$$2 ÷ 2 ÷ 2 = (2 ÷ 2) ÷ 2 \ \ \text{OR}\ \ 2 ÷ (2 ÷ 2) ?$$

Is there any standard rule which is world wide accepted for solving this type of expressions? If I process the expression from left to right then I will get $\dfrac12$. But if I process it from right to left then I get $\dfrac21$, that is $2$.

It might be that it is in invalid expression. But these type of questions are usualy asked in India's exams. E.g. the 82$^{th}$ question of SBI Clerk Exam (Held on 06-07-2008) was:
$$82.Q:\ \ \ \ \ \ \ \ \ \ \ 14400÷64÷9=?$$
The answer given was $25$. They appear to assume the order of execution from left to right.

So is the standard rule is to execute the order of operations is from left to right?

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    $\begingroup$ If such a question is given, I would solve it from left to right. That said, such questions are obviously not written by mathematicians. $\endgroup$ – Carl May 31 '14 at 8:03
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    $\begingroup$ I agree with Carl but people who write things like that are probably sadistic but surely not mathematicians ! $\endgroup$ – Claude Leibovici May 31 '14 at 8:06
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    $\begingroup$ Most calculators and computer languages will execute multiplication and division left-to-right. Similarly for addition and subtraction. But for other expressions such as 1+2*3 and 2^3^2, answers vary between implementations: so either 9 or 7 and either 64 or 512, and most mathematicians would choose $1+2\times 3= 7$ and $2^{3^2}=512$, in the former case doing multiplication before addition, and in the latter case operating right-to-left for exponentiation. $\endgroup$ – Henry May 31 '14 at 8:16
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    $\begingroup$ I see it as $\frac{a}{b/c}=\frac{ac}{b}$, +1 for letting others know what "type"(insert your fav. word here) of questions are asked in India and I totally agree with Carl and Claude $\endgroup$ – Vikram May 31 '14 at 8:19
  • $\begingroup$ @Henry: Could you point out an "implementation" that calculates 1+2*3 = 9? I would be most curious to find out which computer language / compiler (interpreter) combo you had in mind when making your point!! $\endgroup$ – gnometorule Jun 2 '14 at 19:28

$$2 ÷ 2 ÷ 2 = 2 \cdot \frac{1}{2} \cdot \frac{1}{2} $$

Also in multiplication and division you have to go left to right then if you have addition and substraction you also have to go from left to right. Also if you have exponentiation you do that first and if you have brackets you have to do what is inside the brackets, before everything else.

Howewer when you have function composition, and exponentiation you go from right to left. For example:

$$f \circ g |_{x} = f(g(x))$$ so you first apply $g$ to $x$ and then you apply $f$ to the result of $g(x)$. When you have exponentiation: $$a^b = a \uparrow b$$ $$a^{b^c} = a \uparrow b \uparrow c$$ In the last case you also go from right to left. So: $$\left(a^b\right)^c\neq a^{(b^c)}$$

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    $\begingroup$ How $2 ÷ 2 ÷ 2 = \frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2}$? $\endgroup$ – user103816 May 31 '14 at 8:24
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    $\begingroup$ Yeah that is wrong, I corrected it. $\endgroup$ – 05storm26 May 31 '14 at 8:24
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    $\begingroup$ @05storm26, OP has asked a simple arithmetical problem, can you add something to the discussion under consideration? $\endgroup$ – Vikram May 31 '14 at 8:28
  • $\begingroup$ @Vikram I understand what "o5strom26" is trying to say. In many cases the order is from right to left as he is saying. But $2 ÷ 2 ÷ 2 = 2 \cdot \frac{1}{2} \cdot \frac{1}{2}$ is still wrong :-/ . Nonetheless '+1'. $\endgroup$ – user103816 May 31 '14 at 8:40

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