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I was recently shown the equation $6 \div 2(1 + 2) = ?$, and it was disputed whether this equation equals $1$ or $9$.

To solve for $1$:

$$ 6 \div 2(1 + 2) \\ 6 \div 2(3) \\ 6 \div 6 \\ 1 $$

To solve for $9$:

$$ 6 \div 2(1 + 2) \\ 6 \div 2 \cdot 3 \\ 3 \cdot 3 \\ 9 $$

While it is more intuitive for the parentheses multiplication to come first, leading to $1$, PEMDAS dictates that the answer is $9$ due to left-to-right operation of multiplication and division.

What is the correct way to solve the problem? Why? What leads to this dispute? How could this ambiguity be prevented in the future?

Aly
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    Since division and multiplication have the same rank, the expression is calculated from left to right, hence $9$ is clearly the correct result. – Peter Sep 29 '18 at 01:24
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    @Peter what do you think leads to this ambiguity? i.e. why do I and others fall for the parentheses coming first? – Aly Sep 29 '18 at 01:33
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    @Soren: I would ban the use of "PEMDAS" as it leads to confusion because multiplication is before division and addition is before subtraction even though they have the same rank! – Vasili Sep 29 '18 at 02:43
  • That is not a math question; it's a linguistics question. There are at least 4 different conventions. In 2 of the 4, the expression evaluates to 1. However, a larger number of users seem to adhere to one of the others. I had thought the conventions were geographical and temporal. For example, everyone from Australia or anyone who attended grade school before 1975 seems to have been taught to evaluate that to 1. However, a recent report is that it was taught in New York as late as 1995. – Steve B Sep 29 '18 at 02:48
  • @Soren Writing the expression as a fraction would make things much clearer. In this case either $$\frac{6}{2(1+2)}$$ or $$\frac{6}{2}(1+2)$$ would be written down. – Peter Sep 29 '18 at 06:36
  • I don't see a question here, only statements. Anyway, this has been discussed many times before on this site. – Hans Lundmark Sep 29 '18 at 09:03
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    Possible duplicate of What is 48÷2(9+3)? – Hans Lundmark Sep 29 '18 at 09:04

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Although Peter's left-to-right comment is correct, there's a temptation to make division the "top" operation because of all the times in physics and chemistry and engineering we see formulas like $$ z = \dfrac{2\pi f}{m g}$$ (which isn't real but sure looks like something you'd see in science class, doesn't it?). When you write this on a single line you might write $z = 2\pi f / m g$, even though you should write $z = 2\pi f /( m g)$ to be technically correct. If you actually intended the technically correct interpretation of $z = 2\pi f / m g$, which is $$ z = \dfrac{2\pi f} { m} g$$ you would almost certainly write it as $$ z = \dfrac{2\pi f g} { m} $$ or $ z = 2\pi f g/ m $ when written on a single line.

TL;DR The division operator makes us think that we're looking at single line version of a stacked fraction and gets us confused and uncertain. People who aren't trying to intentionally confuse us will use parentheses to be clear.

JonathanZ
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