Вычислить:
$$ \frac{(12\frac{1}{6} - 6\frac{1}{27} - 5\frac{1}{4}) \cdot 13,5 + 0,111}{0,002}.$$
Решение:
\( \frac{(12\frac{1}{6} - 6\frac{1}{27} - 5\frac{1}{4}) \cdot 13,5 + 0,111}{0,002} = 599,3.\)
\(
1) 12\frac{1}{6} - 6\frac{1}{27} = 12\frac{9}{54} - 6\frac{2}{54} = 6\frac{7}{54};
\)
\(
2) 6\frac{7}{54} - 5\frac{1}{4} = 6\frac{14}{108} - 5\frac{27}{108} = 5 + 1 + \frac{14}{108} - 5\frac{27}{108} = 5 + \frac{108}{108} + \frac{14}{108} - 5\frac{27}{108} = 5 - 5 +
\)
\( \frac{(12\frac{1}{6} - 6\frac{1}{27} - 5\frac{1}{4}) \cdot 13,5 + 0,111}{0,002} = 599,3.\)
\(
1) 12\frac{1}{6} - 6\frac{1}{27} = 12\frac{9}{54} - 6\frac{2}{54} = 6\frac{7}{54};
\)
\(
2) 6\frac{7}{54} - 5\frac{1}{4} = 6\frac{14}{108} - 5\frac{27}{108} = 5 + 1 + \frac{14}{108} - 5\frac{27}{108} = 5 + \frac{108}{108} + \frac{14}{108} - 5\frac{27}{108} = 5 - 5 +
\)
\(
+ (\frac{108}{108} + \frac{14}{108} - \frac{27}{108}) = \frac{108 + 14 - 27}{108} = \frac{95}{108};
\)
\(
3) \frac{95}{108} \cdot 13,5 = \frac{95}{108} \cdot 13\frac{1}{2} = \frac{95}{108} \cdot \frac{27}{2} = \frac{95 \cdot 27}{108 \cdot 2} = \frac{95 \cdot 1}{4 \cdot 2} = \frac{95}{8} = 11\frac{7}{8};
\)
\(
4) 11\frac{7}{8} + 0,111 = 11\frac{7}{8} + \frac{111}{1000} = 11\frac{875}{1000} + \frac{111}{1000} = 11\frac{986}{1000} = 11\frac{493}{500};
\)
\(
5) 11\frac{493}{500} : 0,02 = \frac{5993}{500} : \frac{1}{50} = \frac{5993}{500} \cdot \frac{50}{1} = \frac{5993 \cdot 50}{500 \cdot 1} = \frac{5993 \cdot 1}{10 \cdot 1} = \frac{5993}{10} = 599,3.
\)
Ответ: 599,3.
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