$8 \frac{7}{11} \div \frac{5}{11}$

Step 1 ：Convert the mixed number $8 \frac{7}{11}$ to an improper fraction. This is done by multiplying the whole number by the denominator and adding the numerator. So, $8 \times 11 + 7 = 95$. Therefore, $8 \frac{7}{11}$ is equivalent to $\frac{95}{11}$.

Step 2 ：Next, divide the two fractions by multiplying the first fraction by the reciprocal of the second fraction. The reciprocal of $\frac{5}{11}$ is $\frac{11}{5}$. So, $\frac{95}{11} \div \frac{5}{11} = \frac{95}{11} \times \frac{11}{5} = \frac{1045}{55}$.

Step 3 ：The result of the division is an improper fraction. To make it easier to understand, we can convert it back to a mixed number. The whole number is the quotient of the division of the numerator by the denominator, and the remainder is the new numerator. So, $\frac{1045}{55} = 19 \frac{0}{11}$.

Step 4 ：Finally, since the numerator of the fraction part is 0, we can simplify $19 \frac{0}{11}$ to $19$.