{{12}}{{31.43}} + \dfrac{{26}}{6}.\dfrac{6}{{43.49}}\)

\(= \dfrac{{13}}{3}.\left( {\dfrac{1}{7} - \dfrac{1}{{16}}} \right) + \dfrac{{13}}{3}.\left( {\dfrac{1}{{16}} - \dfrac{1}{{31}}} \right) \)\(\,+ \dfrac{{13}}{3}.\left( {\dfrac{1}{{31}} - \dfrac{1}{{43}}} \right) \)\(\,+ \dfrac{{13}}{3}.\left( {\dfrac{1}{{43}} - \dfrac{1}{{49}}} \right)\)

\(= \dfrac{{13}}{3}.\dfrac{1}{7} - \dfrac{{13}}{3}.\dfrac{1}{{16}} + \dfrac{{13}}{3}.\dfrac{1}{{16}} \)\(\,- \dfrac{{13}}{3}.\dfrac{1}{{31}} + \dfrac{{13}}{3}.\dfrac{1}{{31}} - \dfrac{{13}}{3}.\dfrac{1}{{43}}\)\(\, + \dfrac{{13}}{3}.\dfrac{1}{{43}} - \dfrac{{13}}{3}.\dfrac{1}{{49}}\)

\(= \dfrac{{13}}{3}.\dfrac{1}{7} - \dfrac{{13}}{3}.\dfrac{1}{{49}}\)\(\, = \dfrac{{13}}{3}.\left( {\dfrac{1}{7} - \dfrac{1}{{49}}} \right)\)

\(= \dfrac{{13}}{3}.\left( {\dfrac{7}{{49}} - \dfrac{1}{{49}}} \right) = \dfrac{{13}}{3}.\dfrac{6}{{49}} = \dfrac{{26}}{{49}}\) 

\( \Rightarrow \dfrac{A}{B} = \dfrac{{34}}{{49}}:\dfrac{{26}}{{49}} = \dfrac{{34}}{{49}}.\dfrac{{49}}{{26}} = \dfrac{{17}}{{13}}\)