# How to solve 2 ÷ 2 ÷ 2 ? ${}{}{}{}$

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

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?

• If such a question is given, I would solve it from left to right. That said, such questions are obviously not written by mathematicians. – Carl May 31 '14 at 8:03
• I agree with Carl but people who write things like that are probably sadistic but surely not mathematicians ! – Claude Leibovici May 31 '14 at 8:06
• 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. – Henry May 31 '14 at 8:16
• 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 – Vikram May 31 '14 at 8:19
• @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!! – gnometorule Jun 2 '14 at 19:28

$$2 ÷ 2 ÷ 2 = 2 \cdot \frac{1}{2} \cdot \frac{1}{2}$$
$$f \circ g |_{x} = f(g(x))$$ 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: $$\left(a^b\right)^c\neq a^{(b^c)}$$
• How $2 ÷ 2 ÷ 2 = \frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2}$? – user103816 May 31 '14 at 8:24
• @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'. – user103816 May 31 '14 at 8:40