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There is a huge debate on the internet on the value of $48\div2(9+3)$.

I believe the answer $2$ as I believe it is part of the bracket operation in BEDMAS. Mathway yields the same answer. I also believe that if $48\div2\times(9+3)$ was asked it would be $288$ which Mathway agrees with as well.

However, WolframAlpha says it is $288$ either way.

A friend of mine (who is better at math) told me that there is no such thing as 'implicit multiplication', only shorthand so that is in fact done after the division (going left to right, not necessarily because division occurs before multiplication. But he didn't explicitly give a reason)

What is the answer and why?

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    $\begingroup$ I have never understood why one would want to establish arbitrary conventions on the order of operations. Just place appropriate parentheses and remove any ambiguity. Why would one ever want to write something like $5+8\cdot 6:4+3-2:5$? $\endgroup$
    – wildildildlife
    Apr 9, 2011 at 15:34
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    $\begingroup$ @wildildildlife: Well, the reason we put a convention on the order of operations is to make our life simpler. Not for problems like this, mind you, but for general expressions. E.g., if you had to write all the necessary parentheses every time you write out a polynomial, it would obscure things rather than clarify them. Instead of $2x^2+7x+1$, you would have to write$$\Bigl(2\bigl(x^2\bigr)\Bigl) + \Bigl(7x\Bigr) + 1.$$ $\endgroup$
    – Arturo Magidin
    Apr 9, 2011 at 19:36
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    $\begingroup$ «There is a huge debate on the internet»?! $\endgroup$
    – Mariano Suárez-Álvarez
    Apr 15, 2011 at 21:27
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    $\begingroup$ I must be crazy. I wasted 5 min of my life on this ... $\endgroup$
    – GWu
    Apr 15, 2011 at 21:59
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    $\begingroup$ I voted to reopen. I don't see why this is not a real question. The fact that someone picked it up as a meme on the net and wrote "there's a huge debate on the internet" doesn't make the question whether division or multiplication takes precedence here invalid; this is a perfectly legitimate question about mathematical notation. $\endgroup$
    – joriki
    Feb 25, 2012 at 17:22

4 Answers 4

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There is no Supreme Court for mathematical notation; there were no commandments handed down on Sinai concerning operational precedence; all there is, is convention, and different people are free to adhere to different conventions. Wise people will stick in enough parentheses to make it impossible for anyone to mistake the meaning. If they mean, $(48\div2)(9+3)$, they'll write it that way; if they mean $48\div\bigl(2(9+3)\bigr)$, they'll write it that way.

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    $\begingroup$ Recommended reading on this topic: "Order of operations" and other oddities in school mathematics by Hung-Hsi Wu, a Berkeley mathematician who has given much thought to mathematics education. $\endgroup$
    – Bill Dubuque
    May 4, 2011 at 14:30
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    $\begingroup$ My teacher adheres opposite to the convention I like. And she writes it so that there's ambiguity, resulting in rare "mistakes" on my part. Whenever I see $48÷2(9+3)$, I automatically think $2$. $\endgroup$
    – Mateen Ulhaq
    May 7, 2011 at 0:01
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    $\begingroup$ @muntoo, your teacher isn't wise. :) $\endgroup$
    – JMCF125
    Jun 13, 2013 at 22:19
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    $\begingroup$ I don't agree with this type of answer: even if this expression is written by mixing old style symbols and new style ones, you cannot let the idea that you can evaluate an expression by your mood, or convention. There is ONE convention which should be agreed on by any mathematician: multiplication and division have the same precedence. $\endgroup$
    – Emanuele Paolini
    Oct 14, 2013 at 5:35
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    $\begingroup$ @EmanuelePaolini: "ONE convention which should be agreed on by any mathematician"? I think the problem is exactly the fact that most mathematicians don't use the $\div$ sign, so (from lack of use) we haven't really agreed on a convention. (In fact, just to write that sentence I had to look up how to write $\div$ in TeX because I never use it.) $\endgroup$
    – Jesse Madnick
    Mar 4, 2014 at 13:58
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It's ambiguous, there is not one right answer in this case, other than possibly that it is undefined. You may have $$\frac{48}{2(9+3)} = 2$$
or
$$\frac{48}{2}(9+3) = 288$$

Therefore, there is no point in debating this.

Note that the reason you are get different answers from mathway and google calculator is that the algorithms they use for parsing input are different. These algorithms apparently (and understandably) leave it up to the user in this case to give input that can only be interpreted in one way. This is not the case, and is therefore why the two's answers differ.

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  • $\begingroup$ Is there nothing that says if "2(" is part of the bracket operation or not while doing order of operation? If not then i'll agree. $\endgroup$
    – user9638
    Apr 15, 2011 at 21:42
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    $\begingroup$ @acidzombie24: No, there is not. The standard notation to use in this case if you are going to use brackets is to use "fractions" rather than $÷$, as shown. Hope this clears things up =D $\endgroup$
    – Justin
    Apr 15, 2011 at 21:44
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I would say it isn't even well-defined. In Group Theory or such, you usually pass by a statement that says "associativity means that $(1 + 2) + 3$ is the same as $1 + (2+3)$, so we can write $1 + 2 + 3$ without ambiguity." $\div$ doesn't have this property of being unambiguous.

This is one of the advantages of using $\frac{48}{2}(9+3)$ or $\frac{48}{2(9+3)}$ - it's not quite associative, but it isn't ambiguous. I haven't seen $\div$ since elementary school probably for this very reason.

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    $\begingroup$ And personally, I like to think of the obelus as a symbol meaning that "something" goes on top and bottom of the line, and the dots are just placeholders for the things on the left and right. $\endgroup$
    – Joe Z.
    Apr 2, 2014 at 4:00
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    $\begingroup$ Would it really be any different if ÷ were replaced with /, the symbol most programming languages use for divide? $\endgroup$
    – Alec Jacobson
    Dec 5, 2017 at 3:05
  • $\begingroup$ forward slash would then put everything following into the denominator. and hence would make it obvious $\frac{48}{2(9+3)}$ $\endgroup$
    – TheArchitecta
    Nov 26, 2022 at 10:05
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There is no order difference between implicit and explicit multiplications. Purplemath suggests that implied multiplication outside of parentheses also gets parenthetical order priority over all other multiplication(division). So they would interpret the implied multiplied parenthetical as $48\div(2\times(9+3)) = 2$.

Alternatively, all implicit-capable calculators I've tried give the same results as Wolfram Alpha which interprets the implication as $(48\div2)\cdot(9+3) = 288$

Some confusion also seems to be the $\div$ division symbol itself as $48/2(9+3)$ visually supports the mutual parenthetical implied multiplication of Wolfram Alpha: $\frac{48}{2}\cdot(9+3) = 288$

The short answer is that the formula as written is too easily misinterpreted and the author should clarify it to ensure its proper calculation.

PS: to further furrow your brow - employing the parenthetical as a variable yields different results. So, $48\div2c$ where $c=9+3$ yields $2$ - but this conflates the distinction from parenthetical to coefficient-variable syntax. $48\div2\cdot c$ where $c=9+3$ yields $288$.

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    $\begingroup$ it seems like your confusing link to wolfram alpha that yielded 2 was a bug, since now it also yields 288. $\endgroup$
    – Jens Timmerman
    Dec 21, 2015 at 16:17
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    $\begingroup$ Some calculators parse implicit multiplication differently than explicit. This answer is incorrect. TI calculators do this. mathforum.org/library/drmath/view/72166.html $\endgroup$
    – Sue VanHattum
    Aug 2, 2019 at 20:07
  • $\begingroup$ @SueVanHattum, your link is broken. From what I have seen TI calculators use a forward slash. instead of a division symbol, and yet if you convert to long hand they are getting the answer wrong, so I would suggest TI is computing wrong, because everything after the forward slash should be in the denominator TI calculator is wrong. $\endgroup$
    – TheArchitecta
    Nov 26, 2022 at 9:59
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    $\begingroup$ Oddly, my link goes to something other than what it says. Here's a link to a good article: themathdoctors.org/order-of-operations-implicit-multiplication I'm not sure how you can call a calculator wrong. No, everything after the slash is not required to be in the denominator. It depends on interpretation. Amusingly, my TI-30X IIS shows the bar and dots symbol on the keypad, but uses the / on the screen. $\endgroup$
    – Sue VanHattum
    Nov 26, 2022 at 21:14