14. Тин 13 № 2068 Вычислите: 4: (\frac{16}{19}+3\frac{5}{12}-3\frac{13}{24}). Запишите полностью решение и ответ.

Ответ: 19/2

\[4 : (\frac{16}{19} + 3\frac{5}{12} - 3\frac{13}{24}) = 4 : (\frac{16}{19} + \frac{41}{12} - \frac{85}{24}) = 4 : (\frac{16 \cdot 12 \cdot 24 + 41 \cdot 19 \cdot 24 - 85 \cdot 19 \cdot 12}{19 \cdot 12 \cdot 24}) = 4 : (\frac{4608 + 18696 - 19380}{5472}) = 4 : (\frac{3904}{5472}) = 4 : (\frac{16 \cdot 244}{16 \cdot 342}) = 4 : (\frac{244}{342}) = 4 : (\frac{2 \cdot 122}{2 \cdot 171}) = 4 : (\frac{122}{171}) = 4 \cdot \frac{171}{122} = \frac{4 \cdot 171}{122} = \frac{2 \cdot 171}{61} = \frac{342}{61} = \frac{18 \cdot 19}{32 \cdot 19} = 5.606...\] \[4 : (\frac{16}{19}+3\frac{5}{12}-3\frac{13}{24})=4 : (\frac{16}{19}+\frac{41}{12}-\frac{85}{24})\] \(3 \frac{5}{12} = 3 + \frac{5}{12} = \frac{36}{12} + \frac{5}{12} = \frac{41}{12}\) \(3 \frac{13}{24}= 3 + \frac{13}{24} = \frac{72}{24} + \frac{13}{24} = \frac{85}{24}\) \[4 : (\frac{16}{19}+\frac{41}{12}-\frac{85}{24}) = 4 : (\frac{16 \cdot 12 \cdot 24}{19 \cdot 12 \cdot 24}+\frac{41 \cdot 19 \cdot 24}{19 \cdot 12 \cdot 24}-\frac{85 \cdot 19 \cdot 12}{19 \cdot 12 \cdot 24}) = 4 : (\frac{4608}{5472} + \frac{18696}{5472} - \frac{19380}{5472})\] \[4 : (\frac{4608 + 18696 - 19380}{5472}) = 4 : (\frac{3924}{5472}) = 4 : (\frac{61 \cdot 64}{61 \cdot 88}) = 4 : \frac{64}{88} = 4 : \frac{8 \cdot 8}{8 \cdot 11} = 4 : \frac{8}{11} = 4 \cdot \frac{11}{8} = \frac{4 \cdot 11}{8} = \frac{44}{8} = \frac{4 \cdot 11}{4 \cdot 2} = \frac{11}{2}\] \[4 : (\frac{16}{19}+\frac{41}{12}-\frac{85}{24}) = 4 : (\frac{16 \cdot 144 + 41 \cdot 361 - 85 \cdot 228}{4616}) = 4 : (\frac{2304+14801-19380}{4616}) = 4 : \frac{-2275}{4616}\] \[4 : (\frac{16}{19}+\frac{41}{12}-\frac{85}{24}) = 4 : (\frac{16 \cdot 288 + 41 \cdot 456 - 85 \cdot 228}{5472}) = 4 : (\frac{4608+18696-19380}{5472})\] \[4 : (\frac{16}{19}+\frac{41}{12}-\frac{85}{24}) = 4 : (\frac{16 \cdot 12 \cdot 2 + 41 \cdot 19 \cdot 2 - 85 \cdot 19}{456}) = 4 : (\frac{384 + 1558 - 1615}{456}) = 4 : (\frac{384 + 1558 - 1615}{456}) = 4 : (\frac{327}{456})\] \[4:(\frac{16}{19}+3\frac{5}{12}-3\frac{13}{24}) = 4:(\frac{16}{19}+\frac{41}{12}-\frac{85}{24}) = 4:(\frac{16*12*2+41*19*2-85*19}{19*12*2}) = 4:(\frac{384+1558-1615}{456}) = 4:(\frac{327}{456}) = 4*(\frac{456}{327}) = 4*(\frac{152}{109}) = \frac{608}{109}\]

Ответ: 608/109