514. Найдите значение выражения: а) \(\frac14\cdot4\frac34\cdot\frac{16}{57}+\left(4\frac34+1\frac23\right)\cdot\frac{16}{21}+\frac{2}{27}\cdot4\frac12\); б) \(\left(\frac45+\frac16\right)\cdot\left(23\frac23-15\frac59\right):\frac{45}{58}-\frac12\).

а) Преобразуем смешанные числа: \(4\frac34=\frac{19}{4}\), \(1\frac23=\frac53\), \(4\frac12=\frac92\).

\[\frac14\cdot\frac{19}{4}\cdot\frac{16}{57}+\left(\frac{19}{4}+\frac53\right)\cdot\frac{16}{21}+\frac{2}{27}\cdot\frac92\]

\[=\frac{19}{57}+\frac{77}{12}\cdot\frac{16}{21}+\frac13=\frac{19}{57}+\frac{44}{9}+\frac13=\frac{19}{57}+\frac{47}{9}=\frac{106}{19}.\]

Ответ: \(\frac{106}{19}=5\frac{11}{19}\).

б) \(23\frac23=\frac{71}{3}\), \(15\frac59=\frac{140}{9}\), поэтому

\[\left(\frac45+\frac16\right)\cdot\left(\frac{71}{3}-\frac{140}{9}\right):\frac{45}{58}-\frac12\]

\[=\frac{29}{30}\cdot\frac{73}{9}\cdot\frac{58}{45}-\frac12=\frac{15341}{6075}-\frac12=\frac{27607}{12150}.\]

Ответ: \(\frac{27607}{12150}=2\frac{3307}{12150}\).