Вычислите: \(\dfrac{\left(\left(3\dfrac{7}{12}-2\dfrac{11}{18}+2\dfrac{1}{24}\right)\cdot1\dfrac{5}{31}-\dfrac{3}{52}\left(3\dfrac{1}{2}+\dfrac{5}{6}\right)\right)\cdot1\dfrac{7}{13}}{\dfrac{19}{84}:\left(5\dfrac{13}{42}-2\dfrac{13}{28}+\dfrac{5}{24}\right)+1\dfrac{2}{27}-\dfrac{1}{3}\cdot\dfrac{4}{9}}\)

Вычислим числитель:

\[3\dfrac{7}{12}-2\dfrac{11}{18}+2\dfrac{1}{24}=\frac{43}{12}-\frac{47}{18}+\frac{49}{24}=\frac{217}{72}.\]

\[\frac{217}{72}\cdot1\dfrac{5}{31}=\frac{217}{72}\cdot\frac{36}{31}=\frac{217}{62}.\]

\[3\dfrac{1}{2}+\frac{5}{6}=\frac{7}{2}+\frac{5}{6}=\frac{13}{3},\qquad \frac{3}{52}\cdot\frac{13}{3}=\frac14.\]

\[\left(\frac{217}{62}-\frac14\right)\cdot1\dfrac{7}{13}=\frac{403}{124}\cdot\frac{20}{13}=5.\]

Вычислим знаменатель:

\[5\dfrac{13}{42}-2\dfrac{13}{28}+\frac{5}{24}=\frac{223}{42}-\frac{69}{28}+\frac{5}{24}=\frac{171}{56}.\]

\[\frac{19}{84}:\frac{171}{56}=\frac{19}{84}\cdot\frac{56}{171}=\frac{38}{513}.\]

\[1\dfrac{2}{27}-\frac13\cdot\frac49=\frac{29}{27}-\frac{4}{27}=\frac{25}{27}=\frac{475}{513}.\]

\[\frac{38}{513}+\frac{475}{513}=1.\]

Следовательно, всё выражение равно:

\[\frac{5}{1}=5.\]

Ответ: 5.