The problem is to evaluate the expression $25\frac{3}{4} + 6[7\frac{1}{2} - 8\{\frac{7}{15} \div (1\frac{2}{15} - \frac{2}{15})\}]$.

1. Problem Description

The problem is to evaluate the expression $25\frac{3}{4} + 6[7\frac{1}{2} - 8\{\frac{7}{15} \div (1\frac{2}{15} - \frac{2}{15})\}]$.

2. Solution Steps

First, convert the mixed numbers to improper fractions.

$25\frac{3}{4} = \frac{25 \times 4 + 3}{4} = \frac{100 + 3}{4} = \frac{103}{4}$

$7\frac{1}{2} = \frac{7 \times 2 + 1}{2} = \frac{14 + 1}{2} = \frac{15}{2}$

$1\frac{2}{15} = \frac{1 \times 15 + 2}{15} = \frac{15 + 2}{15} = \frac{17}{15}$

Substitute these values into the expression.

$\frac{103}{4} + 6[\frac{15}{2} - 8\{\frac{7}{15} \div (\frac{17}{15} - \frac{2}{15})\}]$

Next, evaluate the expression within the innermost parentheses.

$\frac{17}{15} - \frac{2}{15} = \frac{17 - 2}{15} = \frac{15}{15} = 1$

Now the expression becomes:

$\frac{103}{4} + 6[\frac{15}{2} - 8\{\frac{7}{15} \div 1\}]$

Evaluate the expression within the curly braces.

$\frac{7}{15} \div 1 = \frac{7}{15}$

Now the expression becomes:

$\frac{103}{4} + 6[\frac{15}{2} - 8(\frac{7}{15})]$

Evaluate the expression within the square brackets.

$8(\frac{7}{15}) = \frac{8 \times 7}{15} = \frac{56}{15}$

$\frac{15}{2} - \frac{56}{15} = \frac{15 \times 15}{2 \times 15} - \frac{56 \times 2}{15 \times 2} = \frac{225}{30} - \frac{112}{30} = \frac{225 - 112}{30} = \frac{113}{30}$

Now the expression becomes:

$\frac{103}{4} + 6(\frac{113}{30})$

$6(\frac{113}{30}) = \frac{6 \times 113}{30} = \frac{678}{30} = \frac{113}{5}$

Finally,

$\frac{103}{4} + \frac{113}{5} = \frac{103 \times 5}{4 \times 5} + \frac{113 \times 4}{5 \times 4} = \frac{515}{20} + \frac{452}{20} = \frac{515 + 452}{20} = \frac{967}{20}$

3. Final Answer

$\frac{967}{20}$