[tex]7\frac{1}{5}:2\frac{4}{7} -8\frac{3}{4} :[10-(5\frac{1}{2})^{2} :4\frac{2}{5} ]+(3\frac{1}{8}*2):(8\frac{1}{12} -1\frac{5}{6} )[/tex]

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Let's simplify this expression step by step:

First, start with the innermost parentheses:

[10-(5\frac{1}{2})^{2} = 10 - \left(\frac{11}{2}\right)^2 = 10 - \frac{121}{4} = 10 - 30.25 = -20.25]

Next, divide -20.25 by 4 2/5:

[-20.25 : 4\frac{2}{5} = -20.25 \div \frac{22}{5} = -20.25 \times \frac{5}{22} = -4.59]

Now, substitute this into the original expression:

[7\frac{1}{5} : 2\frac{4}{7} - 8\frac{3}{4} : (-4.59) + \left(\frac{25}{8} \times 2\right) : \left(8\frac{1}{12} - 1\frac{5}{6}\right)]

Next, divide and multiply:

[7\frac{1}{5} : 2\frac{4}{7} = \frac{36}{5} : \frac{18}{7} = \frac{36}{5} \times \frac{7}{18} = \frac{7}{5}]

[8\frac{3}{4} : (-4.59) = \frac{35}{4} : -4.59 = \frac{35}{4} \div -4.59 = \frac{35}{4} \div -4 59/100 = \frac{175}{20} \div -459/100 = -0.38125325]

[\left(\frac{25}{8} \times 2\right) = 25/4 = 6.25]

[\left(8\frac{1}{12} - 1\frac{5}{6}\right) = 97/12 - 11/6 = -49/12]

Finally, substitute these simplified values back into the expression:

[\frac{7}{5} - 0.38125325 + 6.25 : \left(-\frac{49}{12}\right)]

Perform the division and addition first, then the subtraction:

[6.25 : \left(-\frac{49}{12}\right) = 6.25 \div -49/12 = 6.25 \div -4.08333333 = -1.53125]

[\frac{7}{5} - 0.38125325 - 1.53125 = 1.4 - 0.38125325 - 1.53125 = -0.51125325]

Therefore, the final result is approximately -0.51125325.