a: \(\dfrac{A}{B}=\dfrac{\dfrac{4}{7\cdot31}+\dfrac{6}{7\cdot41}+\dfrac{9}{10\cdot41}+\dfrac{7}{10\cdot57}}{\dfrac{7}{19\cdot31}+\dfrac{5}{19\cdot43}+\dfrac{3}{23\cdot43}+\dfrac{11}{23\cdot57}}\)
\(=\dfrac{\dfrac{20}{31\cdot35}+\dfrac{30}{35\cdot41}+\dfrac{45}{41\cdot50}+\dfrac{35}{50\cdot57}}{\dfrac{14}{31\cdot38}+\dfrac{10}{38\cdot43}+\dfrac{6}{43\cdot46}+\dfrac{22}{46\cdot57}}\)
\(=\dfrac{5\left(\dfrac{4}{31\cdot35}+\dfrac{6}{35\cdot41}+\dfrac{9}{41\cdot50}+\dfrac{7}{50\cdot57}\right)}{2\left(\dfrac{7}{31\cdot38}+\dfrac{5}{38\cdot43}+\dfrac{3}{43\cdot46}+\dfrac{11}{46\cdot57}\right)}\)
\(=\dfrac{5\left(\dfrac{1}{31}-\dfrac{1}{35}+\dfrac{1}{35}-\dfrac{1}{41}+\dfrac{1}{41}-\dfrac{1}{50}+\dfrac{1}{50}-\dfrac{1}{57}\right)}{2\left(\dfrac{1}{31}-\dfrac{1}{38}+\dfrac{1}{38}-\dfrac{1}{43}+\dfrac{1}{43}-\dfrac{1}{46}+\dfrac{1}{46}-\dfrac{1}{57}\right)}\)
\(=\dfrac{5\left(\dfrac{1}{31}-\dfrac{1}{57}\right)}{2\left(\dfrac{1}{31}-\dfrac{1}{57}\right)}=\dfrac{5}{2}\)
b: \(\dfrac{A}{B}=\dfrac{\dfrac{34}{7\cdot13}+\dfrac{51}{13\cdot22}+\dfrac{85}{22\cdot37}+\dfrac{68}{37\cdot49}}{\dfrac{39}{7\cdot16}+\dfrac{65}{16\cdot31}+\dfrac{52}{31\cdot43}+\dfrac{26}{43\cdot49}}\)
\(=\dfrac{\dfrac{17}{3}\left(\dfrac{6}{7\cdot13}+\dfrac{9}{13\cdot22}+\dfrac{15}{22\cdot37}+\dfrac{12}{37\cdot49}\right)}{\dfrac{13}{3}\left(\dfrac{9}{7\cdot16}+\dfrac{15}{16\cdot31}+\dfrac{12}{31\cdot43}+\dfrac{6}{43\cdot49}\right)}\)
\(=\dfrac{\dfrac{17}{3}\left(\dfrac{1}{7}-\dfrac{1}{13}+\dfrac{1}{13}-\dfrac{1}{22}+\dfrac{1}{22}-\dfrac{1}{37}+\dfrac{1}{37}-\dfrac{1}{49}\right)}{\dfrac{13}{3}\left(\dfrac{1}{7}-\dfrac{1}{16}+\dfrac{1}{16}-\dfrac{1}{31}+\dfrac{1}{31}-\dfrac{1}{43}+\dfrac{1}{43}-\dfrac{1}{49}\right)}\)
\(=\dfrac{\dfrac{17}{3}\left(\dfrac{1}{7}-\dfrac{1}{49}\right)}{\dfrac{13}{3}\left(\dfrac{1}{7}-\dfrac{1}{49}\right)}=\dfrac{17}{3}:\dfrac{13}{3}=\dfrac{17}{13}\)
