48÷2(9+3) = ? Mathematician Explains The Correct Answer

What is 48÷2(9+3) = ? This problem went viral and generated millions of comments on Facebook, Twitter, YouTube and social media sites. I explain how to get the correct answer by using the modern interpretation of the order of operations. I also explain how you would get a different answer under historical usage of the division symbol.

I studied Mathematics and Economics at Stanford. Press coverage of my work:

*I get many, many emails about this problem and am unable to reply to them.

Here is a 1917 article from "The American Mathematical Monthly" that explains the usage of the division symbol as an exception to the order of operations

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[shanamchenry]

The historical 1917 version was taught when I was in high school. I graduated in 2007. We would have gotten full credit if we had used either method shown here. College required me to learn the more modern method so now I just do math expressions that way. Things would be far less confusing if everyone was taught the same method in high school though.

[rovicalwencandava]

It's sad how this problem became so viral.

[Aziraphale]

I think the problem stems from the fact that almost nowhere do you actually see a division symbol used, precisely because of this ambiguity. Get rid of the division sign and write it as a fraction and there is no more ambiguity.

[aquss]

From 1st to 4th grade we had a really old teacher (in 1st to 4th grade 1 teacher is for all subjects, in 5th and above each subject had a special professor, idk how it is in your country), she thought us that multiplication came before division and addition before subtraction, we were all shocked when we got an actual math teacher in 5th grade when she told us we were wrong on the first when we had an "initiation exam", glad I learned that and wasn't stuck with the incorrect order of operations.

[pvtbambi]

this has nothing to do with the order of operations... what they are messing with is the distributive property. 18 + 6 = 24, 2(9 + 3) = 24, 3(6 + 2) =24, 6(3 + 1) = 24, 18(1 + 1/3) = 24. that 2 has been factored out but still associated with the 9 and the 3. a better way of looking at that association is (2(9 + 3)). you simply cant yank that 2 out and forget about its relationship to the other terms. replace any of the scenarios above into the equation using their logic and you get a different answer every time. replace any of the scenarios above when keeping the association in tact and you will always get an answer of 2. hence the answer of 288 is poppycock

[angelhelp]

My arithmetic dates to this "historical" answer but the reasoning differs slightly. I was taught that the parentheses had to be, in this example, "eliminated". Perhaps the comment from Steve's Mathy Stuff applies here; I don't recall the exact rule, but that 2(12) had to be evaluated before dealing with the 48.

[hughrutherford]

As somebody who learned Math long after 1917, I have to admit that if I saw the expression x/2y I would most likely interpret it as x/(2y) rather than x/2*y regardless what 'divide' symbol was used. That's the whole reason why this question is contentious. If it were simply 'what is 48/2*(9+3)?' then you could just punch it up on a calculator and it wouldn't be an issue!

[salloom]

When I attended university, my professor taught me "always when in doubt parenthesize !!!" You cannot over parenthesize, it simply makes things clearer...

[michielhorikx]

Just writing things as fractions makes it way less ambiguous.

[nathanisbored]

another example of this is trig fuctions, often times you see cos2x instead of cos(2x), and both are considered to mean the same thing, even though by order of operations, technically the first one should mean cos(2)*x

[TheSkorika]

Thats exactly why i overuse brackets when im entering something into code, wolframalpha, etc

[Dragonblaster]

I'm 58 years old, and did my school maths in the 1970s in the UK. I still instantly said "288" before clicking on the video.

[stacycecil]

I have an argument against the answer being 288: Assume the equation is worked to the form 48÷2(12). Now consider an example such as x÷(2y+2z). We could rewrite this equation as x÷2(y+z), by pulling out 2 from our second term. If we can pull that term out so, it must be that we can put that term back in to reach the same answer. For our form x÷(2y+2z), we'll use the values x=48, y=12, and z=0. Working the math by order of operations, we get 48÷(24+0), or 48÷(24), which evaluates to 2. We've seen from rewriting the form with 2 factored out of our second term group, that that second form must evaluate to the same answer. If we take our first form, substitute the values of x=48, y=12, z=0, and rework the form by factoring out a 2 from the second group first we get 48÷2(12+0) or 48÷2(12). The only way our second form evaluates to the same value is if the correct order of operands for our second form is to evaluate 2(12) first. This gives us 48÷24, which is 2.

[m.evanwillis]

We are dealing with the mathematics of a field here. Consequently, only + and x are directly defined, division and subtraction are defined by multiplication by the multiplicative inverse or the addition of the additive inverse, respectively. Thus, any expression should be expressible in terms of numbers, parentheses, x (which will include additive inversion), +, and exponentiation (which will include multiplicative inversion ). Associativity and distributivity are properties of how we use parentheses, and consequently form a higher order of operation than exponentiation, multiplication, and division.

÷a is translated as x(a)^-1. Per order of operations the inversion happens first, then the multiplication. Thus 48÷2(9+3) becomes 48x(2(9+3))^-1 or 48x2^-1x(9+3). Which do we choose? Here use of traditional order of operations gives the answer. ÷ is an exponential and multiplicative nested function. Consequently evaluating ÷ must follow parenthesis use, particularly parenthesis rules. Thus from 48÷2(9+3) we start with parentheses rules, yielding 48÷(18+6) by the distributive property, and then we evaluate within the parentheses to form 48÷24, giving us a single term to exponentiate then inverse. Now taking the exponentiation 48x(1/24). Then multiplying: 2.

If, however, we had started with 48÷2x(9+3), the (9+3) term is rendered separate from the 2 by the added x, thus indicating that distributivity is not intended. Thus it becomes 48÷2x(12) to 48x(1/2)x(12), to 288.

Thus, unless we use an order of operations that includes distributivity and associativity as aspects of parentheses use, the original 48÷2(9+3) is poorly defined as it is notationally unclear upon what we are to apply the inverse exponentiation that will occur before any multiplication at all.

[waynebennett]

1:13 is where the argument comes. Since it's STILL WRITTEN with (), that is the 1st step. Therefore making 48 <divided by> 2(12) as 48 <divided by> 24 = 2.

[aleksandersaski]

All the problems come because of no multiplication sign between 2 and 9... And I think the problem is similar to the difference between "48÷2x" and "48÷2*x".

[laylaalder]

I interpreted this as an algebraic equation:

A / B(C+D) → A / (BC + BD) thus, 48 / 2(9+3) → 48 / (18+6) → 48/24 → 2. Never even knew that the division sign had a different meaning.

[anneromero]

I relate to this one. Sometimes teachers didn't explain it carefully. Some of them just give an example then goodbye. No more further details or explanations

[m.h.]

I learned a different way to interpret the formula in university math class: implicit brackets.
My professor stated, that a number directly in front of the bracket (without a multiply operator) implies an additional bracket around the number and the bracket. Meaning: "Since the number is directly attached to the brackets, they belong together and have to be resolved together with the brackets". This interpretation also leads to the result of 1.

[Chris-hfsl]

The problem arises because PEMDAS does not cover implicit multiplication. In many textbooks, you'll come across expressions such as A/2𝛑r. Everyone knows that this means A / (2𝛑r). This is fine until you substitute numbers in for the symbols and then blindly punch the resulting numerical expression into a calculator. The solution to the problem is to THINK about what the expression actually means before acting.