The problem is to evaluate the expression $(2\frac{1}{27} \times 1\frac{1}{9}) \div 1\frac{5}{6}$. This involves mixed numbers, multiplication, and division.

1. Problem Description

The problem is to evaluate the expression $(2\frac{1}{27} \times 1\frac{1}{9}) \div 1\frac{5}{6}$. This involves mixed numbers, multiplication, and division.

2. Solution Steps

First, convert the mixed numbers to improper fractions.

$2\frac{1}{27} = \frac{2 \times 27 + 1}{27} = \frac{54 + 1}{27} = \frac{55}{27}$

$1\frac{1}{9} = \frac{1 \times 9 + 1}{9} = \frac{9 + 1}{9} = \frac{10}{9}$

$1\frac{5}{6} = \frac{1 \times 6 + 5}{6} = \frac{6 + 5}{6} = \frac{11}{6}$

Now, substitute the improper fractions into the original expression.

$(\frac{55}{27} \times \frac{10}{9}) \div \frac{11}{6}$

Multiply the first two fractions:

$\frac{55}{27} \times \frac{10}{9} = \frac{55 \times 10}{27 \times 9} = \frac{550}{243}$

Now, divide the result by $\frac{11}{6}$. Dividing by a fraction is the same as multiplying by its reciprocal.

$\frac{550}{243} \div \frac{11}{6} = \frac{550}{243} \times \frac{6}{11}$

$\frac{550}{243} \times \frac{6}{11} = \frac{550 \times 6}{243 \times 11} = \frac{3300}{2673}$

Now, simplify the fraction. We can divide both the numerator and the denominator by

3

3. $\frac{3300 \div 33}{2673 \div 33} = \frac{100}{81}$

3. Final Answer

$\frac{100}{81}$