Решение:
\( \frac{4,5 : [47,375 - (26\frac{1}{3} - 18 \cdot 0,75) \cdot 2,4 : 0,88]}{17,81 : 1,37 - 23\frac{2}{3} : 1\frac{5}{6}} = 4.\)

\(
1) 18 \cdot 0,75 = 13,5 = 13\frac{5}{10} = 13\frac{1}{2};
\)

\(
2) 26\frac{1}{3} - 13\frac{1}{2} = 26\frac{2}{6} - 13\frac{3}{6} = 25 + 1 + \frac{2}{6} - 13 - \frac{3}{6} = 25 - 13 + (\frac{6}{6} + \frac{2}{6} - \frac{3}{6}) = 12 + \frac{6 + 2 - 3}{6} = 12 + \frac{5}{6} = 12\frac{5}{6};
\)

\(
3) 12\frac{5}{6} \cdot 2,4 = 12\frac{5}{6} \cdot 2\frac{4}{10} = 12\frac{5}{6} \cdot 2\frac{2}{5} = \frac{77}{6} \cdot \frac{12}{5} = \frac{77 \cdot 12}{6 \cdot 5} = \frac{77 \cdot 2}{1 \cdot 5} = \frac{154}{5} = 30\frac{4}{5} = 30,8;
\)

\(
4) 30,8 : 0,88 = 35;
\)

\(
5) 47,375 - 35 = 12,375;
\)

\(
6) 4,5 : 12,375 = 4\frac{5}{10} : 12\frac{375}{1000} = 4\frac{1}{2} : 12\frac{3}{8} = \frac{9}{2} \cdot \frac{8}{99} = \frac{9 \cdot 8}{2 \cdot 99} = \frac{1 \cdot 4}{1 \cdot 11} = \frac{4}{11};
\)

\(
7) 17,81 : 1,37 = 13;
\)

\(
8) 23\frac{2}{3} : 1\frac{5}{6} = \frac{71}{3} \cdot \frac{6}{11} = \frac{71 \cdot 6}{3 \cdot 11} = \frac{71 \cdot 2}{1 \cdot 11} = \frac{142}{11} = 12\frac{10}{11};
\)

\(
9) 13 - 12\frac{10}{11} = 12 + \frac{11}{11} - 12 - \frac{10}{11} = \frac{11 - 10}{11} = \frac{1}{11};
\)

\(
10) \frac{4}{11} : \frac{1}{11} = \frac{4}{11} \cdot \frac{11}{1} = \frac{4 \cdot 11}{11 \cdot 1} = \frac{4 \cdot 1}{1 \cdot 1} = \frac{4}{1} = 4.
\)
Ответ: 4.

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