Determine the value of the expression: $(\frac{1}{5}÷\frac{1}{3})÷(\frac{1}{10}+1-\frac{8}{10})$

The correct answer is 2.

To evaluate the given mathematical expression, we follow the order of operations by simplifying the expressions inside both sets of parentheses first, and then dividing the two resulting values.

The given expression is:
$(\frac{1}{5}÷\frac{1}{3})÷(\frac{1}{10}+1-\frac{8}{10})$

Step 1: Simplify the first set of parentheses
To divide fractions, multiply the first fraction by the reciprocal of the second fraction:
$\frac{1}{5}÷\frac{1}{3}=\frac{1}{5}\times \frac{3}{1}=\frac{3}{5}$

Step 2: Simplify the second set of parentheses
First, express the whole number 1 as a fraction with a denominator of 10:
$1=\frac{10}{10}$
Now evaluate the addition and subtraction from left to right:
$\frac{1}{10}+1-\frac{8}{10}=\frac{1}{10}+\frac{10}{10}-\frac{8}{10}=\frac{1+10-8}{10}=\frac{3}{10}$

Step 3: Perform the final division
Now divide the result from Step 1 by the result from Step 2:
$\frac{3}{5}÷\frac{3}{10}=\frac{3}{5}\times \frac{10}{3}$
Multiplying the numerator and denominator gives:
$\frac{3\times 10}{5\times 3}=\frac{30}{15}=2$

Thus, the value of the given expression is 2.