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

[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.

 

[jhae2.718]

If you follow the standard order of operations, the expression evaluates to 288.

 

[Perfection]

The answer is to give the maker of the statement a chewing out.

 

[Triple_D]

This topic has the potential to last longer than the DDWFTTW thread.

 

[oraclelive]

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

I think by using or applying the rule of BODMAS this could be better understood and solved well whelther with a calculator or not. That is; 48/(9+3)*2 = 48/(12*2) = 2.

 

[oraclelive]

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.

The best and simplest way of resolving implied multiplication is by proper application of the rule of BODMAS. That is, solving in the presiding order from left to right-Bracket Of Division Multiplication Addition and Subtraction. Thanks.