Solve the following given expression: \[\dfrac{{\dfrac{1}{2} \div 4 + 20}}{{\dfrac{1}{2} \times 4 + 20}}\]

Hint: The division by 4 can be written as the multiplication by reciprocal of 4 and by using the BODMAS rule, we can simplify the expression to get the final answer.

Complete step-by-step answer:
BODMAS rule gives the priority of the arithmetic operations. BODMAS stands for Bracket, Of, Division, Multiplication, Addition, and Subtraction. This is the order of preference too, with brackets being first evaluated and subtraction being evaluated last.
Hence, in the expression \[\dfrac{{\dfrac{1}{2} \div 4 + 20}}{{\dfrac{1}{2} \times 4 + 20}}\], we need to solve the division first.
The division by any number can also be written as a multiplication by its reciprocal. The division by number 4 can be written as a multiplication by the reciprocal of 4, that is, \[\dfrac{1}{4}\]. Hence, we have:
\[\dfrac{{\dfrac{1}{2} \div 4 + 20}}{{\dfrac{1}{2} \times 4 + 20}} = \dfrac{{\dfrac{1}{2} \times \dfrac{1}{4} + 20}}{{\dfrac{1}{2} \times 4 + 20}}\]
Next, we multiply the numbers in the numerator and denominator.
\[\dfrac{{\dfrac{1}{2} \div 4 + 20}}{{\dfrac{1}{2} \times 4 + 20}} = \dfrac{{\dfrac{1}{8} + 20}}{{2 + 20}}\]
As the next step, we add the numbers in the numerator and the denominator. The terms in the numerator can be added by multiplying 20 with 8 and adding to 1 and dividing by 8 again.
\[\dfrac{{\dfrac{1}{2} \div 4 + 20}}{{\dfrac{1}{2} \times 4 + 20}} = \dfrac{{\dfrac{{1 + 20 \times 8}}{8}}}{{22}}\]
Again, solving the multiplication first, we have:
\[\dfrac{{\dfrac{1}{2} \div 4 + 20}}{{\dfrac{1}{2} \times 4 + 20}} = \dfrac{{\dfrac{{1 + 160}}{8}}}{{22}}\]
We now add 1 and 160 to get 161.
\[\dfrac{{\dfrac{1}{2} \div 4 + 20}}{{\dfrac{1}{2} \times 4 + 20}} = \dfrac{{\dfrac{{161}}{8}}}{{22}}\]
We now have fraction inside a fraction, this can be easily solved to write in the simplest form. It is just 161 divided by 8 divided by 22, hence, we just multiply 8 and 22 and divide it with 161.
\[\dfrac{{\dfrac{1}{2} \div 4 + 20}}{{\dfrac{1}{2} \times 4 + 20}} = \dfrac{{161}}{{8 \times 22}}\]
The answer when 8 is multiplied with 22 is 176. Hence, we have:
\[\dfrac{{\dfrac{1}{2} \div 4 + 20}}{{\dfrac{1}{2} \times 4 + 20}} = \dfrac{{161}}{{176}}\]
Hence, the simplest form of the given expression is \[\dfrac{{161}}{{176}}\].

Note: You may commit a mistake when dividing \[\dfrac{1}{2}\] by 4 and write the answer as 2 but the correct answer is \[\dfrac{1}{8}\]. Also first multiply, then do the addition.Whenever we try to solve these types of problems, apply the BODMAS rule for arithmetic operations to get the correct answer.