Problem 2 : Basic Operations

$\bullet$ The operation is for 3 steps, each step has 4 choices. Thus you will seemingly have $64$ different sets of operations.

$\bullet$ But if your consecutive 2 operations were reverses of each other, then in their case only 1 step should give the same number and this thing is there only for number $0$ and the 1 step to be $\times$ or $\div$

$\bullet$ This will happen for following pairs, of consecutive functions " $+$ and $-$ " , " $\times$ and $\div$ ". This gives $16$ triples of functions in which only 1 step answer arrives, the answer will only be 0.

$\bullet$ In the above cases, if the other (1st or 3rd) operation is $+$ or $-$ , along with the opposites, then there are no solutions, reduces 16 cases, leaving 32 cases to look at.

If there is only $+$ and $\times$ , then that will give non-positive solutions , thus remove the 8 additional cases.

When you have a set of the type $\times$ and $\div$ at the ends, due to the middle function, either answer is 0 or you get inconsistent equation, removes $4$ cases.

$\bullet$ If the three operations are containing only $\times$ and $\div$ then answer is 0, there are 8 cases, removing few more !

$\bullet$ Computing the other things (i listed most of them, but did that somewhat smartly), just ignore cases that give negatives and you are left with a set of values

$3,6,9,\dfrac{3}{8} , \dfrac{9}{8} , \dfrac{27}{8}$ .

Their sum is $18+\dfrac{39}{8} = \dfrac{183}{8}$

Hence answer $\boxed{191}$

$Note~ that~ using ~\times~or \div~~~and~~~-~or~+~~gives~same~ result.~Just ~one~illustration ~below.\\ 3x-3-3=x~is~ the~ same~ as~\dfrac{x+3+3} 3=x.~~In ~both~ case~ x=3.\\ ~~~\\ So~an~operation~type~A~will~use~\times 3~operator.\\ and~an~operation~type~B~will~use~- 3~operator.\\ ~~~\\ Also~ note~~that~all~three~options~~of~the~same~type~will~not~give~valid~solution.\\ ~~~\\ So~the~ three ~operations~can ~ be~in~any~of~the~three~order~given~below.\\ First~two~operations~same~type.~~Last~two~same~type.~~First~and~last~same~type.\\ ~~~\\ Since~there~are~two~ type~of~operations,~~A~~and~~B,~~we~shall~have~SIX~~solutions.\\ ~~~\\ A-A-B:-~~~3x*3 -3=x,~\implies~9x-3=x.~~~\therefore~~\color{#3D99F6}{x = \dfrac 3 8.}\\ B-B-A:-~~~(x-3-3)*3=x,~\implies~3x-18=x.~~~\therefore~~\color{#3D99F6}{x = 9 }\\ A-B-B:-~~~3x-3-3=x,~\implies~3x-6=x.~~~\therefore~~\color{#3D99F6}{x = 3 }\\ B-A-A:-~~~(x-3)*3*3=x,~\implies~9x-27=x.~~~\therefore~~\color{#3D99F6}{x = \dfrac {27} 8.}\\ A-B-A:-~~~(3x-3)*3=x,~\implies~8x-9=x.~~~\therefore~~\color{#3D99F6}{x = \dfrac 9 8.} \\ B-A-B:-~~~(x-3)*3-3=x,~\implies~3x-12=x.~~~\therefore~~\color{#3D99F6}{x = 6 }\\ ~~~\\ Sum~of~x~values(in~blue),~=\dfrac{183} 8=\dfrac a b. \\ a+b=\Large~~\color{#D61F06}{191}.$