Evaluate the "of" operation in the numerator.

First, convert the mixed fraction to an improper fraction:

$$2 \frac{2}{5} = \frac{2 \times 5 + 2}{5} = \frac{12}{5}$$

Now, calculate $\frac{5}{6} \text{ of } \frac{12}{5}$:

$$\frac{5}{6} \times \frac{12}{5} = \frac{5 \times 12}{6 \times 5} = \frac{60}{30} = 2$$

Convert the mixed fraction to an improper fraction:

$$1 \frac{3}{4} = \frac{1 \times 4 + 3}{4} = \frac{7}{4}$$

Add this to the result from Step 1:

$$\frac{7}{4} + 2 = \frac{7}{4} + \frac{8}{4} = \frac{7+8}{4} = \frac{15}{4}$$

So, the numerator is $\frac{15}{4}$.

Convert the mixed fraction to an improper fraction:

$$2 \frac{4}{5} = \frac{2 \times 5 + 4}{5} = \frac{14}{5}$$

Perform the division:

$$\frac{14}{5} \div \frac{1}{10} = \frac{14}{5} \times \frac{10}{1} = \frac{14 \times 10}{5 \times 1} = \frac{140}{5} = 28$$

So, the denominator is 28.

Divide the numerator (from Step 2) by the denominator (from Step 3):

$$\frac{\frac{15}{4}}{28} = \frac{15}{4} \div 28 = \frac{15}{4} \times \frac{1}{28} = \frac{15 \times 1}{4 \times 28} = \frac{15}{112}$$

The fraction $\frac{15}{112}$ is in its simplest form as 15 and 112 share no common factors other than 1.