Which is the correct answer for 48÷2(9+3): 2 or 288?

[JaredJames]

I can see what's been done. I was running it in Python as a check but with very strict brackets (wrong order) which gave me the same answer.

 

[uart]

I voted for the answer 2 for the following reason. Everywhere in written mathematics (textbooks, papers, exams etc) that I see implied multiplication it is always is given higher precedence than division.

Something like 8 x^2 \div 2x, for example, invariably means 8 x^2 \div (2x). So I've taken to modifying BIDMAS in the following was to also include "implied" multiplication.

BIIDMAS : (brackets, indices, implied multiplication, division, multiplication, addition, subtraction).

 

[JaredJames]

I voted for the answer 2 for the following reason. Everywhere in written mathematics (textbooks, papers, exams etc) that I see implied multiplication it is always is given high precedence than division.

Something like 8 x^2 \divide 2x, for example, invariably means 8 x^2 \divide (2x). So I've taken to modifying BIDMAS in the following was to also include "implied" multiplication.

BIIDMAS : (brackets, indices, implied multiplication, division, multiplication, addition, subtraction).

Well unless that is official notation it's worthless. I've never seen it given precedence.

I'm curious how does 8x²x = 8x²2x?

 

[uart]

Well unless that is official notation it's worthless

I'm not sure what you mean. I'm talking about how written mathematical equations with implied multiplication are invariably interpreted in my experience.

Implied multiplication is where the "times" symbol is not explicitly written but is implied by algebraic convention.

In written mathematics for example 8 x^2 \div 2x invariably means 8 x^2 \div (2x).

James. I challenge you to find one example in a well written mathematical text or paper where the divide symbol is allow to "break" an implied multiplication. That is an instance where for example 8 x^2 \div 2x is written but (8 x^2 \div 2) \times x is meant.

 

[JaredJames]

I'm not sure what you mean. I'm talking about how written mathematical equations with implied multiplication are invariably interpreted in my experience.

8 x^2 \div 2x, for example, invariably means 8 x^2 \div (2x).

Ah, so there's a divide in there.

Well it's only implied and allows interpretation. If you follow the standard rule on it, the implication is worthless.

The only way to guarantee it is to use brackets - which is something I always do. I'll go overkill on brackets if I need to because I want to make sure everyone knows exactly what I'm doing.

 

[uart]

Ah, so there's a divide in there.

Yes it was always there but a latex error preventing it from displaying properly for the first minute after I posted.

If you follow the standard rule on it, the implication is worthless.

Then please take on my challenge.

James. I challenge you to find one example in a well written mathematical text or paper where the divide symbol is allow to "break" an implied multiplication. That is an instance where for example 8 x^2 \div 2x is written but (8 x^2 \div 2) \times x is what is meant.

 

[jhae2.718]

I don't think the \div symbol is used once in any mathematical text I have.

 

[JaredJames]

Then please take on my challenge.

I don't think the ÷ symbol is used once in any mathematical text I have.

That was about to be my exact response to that challenge.

It's a non-issue if you use the alternate notation seen pretty much everywhere else.

You can do the division and leave the x term there and not have a problem.

Example:

9 / 2x = 4.5 / x

9x / 2x = 4.5

I've done no multiplication what-so-ever and still have simplified equations (or the answer in the latter case).

 

[uart]

That was about to be my exact response to that challenge.

It's a non-issue if you use the alternate notation seen pretty much everywhere else.

You can do the division and leave the x term there and not have a problem.

Example:

9 / 2x = 4.5 / x

9x / 2x = 4.5

I've done no multiplication what-so-ever and still have simplified equations (or the answer in the latter case).

Utter nonsense! If you do the division first in that example you get,

9 / 2x = 4.5 x instead of the usual interpretation which would be 9 / 2x = 4.5 / x.

 

[JaredJames]

Utter nonsense! If you do the division first in that example you get,

9 / 2x = 4.5 x instead of the correct 9 / 2x = 4.5 / x

I put brackets into try and force the multiplication in the calculator.

9/2x = 4.5/x = 18/4x

If you want to simplify the equation, you can just work with the numbers and ignore the x values. Cancel it down in other words.

Not quite what we're going for ey? I see, I've gone the wrong way.

In which case, 9 / 2x = 4.5x, you are correct. But then you've answered your own question of when the division can interrupt the multiplication, surely?

 

[jhae2.718]

I'm assuming that that example is if we follow the implied first interpretation, and that in this case
9/2x \equiv \frac{9}{2x}?

 

[JaredJames]

I'm assuming that that example is if we follow the implied first interpretation, and that in this case
9/2x \equiv \frac{9}{2x}?

Doesn't matter does it? It's reduced the same either way.

 

[jhae2.718]

Well, from what I remember from arithmetic, if we had (9/2)*x it would reduce to 4.5*x, and 9/(2x) would reduce to 4.5/x.

So the 9/2 would always go to 4.5, but the power of x would be either 1 or -1 based on the grouping.

 

[JaredJames]

Well, from what I remember from arithmetic, if we had (9/2)*x it would reduce to 4.5*x, and 9/(2x) would reduce to 4.5/x.

So the 9/2 would always go to 4.5, but the power of x would be either 1 or -1 based on the grouping.

Got me thinking now, think I've got it *** about face.

(Maths isn't my strong point, hence my need to try and follow rules as much as possible - or, well, this happens.)

 

[Borek]

And what about 48÷(9+3)2?

 

[uart]

I'm assuming that that example is if we follow the implied first interpretation, and that in this case
9/2x \equiv \frac{9}{2x}?

Yes 9 \div 2x \equiv \frac{9}{2x}. In my experience that is how written mathematics is invariably interpreted.

Also if you have a calculator that can handle implied multiplication (most made in the last few years should allow this) then try typing in something like 12 \div 2\pi, you'll find that it is interpreted exactly as I say.

BTW. I just checked on my aging "Casio fx-82MS" and

12 \div 2\pi returned 1.909859 and 48 \div 2(9+3) returned 2.

In other words, don't just try this on C or MATLAB or python or anything else that doesn't allow algebraic implied multiplication, because it's irrelevant. Try it on a calculator that does allow algebraic implied multiplication if you really want to do a proof by calculator.

 

[jhae2.718]

48/(9+3)*2 = 48/12*2 = 4*2 = 8

 

[uart]

And what about 48÷(9+3)2?

Hey I don't like this notation either Borek but I'm just calling it as I see it commonly interpreted. This is why the fraction notation is preferred by most people as the "fraction bar" provides a well recognized "grouping symbol" and removes any ambiguity.

For that one my old calculator says "syntax error" (it wants either an explicit divide or times symbol after the bracketed expression). But if I was forced to make a call I still say implied multiplication and the answer is still 2, but I really wouldn't use that notation myself.

 

[uart]

48/(9+3)*2 = 48/12*2 = 4*2 = 8

Yep, it certain does if you put an explicit multiplication symbol in there. The whole point of this thread though is about what happens when (and what are the potential ambiguities that can occur when) we use the algebraic implied multiplication in an expression.

 

[Dembadon]

After thought, you might actually be right. For example:
a/bc

Is it \frac{ac}{b} or is it \frac{a}{bc}?

EDIT: Perhaps the ambiguity of the question is getting to me, and I was correct initially: a/bc = \frac{ac}{b}

Since parenthesis weren't used around b and c, I would interpret a/bc as \frac{ac}{b}.

If parenthesis had been used, I would assume \frac{a}{b}*\frac{1}{c}.

 

[jhae2.718]

I hold that implicit multiplication is evaluated as any explicit operation. Unfortunately, cases like this are extremely ill-defined.

However, the best way is to avoid the issue entirely and use \frac{}{} or additional parentheses for grouping.

 

[uart]

Since parenthesis weren't used around b and c, I would interpret a/bc as \frac{ac}{b}.

If parenthesis had been used, I would assume \frac{a}{b}*\frac{1}{c}.

What about a \div bc which was the notation used in the original question here?

 

[jhae2.718]

What about a \div bc which was the notation used in the original question here?

Same thing as a/bc.

 

[uart]

Same thing as a/bc.

Ok then I say your interpretation differs from that of at least 99% of written mathematics (maths science engineering textbooks and papers etc).

I don't like this notation either, I also find it wide open to ambiguity and of course many books and papers etc will avoid using it for that very reason. But if you look hard enough you will find textbooks or papers etc that do use notations like f = \omega \div 2\pi and when they do so then it pretty much always means f = \omega \div (2\pi) and not f = (\omega \div 2) \times \pi

 

[jhae2.718]

Then I would argue that such usage is contrary to conventional interpretation of order of operations.

Of course, as long such books/papers/etc. are consistent in their convention of operator precedence, I see no problem.

 

[uart]

Then I would argue that such usage is contrary to conventional interpretation of order of operations.

Of course, as long such books/papers/etc. are consistent in their convention of operator precedence, I see no problem.

jhae, do you own a calculator that is less than about 5 years old? If so try something like 12 \div 2\pi (without any explicit multiplication symbol between the 2 and the pi). You may get a surprise.

 

[Studiot]

Micosoft comes up with 288 in the windows calculator.

Excel is more interesting in that if you try to type it straight in that *** paperclip corrects you.

If you follow clippy's advice you get 288

 

[jhae2.718]

I have a TI-84 that is about 6 years old, I think, and a cheap Casio scientific calculator I bought a few months ago. The TI-84 gives 6pi, and the Casio can't handle 12/2pi being input without an explicit operator. Calculators vary on precedence used; how repeated exponentiation is treated is a good example.

I prefer to use the standard* order of operations I learned years ago. *I think we've learned from this thread that there really isn't a uniform standard defined for operator precedence.

 

[uart]

I have a TI-84 that is about 6 years old, I think, and a cheap Casio scientific calculator I bought a few months ago. The TI-84 gives 6pi, and the Casio can't handle 12/2pi being input without an explicit operator. Calculators vary on precedence used; how repeated exponentiation is treated is a good example.

I prefer to use the standard* order of operations I learned years ago. *I think we've learned from this thread that there really isn't a uniform standard defined for operator precedence.

I'm surprised about that. All recent scientific calculators from both Casio and Sharpe that I've seen have been able to handle that type of operation. Are you sure you used the divide "\div" key to enter that expression and not some kind of calculator fraction notation? I don't think your Casio calculator would even have a "/" key.

BTW. In all of this discussion can we please stick with the divide "\div" notation as per the original question. The alternative "/" is not used in well formatted typeset text such as in papers or textbooks. It exacerbates the ambiguity even further as it doubles as both a divide symbol and a half baked fraction bar as well. The original question was expressly about the divide "\div" symbol.

 

[jhae2.718]

I did use the \div key. Keep in mind my Casio is a really cheap and basic scientific calculator; what happens is that it replaces the 2 in the expression with \pi unless an explicit operator is used.

I have never seen "\div" used in any paper I have read.

 

[uart]

I did use the \div key. Keep in mind my Casio is a really cheap and basic scientific calculator; what happens is that it replaces the 2 in the expression with \pi unless an explicit operator is used.

I have never seen "\div" used in any paper I have read.

Ok but just to make that clear, are you saying that they always use the alternate "/" symbol instead, or are you saying that they always forgo the divide symbol for a proper well formatted fraction bar?

AFAIK "\div" is the correct symbol for divide so I can't think of any good reason to completely forgo it.

 

[jhae2.718]

I recall seeing \frac{}{} used most of the time*, though there have been a few uses of "/" I can recall, but almost always with something in the form a/b, i.e. only two arguments.

Of course, I'm sure there are plenty of authors who use "\div"; I just haven't read any.

Wolfram Mathworld lists both symbols for division. http://mathworld.wolfram.com/Division.html

*Most of the expressions are typeset in the equation environment in the papers I've read.

 

[Temad]

48÷2(9+3)=288

48÷[2(9+3)]=2

 

[tak08810]

Simplify 16 ÷ 2[8 – 3(4 – 2)] + 1.

The confusing part in the above calculation is how "16 divided by 2[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 [2], 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! - http://www.purplemath.com/modules/orderops2.htm

 

[Mark44]

Registered on this fine forum just because of this. :)

Regarding calculators, here is something interesting:

See the thumbnails in post #46.

This is a sad state of affairs when two models of calculators (TI 85 and TI 86) from the same company report different answers for exactly the same simple arithmetic expression.

 

[uart]

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.

Arh, proof by large font. I think that even trumps my proof by calculator.

I agree with it though.

 

[RJS]

And again it's a 50/50 split. This is obviously the most difficult math problem ever conceived.

 

[DR13]

And again it's a 50/50 split. This is obviously the most difficult math problem ever conceived.

On Tuesday I'm going to go to office hours and ask my math professor.

 

[DR13]

I think the problem comes down to whether or not 2(9+3) is the same as 2x(9+3)

 

[fraga]

See the thumbnails in post #46.

This is a sad state of affairs when two models of calculators (TI 85 and TI 86) from the same company report different answers for exactly the same simple arithmetic expression.

Thank you.
At least some one noticed.

 

[Studiot]

This is obviously the most difficult math problem ever conceived.

No that would be "How long is a piece of string?"

 

[gb7nash]

And again it's a 50/50 split. This is obviously the most difficult math problem ever conceived.

Goldbach's Conjecture, move out of the way. We have a new problem!

 

[Mirin]

U aware OP?


My vote goes to 288

 

[itsgotime]

I think it's 2! I hope I'm right :(

 

[Eldar]

My vote is on 2 as well, if it were 2*(9+3) then I'd probably be more inclined to say 288.

 

[RJS]

U aware OP?


My vote goes to 288

OP is aware, and mirin.

 

[RandomName]

288. I disagree, never have I heard that implied multiplation is of higher precedence.

 

[jhae2.718]

Implied multiplication is not a feature of the standard order of operations. Some texts may define implied multiplication as having a higher precedence than explicit multiplication it seems, however, I have not come across any to date.

 

[Mark44]

I think the problem comes down to whether or not 2(9+3) is the same as 2x(9+3)

Assuming you are using 'x' to mean multiplication, which I'm not aware is done in any books past arithmetic, 2(9+3) is exactly the same as 2 x (9 + 3). In both cases you are multiplying 12 by 2.

 

[DeadOriginal]

I created an account on the forum just for this.

I thought the answer was 2 when I first saw the equation but now that I've read some of the posts, I'm not so sure anymore.

I played around with the equation and set 2 to x like the OP said. From what I can see I think it really comes down to if x should be distributed into the parentheses or not.

If you multiplied the 2 with the (9+3) first and then divided with 48 with the 24 then your answer would become 2. If you set the 2 to x then you would first distribute the x within the parentheses which in the end would leave your x = 2 if you make your equation equal to 2.

If you divided the 48 by 2 and then multiplied by 12 you would get 288 so if you substituted your 2 with x you would first divide your 48 with x before multiplying it with 12 which if written on paper would be the same as 48 multiplied by 12 divided by x. In that case if your set your equation equal to 288 you would still get x = 2.

In other words, setting it to x doesn't really help. Now that this equation has melted my brain, can someone please clarify? Haha.