Simplify 8 2/3 ÷ (23 7/8 of 3 1/4 - 22)

$$8\frac{2}{3} = \frac{8 \times 3 + 2}{3} = \frac{24 + 2}{3} = \frac{26}{3}$$ $$23\frac{2}{5} = \frac{23 \times 5 + 2}{5} = \frac{115 + 2}{5} = \frac{117}{5}$$ $$3\frac{1}{4} = \frac{3 \times 4 + 1}{4} = \frac{12 + 1}{4} = \frac{13}{4}$$

The expression becomes:

$$\frac{26}{3} \div \left(\frac{117}{5} \text{ OF } \frac{13}{4} - 22\right)$$

$$\frac{117}{5} \times \frac{13}{4} = \frac{117 \times 13}{5 \times 4} = \frac{1521}{20}$$

The expression now is:

$$\frac{26}{3} \div \left(\frac{1521}{20} - 22\right)$$

To subtract 22 from $\frac{1521}{20}$, we convert 22 to a fraction with a denominator of 20.

$$22 = \frac{22 \times 20}{20} = \frac{440}{20}$$

Now subtract:

$$\frac{1521}{20} - \frac{440}{20} = \frac{1521 - 440}{20} = \frac{1081}{20}$$

The expression now is:

$$\frac{26}{3} \div \frac{1081}{20}$$

To divide by a fraction, multiply by its reciprocal.

$$\frac{26}{3} \div \frac{1081}{20} = \frac{26}{3} \times \frac{20}{1081}$$

Multiply the numerators and the denominators:

$$\frac{26 \times 20}{3 \times 1081} = \frac{520}{3243}$$

The fraction $\frac{520}{3243}$ cannot be simplified further as there are no common factors between the numerator and the denominator ($520 = 2^3 \times 5 \times 13$ and $3243 = 3 \times 23 \times 47$).