What complexity class decision problems can be solved by only addition, multiplication, division and subtraction? So in what complexity class would all decision problems in that class can be transformed into a problem (by many-one reduction) that its solution only involves basic arithmetic operation of multiplication, division, addition and subtraction? 

So, like this: for every problem, there is input and output (we consider only those that halt) and my question is that is there any complexity class that all problems in the class can be transformed into a problem that involves only elementary arithmetic of the form of arithmetic function involving only arithmetic like $2^x+x^5/4-4x+2$ where $x$ is input coded as natural number.
The question isn't exactly well-defined, but seems like a good/interesting question to me. I believe the answer you might be looking for is the class of Elementary Recursive functions. This class consists of almost exactly what you described: Problems solvable by functions that can be written with only addition, subtraction, and multiplication of integers. (Except, we also require a zero function and a "projection" function that outputs one of the inputs.)

You can get more functions (the Primitive Recursive functions) by additionally allowing functions that call themselves (but are guaranteed to terminate). Finally, you can get the class of all Turing-computable functions by introducing a minimization operator: find the minimum $x$ such that $f(x) = 0$.

Anyway, it turns out that $\mathsf{EXPTIME} \subsetneq \mathsf{ELEMENTARY}$, so it includes many (most?) problems that we'd usually consider interesting.