# 48÷2(9+3) equation

• ### 288

• Total voters
50
Every forum I've searched this on is split 50/50 between 2 and 288 as the answer. What is your consensus?

People arguing PEMDAS say 288

But if you set the 2 in the equation to x and set the equation equal to 288, x then = 1/72. Thus proving 2 is the correct answer.

Thoughts?

gb7nash
Homework Helper

288? Following PEMDAS, I get 2:

48÷2*(9+3) = 48÷2*(12) = 48÷24 = 2

The reason why some people are getting 288 is that they're forgetting that there's an invisible multiply sign in the expression.

Honestly though, if I ever saw this I would apply a facepalm. It's not good notation and like you're seeing, it's ambiguous without proper use of parentheses and will confuse people. It's better notation to say $$\frac{48}{2(9+3)}$$

288? Following PEMDAS, I get 2:

48÷2*(9+3) = 48÷2*(12) = 48÷24 = 2

The reason why some people are getting 288 is that they're forgetting that there's an invisible multiply sign in the expression.

Honestly though, if I ever saw this I would apply a facepalm. It's not good notation and like you're seeing, it's ambiguous without proper use of parentheses and will confuse people. It's better notation to say $$\frac{48}{2(9+3)}$$
If you follow the order of operations, why are you multiplying 2 by twelve before dividing 48 by two? Multiplication doesn't take precedence over division, they're performed from left to right: $$\frac{48}{2}(9+3)$$

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gb7nash
Homework Helper

You go left to right? I didn't know that.

jhae2.718
Gold Member

And this is why you never see ÷ used for any serious purpose...the notation is just awful.

MATLAB and Python say 288.

PEMDAS = Parenthesis -> exponents > multiply -> divide -> add -> subtract
That's simply the order in which they're stated. Multiplication and division are equal, so are addition and subtraction. I could just as easily say that PEDMSA represents the order of operations. One way to look at it is to say that division is simply multiplication of the reciprocal, and subtraction to be addition of the opposite.

jhae2.718
Gold Member

You go left to right? I didn't know that.
I think it comes from viewing division as multiplication by the reciprocal. I wasn't sure, either.

Edit: too late...

gb7nash
Homework Helper

And this is why you never see ÷ used for any serious purpose...the notation is just awful.

MATLAB and Python say 288.
Even I'm getting confused. :tongue: It's just bad notation.

Mark44
Mentor

I get 288. In the acronym PEMDAS, the M and D operations are at the same priority and the A and S are at the same priority. Arithmetic operations at the same priority are evaluated left to right. So for example, 2 + 5 - 3 is the same as (2 + 5) - 3 = 7 - 3 = 4, while 2 - 5 + 3 is the same as (2 - 5) + 3 = -3 + 3 = 0.

I can't say that I remember my algebra teacher in ninth grade going into quite such detail (in fact, all I remember her telling us was the acronym MDAS, with a mnemonic device of My Dear Aunt Sally), but programming languages such as C, C++, C#, Fortran, Pascal, and others are very specific about operator precedence.

For this reason, 48÷2*(12) should be evaluated as if it were written (48÷2)*12 = 24 * 12 = 288.

If you really meant
$$\frac{48}{2(9+3)}$$

it should be written as 48/(2(9 + 3)). That forces the multiplication to be performed before the division.

Mark44
Mentor

MATLAB and Python say 288.
So does C#. I'm positive that C and C++ would report the same result.

We're taught BODMAS in school.

Same basic principle.

Brackets, Orders, Division / Multiplication, Addition / Subtraction.

I will add though that my secondary school teachers told us to use that exact order and not that D and M held equal value (and A and S the same).

So we'd always do division before multiplication and addition before subtraction. But I doubt that affects things.

EDIT: So if I give division priority (as per strictly following the order of BODMAS) after the brackets I get:

48 / 2*(9+3) = 48 / 2*(12) = 24*12 = 288

But if I give multiplication priority (which I'd never do):

48 / 2*(9+3) = 48 / 2*(12) = 48 / 24 = 2

Have I missed something here?

Based on BODMAS and following that exact order I agree with 288. So I'd say they were right to say follow the order strictly.

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So does C#. I'm positive that C and C++ would report the same result.
They do. So do Wolfram Alpha, Google, PHP, Perl, and Ruby.

Mark44
Mentor

And this is why you never see ÷ used for any serious purpose...the notation is just awful.
Even I'm getting confused. :tongue: It's just bad notation.
No, the notation is fine - 48/2*12 really isn't ambiguous if you understand that arithmetic operators at the same precedence level are evaluated left to right. It seems clear from this thread that not everyone is taught this fine point.

jhae2.718
Gold Member

I wrote a quick C program (it's been a looong time...thankfully I have my K&R) and C also says 288.

No, the notation is fine - 48/2*12 really isn't ambiguous if you understand that arithmetic operators at the same precedence level are evaluated left to right. It seems clear from this thread that not everyone is taught this fine point.
I agree it's not ambiguous, but I just hate the ÷ symbol.

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This problem also face at my Programming called Tree.

Actually, i am no agree 288 with this answer but i haven't any good reasons because normally you can press the calculator would get 288.Therefore, our tutor said that we have to considered going with the left-hand side when facing the problem with time or divide which going the first, so depended on the question given,for this, we considered the division going first then the answer would get 288.

But for me , i will say that the question problem because it haven't make that clear for the packaging. Last edited:

i will say that the question problem because it haven't make that clear for the
But if you follow the rules, you'll always get the correct answer - which I've just ran in python (as above have) to get 288.

Looking at it like this though:

48 ÷ x(9 + 3) = 288
48 ÷ 9x + 3x = 288
48/12x = 288
4/x = 288
4 = 288x
4/288 = x
1/72 = x

Would suggest that 288 is wrong, and that 2 is correct.

Also, with the order of operations aren't we using the distributive property which states as an example:

Simplify 16 ÷ 2[8 – 3(4 – 2)] + 1.
16 ÷ 2[8 – 3(4 – 2)] + 1
= 16 ÷ 2[8 – 3(2)] + 1
= 16 ÷ 2[8 – 6] + 1
= 16 ÷ 2 + 1 (**)
= 16 ÷ 4 + 1
= 4 + 1
= 5

Then, looking at what I have bolded below:

The confusing part in the above calculation is how "16 divided by 2 + 1" (in the line marked with the double-star) becomes "16 divided by 4 + 1", instead of "8 times by 2 + 1". That's because, even though multiplication and division are at the same level (so the left-to-right rule should apply), parentheses outrank division, so the first 2 goes with the , rather than with the "16 divided by". That is, multiplication that is indicated by placement against parentheses (or brackets, etc) is "stronger" than "regular" multiplication. Typesetting the entire problem in a graphing calculator verifies this hierarchy.

Note that different software will process this differently; even different models of Texas Instruments graphing calculators will process this differently. In cases of ambiguity, be very careful of your parentheses, and make your meaning clear. The general consensus among math people is that "multiplication by juxtaposition" (that is, multiplying by just putting things next to each other, rather than using the "×" sign) indicates that the juxtaposed values must be multiplied together before processing other operations. But not all software is programmed this way, and sometimes teachers view things differently. If in doubt, ask!

This all seems to point toward 2 being the correct answer.

Looking at it like this though:

48 ÷ x(9 + 3) = 288
48 ÷ 9x + 3x = 288
48/12x = 288
4/x = 288
4 = 288x
4/288 = x
1/72 = x

Would suggest that 288 is wrong, and that 2 is correct.

Also, with the order of operations aren't we using the distributive property which states as an example:

Simplify 16 ÷ 2[8 – 3(4 – 2)] + 1.
16 ÷ 2[8 – 3(4 – 2)] + 1
= 16 ÷ 2[8 – 3(2)] + 1
= 16 ÷ 2[8 – 6] + 1
= 16 ÷ 2 + 1 (**)
= 16 ÷ 4 + 1
= 4 + 1
= 5

Then, looking at what I have bolded below:

The confusing part in the above calculation is how "16 divided by 2 + 1" (in the line marked with the double-star) becomes "16 divided by 4 + 1", instead of "8 times by 2 + 1". That's because, even though multiplication and division are at the same level (so the left-to-right rule should apply), parentheses outrank division, so the first 2 goes with the , rather than with the "16 divided by". That is, multiplication that is indicated by placement against parentheses (or brackets, etc) is "stronger" than "regular" multiplication. Typesetting the entire problem in a graphing calculator verifies this hierarchy.

Note that different software will process this differently; even different models of Texas Instruments graphing calculators will process this differently. In cases of ambiguity, be very careful of your parentheses, and make your meaning clear. The general consensus among math people is that "multiplication by juxtaposition" (that is, multiplying by just putting things next to each other, rather than using the "×" sign) indicates that the juxtaposed values must be multiplied together before processing other operations. But not all software is programmed this way, and sometimes teachers view things differently. If in doubt, ask!

This all seems to point toward 2 being the correct answer.
Your entire argument is based on the point that multiplication by juxtaposition takes precedence over multiplication by operator, despite the fact that they're just two ways of expressing one operation. It's like saying that ÷ takes precedence over / or vice versa.

Mark44
Mentor

Looking at it like this though:

48 ÷ x(9 + 3) = 288
48 ÷ 9x + 3x = 288
48/12x = 288
You are tacitly assuming that 48 ÷ x(9 + 3) means 48 ÷ [x(12)]. I maintain that it means (48 ÷ x)* 12. Writing 12 as 9 + 3 needlessly complicates things.

Your second line does not follow from the first. That would be interpreted as
(48 ÷ 9x) + 3x = 288, instead of what you intended, which was 48 ÷ (12x) = 288. Again, you are assuming that the multiplication of x and 12 is somehow of higher precedence than the division of 48 and x.

4/x = 288
4 = 288x
4/288 = x
1/72 = x

Would suggest that 288 is wrong, and that 2 is correct.

Also, with the order of operations aren't we using the distributive property which states as an example:

Simplify 16 ÷ 2[8 – 3(4 – 2)] + 1.
16 ÷ 2[8 – 3(4 – 2)] + 1
= 16 ÷ 2[8 – 3(2)] + 1
= 16 ÷ 2[8 – 6] + 1
= 16 ÷ 2 + 1 (**)
= 16 ÷ 4 + 1
= 4 + 1
= 5

Then, looking at what I have bolded below:

The confusing part in the above calculation is how "16 divided by 2 + 1" (in the line marked with the double-star) becomes "16 divided by 4 + 1", instead of "8 times by 2 + 1". That's because, even though multiplication and division are at the same level (so the left-to-right rule should apply), parentheses outrank division, so the first 2 goes with the , rather than with the "16 divided by". That is, multiplication that is indicated by placement against parentheses (or brackets, etc) is "stronger" than "regular" multiplication. Typesetting the entire problem in a graphing calculator verifies this hierarchy.

Note that different software will process this differently; even different models of Texas Instruments graphing calculators will process this differently. In cases of ambiguity, be very careful of your parentheses, and make your meaning clear. The general consensus among math people is that "multiplication by juxtaposition" (that is, multiplying by just putting things next to each other, rather than using the "×" sign) indicates that the juxtaposed values must be multiplied together before processing other operations. But not all software is programmed this way, and sometimes teachers view things differently. If in doubt, ask!

This all seems to point toward 2 being the correct answer.

Your entire argument is based on the point that multiplication by juxtaposition takes precedence over multiplication by operator, despite the fact that they're just two ways of expressing one operation. It's like saying that ÷ takes precedence over / or vice versa.
I agree, but in this particular case, doesn't juxtaposition take precedence?

I agree, but in this particular case, doesn't juxtaposition take precedence?
It never does, multiplication is multiplication.

It never does, multiplication is multiplication.
Ok, but then what do you take this statement to mean: The general consensus among math people is that "multiplication by juxtaposition" (that is, multiplying by just putting things next to each other, rather than using the "×" sign) indicates that the juxtaposed values must be multiplied together before processing other operations.

Ok, but then what do you take this statement to mean: The general consensus among math people is that "multiplication by juxtaposition" (that is, multiplying by just putting things next to each other, rather than using the "×" sign) indicates that the juxtaposed values must be multiplied together before processing other operations.
It means that multiplication by juxtaposition takes precedence over multiplication by operator. That statement is false.

It means that multiplication by juxtaposition takes precedence over multiplication by operator. That statement is false.
I'm not saying your wrong, I just haven't seen the answer explained to my satisfaction yet. I see arguments from both sides backed up by sources so I'm getting confused. It's a pretty interesting problem though, seeing the division being nearly perfectly 50% on other polls is fascinating.

Follow the rules and you get 288. There's no way to split it without ignoring the basics, making you wrong.