Simplify the following. $\frac{2\{3+3(\frac{1}{4}÷\frac{15}{20}+2)-18\times \frac{1}{3}\}}{2(4+20÷4)}$

The correct answer is 4/9.

Let us simplify the given mathematical expression step-by-step using the BODMAS (Brackets, Orders, Division, Multiplication, Addition, Subtraction) rule.

The given expression is:
$\frac{2\{3+3(\frac{1}{4}÷\frac{15}{20}+2)-18\times \frac{1}{3}\}}{2(4+20÷4)}$

Step 1: Simplify the innermost parentheses in the numerator
Let's first simplify the term inside the parenthesis: $\frac{1}{4}÷\frac{15}{20}+2$ .
Dividing by a fraction is the same as multiplying by its reciprocal:
$\frac{1}{4}÷\frac{15}{20}=\frac{1}{4}\times \frac{20}{15}=\frac{20}{60}=\frac{1}{3}$
Now, add 2 to this result:
$\frac{1}{3}+2=\frac{1+6}{3}=\frac{7}{3}$

Step 2: Simplify the curly brackets in the numerator
Substitute $\frac{7}{3}$ back into the numerator's expression:
$3+3\left(\frac{7}{3}\right)-18\times \frac{1}{3}$
Multiply the terms:
$3\times \frac{7}{3}=7$
and
$18\times \frac{1}{3}=6$
So, the terms inside the curly brackets become:
$3+7-6=10-6=4$

Step 3: Calculate the entire numerator
Now multiply by the 2 outside the curly brackets:
$\text{Numerator}=2\times 4=8$

Step 4: Simplify the denominator
The expression for the denominator is:
$2(4+20÷4)$
First, perform the division inside the parentheses:
$20÷4=5$
Now, add 4 to this value:
$4+5=9$
Finally, multiply by 2:
$\text{Denominator}=2\times 9=18$

Step 5: Divide the numerator by the denominator
Combine the results of the numerator and the denominator:
$\frac{\text{Numerator}}{\text{Denominator}}=\frac{8}{18}$
Reducing the fraction to its simplest form by dividing both the numerator and the denominator by their greatest common divisor, which is 2:
$\frac{8÷2}{18÷2}=\frac{4}{9}$

Thus, the simplified value of the expression is 4/9.