Math poll

[lumberjack_jeff]

48 ÷ 2(9+3)=?

[SteppingRazor]

1. Why the age diff.?

Incidentally, I voted "under 40 and 2" but I really do hate that Sally lady.

[boppers]

7. Because the rules have been changing.

Different generations have different solutions.

[salin]

36. interesting because unlike rules of language - math rules don't really change.

which makes the results potentially very interesting.

[lumberjack_jeff]

43. If the math rules aren't changing, then the answer is unequivocally 2.

And the TI-86 calculator (as well as google and wolfram alpha are wrong)

[salin]

48. I would agree

but the proliferation of an "alternative" view as meaning "both are right" - is really interesting to me - esp if there is (per the poll) a generational difference.

[boppers]

54. Tell that to zero.

He keeps insisting he exists now.
Oh, and the Roman numerals are totally going to mock Arabic numbers now.

(This isn't really about math rules changing, but which rules apply in ambiguous situations)

[salin]

70. not ambiguous unless one decides the distributive property in math - for

some reason no longer applies. Does 2(x+8) = 2x + 2*8 ? if so the answer to the problem is 2. If the answer to the problem is 288 - then someone has disproven the distributive property and not disclosed why the rule is wrong and now should be ignored. If that paper has been published and accepted by the mathematicians - than I have missed it - and I am duly corrected upon seeing it - in math terms - disproven.

[boppers]

91. You're looking at the wrong part of the problem.

It's not about the distributive property causing the issue. That's a red herring.

[U4ikLefty]

94. no they haven't

and no they don't.

Only the mathematically ignorant get the wrong answer.

[slackmaster]

159. I disagree. I am 53 and remember learning the rules of precedence, and left-to-right

Multiplication and division are equal, and take precedence over addition and subtraction. You have to work the problem from left to right, unless it's presented vertically.

[eShirl]

19. decline of education

[Joe Bacon]

294. Doesn't matter --I still HATED math class

People who tell me I'm going to hell, I reply. I've been in hell and it's called High School Algebra.

[ecstatic]

35. Probably calculator crowd vs. non calculator crowd

Relying on calculators can lead one astray at times. Oh, on edit... I'm under 40, but I had to learn math without a calculator.

[tridim]

2. According to this video, it's both.

http://www.youtube.com/watch?v=U706g7_m_78&feature=related

[lumberjack_jeff]

According to the video, it's two.

48/2*(9+3) is a different question.

[ReggieVeggie]

34. you're saying 48/2*(9+3) is not the same as 48/2(9+3)? nt

[salin]

44. of course they are the same - just the understanding that they are the same

is what varies. In such a problem - there can not be "two correct answers". Math is pretty precise. Sort of a strange generational difference artifact that we could settle upon an odd assumption (that two such different answers could equally satisfy the same equation) not based on any math that I can understand (such as a +/- situation such as sqaured factors or absolute values where by definition there will be two different answers.)

[lumberjack_jeff]

47. That is correct. They are not the same.

Journal of Algebra Vol 17: "We shall sometimes use • and sometimes juxtaposition to indicate ring multiplication. When both occur, juxtaposition takes precedence, ie, xy ■ z = (xy)z."

Tamkang Journal of Mathmatics: "Multiplication is indicated both by juxtaposition and by "x". When both forms occur, juxtaposition takes precedence. So (ab x c) x de means ((ab)c)(de) ."

Which is consistent with the video.

[ManiacJoe]

67. This rule does not get the teaching that it needs.

This certainly clears up the suggested ambiguity.

[demwing]

131. "when BOTH occur" ... In the the example you gave, only juxtaposition occured

so the two are the same

[lumberjack_jeff]

147. "both" refers to multiplication OR division.

explicit division and multiplication are interchangeable in terms of order. Multiplication by juxtaposition takes precedence to either.

[demwing]

305. That's not what your quotes say -

your two quotes are each very specific to multiplication.

Your comment that "division and multiplication are interchangeable in terms or order" is true, but not relevant to the specific quote.

[proud2BlibKansan]

3. Always start with the parentheses

9+3=12

That leaves you with:

48 ÷ 2(12)

Which is

48 ÷ 24

Which equals 2

[The Magistrate]

4. That Is How We Were Drilled Too, Ma'am

[proud2BlibKansan]

5. A+ for you

:hi:

[Earth_First]

6. The operation which precedes a the parentheses results in multiplying the two...

If I recall...

[salin]

46. Agreed - and the idea that there can be 2 "right" answers, when the

equation is not defined as one with multiple answers (per parts of the equation, such as an exponential factor) - is ... er... interesting.

[JackBeck]

59. I am under 40 and that was what I learned.

We were talking about the new math rules just the other day during a family gathering.

[Skidmore]

141. Yes, operations in parentheses are always calculated first.

[Appenzell Wars]

191. This nt

[demwing]

151. But the 2 isn't in the parentheses, right?

So wouldn't it look like this?

48÷2(9+3) =

48÷2(12) =

48÷2*12 =

24*12 =

288


-OR, using distributive property-

48÷2(9+3) =

((48÷2)*9)+((48÷2)*3) =

(24*9)+(24*3) =

216+72 =

288


It seems that those in the thread that are getting 2 as the answer are doing one of the following:

1. Assuming parentheses where there are none, and reading the problem as this: (48)÷(2(9+3))
2. Only distributing the 2 on the (9+3), instead of distributing the (48÷2)
3. Giving Multiplication a higher precedence in the order of operations than Division

[Lucky Luciano]

165. Precisely.

Anyone can do this in excel and get the right answer.


Though excel does screw up one order of operations.

type:

-1^2

excel gives 1. It should be -1.

It would be correct for the answer to be1 if you typed (-1)^2.

[Doctor_J]

261. Correct, demwing

the 9+3 is in (), so should be done first

after that, all that's left is mult & div, which is then done from left to right. The answer is 288

Enter it into any calculator as written in the OP, and 288 will come out

[YankmeCrankme]

323. Your mistake is that in mathematical equations parenthesis mean something

Parenthesis are put in place for a reason, if the person writing such an equation wanted you to do the 48/2 first they would have put it in parenthesis. You wouldn't do that mathematical calculation first. Parenthesis should not be assume in a mathematical equation. In math, symbols and numbers are used for a purpose and your examples you're adding them in.

The 2(9+3) is a single term in math. Therefore you don't separate it. This equation is basically saying 48 is the numerator and 2(9+3) is in the denominator. What if the equation read: 48/2(x+3)? The 2 in front of parenthesis means that you have reduce the term of 2x+6 to 2 times (x+3).

[slackmaster]

160. Multiplication and division have equal precedence

You have to work 48 ÷ 2(12) from left to right, so the answer is actually 288.

[hfojvt]

178. not if you start with the parentheses

2*(12) still has a parenthesis, you have to get rid of that first, like so 2*(12) = 24

so the answer is actually 2

[Doctor_J]

262. No, the only parentehtical operation is 9+3

2*(12) is not inside (), and thus has no precedence

[demwing]

317. Only if the 2 is INSIDE the parentheses

48÷(2(9+3))= 2
48÷2(9+3)= 288

[ManiacJoe]

8. The problem with the equation is the shorthand used for the multiplication.

a: 48 ÷ 2(9+3) = 48 ÷ 2 * (9+3) = 48 ÷ 2 * 12 = 288
or
b: 48 ÷ 2(9+3) = 48 ÷ (2 * (9+3)) = 48 ÷ (2 * 12) = 2

I am over 40, my high school and college schooling says the correct answer is 288.

[PVnRT]

140. I don't even think it's that, it's the ambiguous parentheses

Everyone seems to be assuming that parentheses are implied as you have in your second equation, which is how I assumed it to be done. However, the top equation is also technically correct.

Unrec for 1)ageist bullshit and 2)poorly written arithmetic.

[fishwax]

185. I think it has more to do with the "÷"

Perhaps using variables can clarify things.

4(x + 1) / 2(x + 1)

Since it is often difficult to type actual fractions in most programs (or impossible, aside from a few basic ones such as 1/2), it would be reasonable to read that as a fraction with 4(x+1) as the numerator and 2(x+1) as the denominator. The answer, then, would be 2.

But what about 4(x + 1) ÷ 2(x + 1)? Is this the same equation? Does the ÷ sign change how we treat it? Because if we don't treat it as a fraction, and we don't prioritize implied multiplication/juxtaposition/distribution, then the answer would be 2(x+1)(x+1).

4 * (x+1) ÷ 2 * (x+1) -- since the parens can't be simplified, if we multiply starting on the left we get ...
= (4x+4) ÷ 2 * (x+1) -- the parens still can't be simplified, so we move on to the division and get ...
= (2x+2) * (x+1)
= 2(x+2) * (x+1)

Perhaps it's the "÷" sign that is causing confusion ...

[ManiacJoe]

187. The choice of the inline symbol for division is not the problem.

The problem is the juxtaposition "2(" shorthand notation. See post #47.

[fishwax]

190. I agree with 47, but see that juxtaposition as a solution rather than a problem

:)

[ManiacJoe]

202. The problem is that the juxtaposition rule is commonly forgotten.

I am guilty of that in my original response.

[demwing]

9. Cut and Paste "48 ÷ 2(9+3)" into Google

The answer is 288

[boppers]

10. I think we can assume google is under 40, no?

:evilgrin:

[Earth_First]

11. + 1 n/t

[demwing]

13. I so completely should have used that joke first :)

But I'm over 40, and slowing down....

[BillyJack]

124. Perfect *zing*

:rofl:

[WatsonT]

206. Someone should check AOL then

[salin]

69. that works if we pretend there is no distributive property

where 2(9+3) = 2*9+ 2*3 + 18+6 = 24.

Certainly if we ignore the distributive property the answer is 288. However I haven't yet read about the ground breaking paper that dispells the distributive property. Basic algebra. 2(x+4) = 2x + 8. Same principle.

[demwing]

86. That's not correct. There's no conflict, check it...

Either method works:

2(9+3) = (2*9)+(2*3) = 18+6 = 24
-or-
2(9+3) = 2*(12) = 24

Even for the FULL equation

48/2(9+3) = ((48/2)*9)+((48/2)*3) = (24*9)+(24*3) = 216+72 = 288
-or-
48/2(9+3) = (48/2)*(9+3) = 24*12 = 288

[dsc]

106. You have to distribute 48\2, not just 2

there is no plus sign there so the entirety gets distributed.

If you have something like this 4 + 2(9+3) then only the 2 gets distributed. But if you have 48\2(9+3) you have to distribute 48\2 which is 24, not 2. Thus you would get 24(9) + 24(3) which is 216 + 72 or 288.

On edit: another way to think of this is that you actually have the product of two fractions. One fraction is 48/2 while the other is (9+3)/1. You then do the distributive property 48/2 * 12/1 then you multiply the numerators 48*12 and put that over the product of the denominators 2*1 getting 576/2 which is 288.

[demwing]

113. right, exactly what I wrote in post #86

Though I've got to say that having a math teacher confirm the logic makes me feel better :)

[cliffordu]

12. Aunt sally smoked all my medicine.

What was the question???

[Blue-Jay]

14. I know how to do math and the answer is 2.

[lumberjack_jeff]

16. Different people... or calculators, can disagree.

[Bill McBlueState]

23. *shudder*

If a TI-85 and a TI-86 disagree, we're doomed.

[cliffordu]

28. No kidding.....

DOOOOOOOMED...

[Renew Deal]

282. I agree

I think this is a significant problem that puts into question almost all calculations. And I'm not kidding.

[Blue-Jay]

24. 288 is wrong.

Why is there a discussion about this? An equation is not necessarily read left-to-right like a sentence. There's no argument about this.

[lumberjack_jeff]

27. Then I hope rocket scientists don't use a TI-86 n/t

[struggle4progress]

60. The convention that I was taught was:

parentheses rule

exponentiation binds more tightly than multiplication or division, which bind more tightly than addition or subtraction

and when the parentheses don't determine enough, handle multiplications and divisions from left to right before handling additions and subtractions from left to right



According to this, in 48 ÷ 2(9+3) = 48 ÷ 2 x (9+3) we certainly handle the parentheses first 48 ÷ 2 x 12 and then handle multiplications and divisions from left to right: 24 x 12 = 288

But it's just a convention, and hardly anyone ever uses the third part

[demwing]

154. It is read left to right

When parentheses and exponents have been solved, work the Multiplication and Division, left to right, and then work the Addition and Subtraction, left to right.

[Tuesday Afternoon]

33. awesome!

:rofl:

[DCBob]

57. this is fascinating!!

clearly the rules have changed.

[lumberjack_jeff]

64. That... or the guy who designed the TI-85 retired.

And the guy who designed the TI-86 thought differently.

[ThoughtCriminal]

96. TI-85 v TI-86

TI-85: In my day, they taught us how to do equations right.

[demwing]

117. TI-86 replaced the TI-85

wonder why? :)

[Art_from_Ark]

254. "New and improved"

doesn't always mean "improved"

[hfojvt]

177. microsoft works also came up with 288

I guess it makes some sense to do the problem sequentially, but it is still Smokey THE bear.

[Sheepshank]

184. well ,my Sharp EL-W535 says 2

but then I used the divide sign instead of /.

[Shrek]

214. My TI-84 agrees with the TI-86

So apparently Texas Instruments changes its mind from model to model.

FWIW I agree with the TI-85, but the calculator in Windows 7 and the one on my Droid phone both say it's 288.

[NutmegYankee]

222. I've run into this with a TI-83.

You must be careful to ensure that you are performing the correct operations in the correct order. Personally, I'd type 48/(2*(9+3)).

[Motown_Johnny]

235. the TI 86 has flawed software

it is simply not possible that the answer changed

the answer is 2

[demwing]

246. The software is not flawed, TI purposefully made the change

http://epsstore.ti.com/OA_HTML/csksxvm.jsp?nSetId=103110

So, why did the largest manufacturer of calculators in the world purposefully stop giving priority to implied multiplication?

[lumberjack_jeff]

249. You have completely misunderstood the article you posted.

Does implied multiplication and explicit multiplication have the same precedence on TI graphing calculators?

Implied multiplication has a higher priority than explicit multiplication to allow users to enter expressions, in the same manner as they would be written. For example, the TI-80, TI-81, TI-82, and TI-85 evaluate 1/2X as 1/(2*X), while other products may evaluate the same expression as 1/2*X from left to right. Without this feature, it would be necessary to group 2X in parentheses, something that is typically not done when writing the expression on paper.

[demwing]

255. But that's history now

Implied multiplication HAD a higher priority than explicit multiplication, and now TI says it does not.

[lumberjack_jeff]

272. No. TI says that their newer calculators no longer include this feature.

The fact that accuracy is now considered a feature is somewhat vexing.

[demwing]

292. Thats only if you assume that "feature" added accuracy

if it did not, then it's a relief that it was scrapped, not vexing at all

[fishwax]

301. but they don't suggest that the way expressions are actually *written* has changed

they just say that they no longer calculate such expressions in the same way that they're written -- you have to "translate" the expression from the page to the calculator. That's always been true of calculators anyway--the important thing to know is how they handle equations so that you can account for that. Just as one example, if you have a vertical fraction there is no need to write parenthesis around either numerator or denominator. But when you put it into a calculator, you do need to include them or you may get the wrong answer.

[demwing]

307. But the do suggest just that

And I responded to that line of logic in post 257:

To me that seems pretty straightforward. TI does not say:

"Implied multiplication has a higher priority than explicit multiplication to allow users to enter expressions, in the same manner as they SHOULD be written"
-or-
"Implied multiplication has a higher priority than explicit multiplication, as is the standard in algebraic expressions."

No, their technical writer used the term "would," the past tense of will. TI is saying "This is how people would write it out, back in the day."

The proof is in the pudding - if TI thought implied multiplication should maintain its higher priority, they never would have changed the programming.

[fishwax]

313. but "would" needn't be past tense

"No, their technical writer used the term "would," the past tense of will. TI is saying "This is how people would write it out, back in the day." "

Replace "how people would write it out, back in the day" with "how people wrote it out," back in the day, and you've got a point. But, of course, the writer didn't write anything like that.

The phrase the writer uses is "the way it would be written," which doesn't imply that it is no longer written that way. (Otherwise it would be impossible to ask how it would be written now. ;))

And, of course, there is the matter of the present tense ("has") when referring to implied multiplication's priority over explicit multiplication.

[demwing]

315. True enough, but I believe that the added weight of having retired the priority

tips the scale

[fishwax]

320. it's funny that you've highlighted the change in calculators

because reading through the link you posted last night reminded me of my undergrad years, which were during that time. And I hadn't thought about it in years, but there was actually a controversy.

I wish I could remember the exact details, but as I said I haven't thought about it in a while and it's not as though purchasing a calculator was ever a profound event in my life, but the gist of it was that one professor (can't even remember whether it was a calc, statistics, or physics prof) was adamant that we not purchase calculator x (and I'm pretty sure it was one of the new ones, though I could be wrong). But I'd already purchased the forbidden model and so I had to purchase another, and wound up with two calculators. I was trying to find one of them last night, after you posted that link, but alas, could not.

But it made me chuckle, remembering how worked up our prof was about a calculator :)

The thing about notation is that it is a language, and so is subject to evolve over time, which can create ambiguities. In vertical fractions, there is always an implied parenthesis around both the numerator and denominator (and one must remember to actually add those parentheses when putting values in a calculator). The "/" is rarely used precisely because vertical fractions eliminate potential for ambiguity, but when they are used, there is a convention in at least some fields (and I suppose I don't know how widespread it is, but you can see it in physics journals, for instance) where items multiplied after a "/" sign are treated as a unit, with the entire thing as the denominator. Most people familiar with geometry, for instance, would recognize r = C/2π as the way to solve for the radius of a circle when you know the circumference, with the 2π as the denominator.

But clearly that convention is not universal, and perhaps not as widespread as it once may have been. As I've said elsewhere, I would guess that the increasing prevalence/importance of computer programming code (where, as I understand it, that convention is not observed, and so 4/2x would equal (4/2)*x ... ) is driving the shift.

[Electric Monk]

15. BEMDAS

2

[some guy]

20. what is BEMDAS? n/t

[Electric Monk]

25. Brackets, exponents, multiply, divide, add, subtract

The order to do them in. It's a mnemonic I learned in high school in the late 80s. Everything inside the brackets gets done first, etc.

[some guy]

29. Thanks.

I had seen PEMDAS explained in the other thread, but wasn't sure what the B would be.

I don't think I ever learned either of those; if I did, it didn't "take" :hi:

[salin]

51. the issue seems to be whether or not after one does the operation in the bracket

one has to to the operation that is expressed to the bracket before moving on. Just realized the basic distributive property is what dictates that one has to then do the operation expressed as being done to the "quantity" in the brackets before moving to the rest of the mnemonic.

[demwing]

127. Thats fine, but you have to ditribute everything, not just the 2

48/2(9+3) =

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

(24*9)+(24*3) =

216 + 72 =

288

[MindandSoul]

30. No, with BEMDAS it is 288. But I still think 2 makes more sense.

The ONLY way to get to the answer being 288 is by applying BEMDAS (or PEMDAS)

Because if you use BEMDAS and the expression is: 48/2(9+3)=

you do the bracket first (9+3) =12

So you get 48/2*12
Then you do from left to right, by order of BEMDAS

so you get 48/2 =24

Then the multiplication 24*12 = 288

However, if you DON'T use Bemdas, then you get 2.

The real answer is to rewrite the expression CORRECTLY with proper format i.e. 48/<2(9+3)>
which then gives you (bracket first) 48/(2*12)
and bracket again 48/24
and finally the division 2

[Electric Monk]

31. M before D. 48/2*12 = 48/24 = 2 nt

[Lucian]

There is no "M before D."

They're on the same level. In that case, you go from left to right.

288.

[demwing]

123. No. M and D are in the same operation, so you work left to right

1. Parentheses
2. Exponents
3. Multiplication and Division (left to right)
4. Addition and Subtraction (left to right)


the answer is 288

[Electric Monk]

126. I stand corrected on the (MD) and (AS), but the spaces in the OP imply brackets (parenthesis)

See post 52.

http://www.democraticunderground.com/discuss/duboard.php?az=show_mesg&forum=439&topic_id=982937&mesg_id=983374

So, 2.

[demwing]

132. You can't over-ride the Order of Operations by saying "implied parentheses"

That's like yelling "Base" in tag just before you get caught! It's just a made up rule. Its the mathematical equivalent of Calvin Ball!

Such a thing would open math to wild subjectivity! How would we know if there were an "implied parenthesis" or just a mistake in typing?

You wouldn't, and that's why-for accuracy-you use rules, not implications.

[Electric Monk]

134. I love this thread

Since LoZocollo opened the door by posting so much shit without getting banned, we can post almost anything now without it getting removed, if we contextualize it as such. It's a brave new DU!!

You'll need an extra strength irony and sarcasm meter though, to fully appreciate it.

[Lucian]

40. When it's ambiguous, you go PEMDAS.

288

[PufPuf23]

17. I was much better than average in math and recognizing patterns.

My eyesight, hearing, ability to read and comprehend fast, and ability to do math in my head is slipping at age 58.

Plus my Body is in decline.

Part may be a result of calculators and pcs.

I interact with much younger university graduates in my fields and they tend to not understand the mathematics nor do could they do the calculations readily without newer electronic tools (that do greatly expand what can be quantified but are extensions of basic models). Other mathematical techniques such as fractal geometry have become useful and common because of computers but the users tend to not understand the full implications mathematics in their models.

[lumberjack_jeff]

26. My theory is that the rules as taught to us older folks have changed.

That may be the case.

FWIW, I think the answer is two for the reason that proud2blib gave upthread, but (many if not most) newer calculators and math programs say the answer is 288.

[boppers]

52. I was taught to work from the inside out, and parenthesis *can* be implied by spacing

48 / 2(9+3) = 2, because the additional spaces in the formula indicate (48)/(2(9+3))
48/2(9+3) = 288, because the space "operator" isn't used to group (as above) the pieces

[PufPuf23]

75. I agree the answer is 2 by how written and easy in head calculation. nt

[demwing]

157. I'm open to this, can you show me where that rule (parentheses implied by spacing) is written

not in this context anyway.

Implied parentheses apply to vertical fractions (solve the numerator as if it were written within parentheses, same with the denominator), and exponents have implied parentheses as well. Those are the only examples I know of, unless you can provide a source for a better understanding.

[ManiacJoe]

226. See post #47.

[The Philosopher]

18. I suppose saying

forty-eight divided by two times nine plus three doesn't helps any.

[Electric Monk]

21. Makes it worse actually, because you left out the brackets.

[SidDithers]

22. Replace (9+3) with x...

Does 48÷2x = 24/x or 24x?

It's ambiguous, but I would interpret that to be 24/x, and if x=(9+3)=12, then 24/12 = 2

If you interpret 48÷2x = 24x, and x=12, then your answer would be 288.

I'm 43 and my vote is for 2. :hi:

Sid

[demwing]

110. Thats not how you work the algebra

In this equation you wouldn't replace (9+3) with X, you would replace (9+3) with (X+Y)

and the equation would look like this:

48/2(X+Y) = ((48/2)*X)+((48/2)*Y) = (24X)+(24Y)

And there is no answer. X and Y can mean anything here, because there is no outcome.


The equation only makes sense when you know the outcome (such as 288) and you are tasked with finding X,Y

48/2(X+Y) = ((48/2)*X)+((48/2)*Y) = (24X)+(24Y) = 288

X, Y may have a few variable that work, as as long as (X+Y) = 12.


Now take it to the next level, and set the outcome to 2, like this:

48/2(X+Y) = ((48/2)*X)+((48/2)*Y) = (24X)+(24Y) = 2

If you solve for X,Y here, you cannot possibly get 9,3. It does not work.


There's a few ways to get an outcome of 2. For instance, X an Y could both equal 1/24th:

(24*(1/24))+(24*(1/24)) = 1+1 = 2

or X could equal 1/8th, and Y could = (-1/24th):

(24*(1/8))+(24*(-1/24)) = 3+(-1) = 3-1 = 2

X, Y may have a few variable that work to get you to an outcome of 2, as as long as (X+Y) = 1/12 (and isn't that an interesting coincidence?)


Anyway, as you can see, 48/2(9+3) cannot equal 2. It just isn't possible.

[hunter]

167. 48/2(X+Y) = ((48/2)*X)+((48/2)*Y) = (24X)+(24Y)

:applause:

[fishwax]

188. sure you can replace (9+3) with just x

48/2(9+3) = 48/2(12).

You can replace 12 with x. So either you can replace (9+3) with x, or those two aren't actually equal.

But anyway, even if we use (x+y) instead, your analysis still highlights why it's reasonable to get "2" as an answer.

In this equation you wouldn't replace (9+3) with X, you would replace (9+3) with (X+Y)

and the equation would look like this:

48/2(X+Y) = ((48/2)*X)+((48/2)*Y) = (24X)+(24Y)

I noticed that you used the "/" instead of the "÷" -- are these two equivalent? Of course they are, and yet, it introduces ambiguity. Since it is difficult/impossible to create vertical fractions in most programs that people regularly use, including most word processing software and most internet message boards, it's perfectly reasonable to read 48/2(x+y) as a fraction in which 48 is the numerator and 2(x+y) is the denominator. Similarly, it would be reasonable to read 48/2z (I'll stipulate here that z = x+y) as a fraction with numerator 48 and a denominator of 2z.

Since people see the "÷" and the "/" as interchangeable, then, it could be reasonable to conclude that 48 ÷ 2(9+3) = 48 / 2(9+3) = 48 (over) 2(9+3). Of course, the "÷" is often avoided in math texts or functions beyond elementary arithmetic.

Different people might read the problem differently, of course. Some may not interpret 48/2z as a single fraction but rather as a fraction multiplied by z. This might be (I don't know) because of conventions for treating such arrangements in programming code. Perhaps, over time, those conventions will become the standard for everyone. (The order of operations is not set in stone, and conventions have changed over the years, after all.) But in the meantime, for those whose experience is with equations presented in more traditional formats, I think it's perfectly reasonable to read it as 48 over 2z.

That said, though, let's try another experiment.

Let's start with the assumption that 48/2(x+y) = 48/2*(x+y) = (48/2)*(x+y) = 24 * (x+y).

If we alter the equation slightly to 48/x we run into a problem with the coefficient. If a variable doesn't have a coefficient, the assumption is that the coefficient is one, such that x = 1x. Mathematically, then, 48 ÷ x should be equal to 48 ÷ 1x and 48/x should be equal to 48/1x.

But based on our initial assumption, 48/1x should be 48/1*x = (48/1)*x = 48x. That means that 48 multiplied by x is the same as 48 divided by x (that's true, of course, if the value for x is 1 or -1, but otherwise it's obviously false).

[ReggieVeggie]

32. Finally! Something we can all agree on!

:hi:

[Cerridwen]

37. I'm over 40 and my answer is 2. My HP32S RPN is under 40

and if I keyed it correctly its answer is 2; it's been a while since it's been used for more than checkbook balancing.

48 enter
2 enter
9 enter
3
+
*
/
answer = 2

Oh, and RPN means Reverse Polish Notation

Anyone else know RPN can check my entries?

[struggle4progress]

65. The Polish and reverse Polish notations are unambiguous without parentheses

The question is whether 48 ÷ 2 x (9+3) means

÷ 48 x 2 + 9 3

or

x ÷ 48 2 + 9 3

[Cerridwen]

76. Or...

48 2 9 3 + * /

versus

9 3 + 48 2 / *

In short, the whole question could have to do with postfix versus infix. Interesting...in a nerdy sort of way. I found a fun paper about it.

http://digitalcommons.unl.edu/cgi/viewcontent.cgi?article=1045&context=mathmidexppap&sei-redir=1#search=%22history+math+order+of+operation%22

[rufus dog]

38. Over 40, say 2

And I don't even know your Dear Aunt Sally so how could I hate her?

[Lucian]

39. Male under 40, the answer is 288.

I posted this in ATA. I want to know what they think the answer is.

:P

[W_HAMILTON]

41. WHAT THE HELL IS THE ANSWER?!

I first thought it was simple: the answer was 2!

But then when I thought more about it, you should first compute the (9+3). That basically reduces the equation to 48 / 2 x 12, in which case you would compute from left to right, so the answer would be 288.

Now I am being driven insane.

WHICH ONE IS RIGHT?

[Lucian]

42. 288 is right.

[eShirl]

2

[ecstatic]

45. Same here. I've always been great in math

and 2 was my immediate thought. However, I'm talking to a former college classmate who says 288.

[salin]

49. dealing with the paren first = both the operation inside the paren... and then

only discarding the paren after performing the operation expressed as being done to the "quantity" in the paren. That is - to get rid of the paren one not only has to add the 9+3 (then have 48/2(12). To get rid of the paren - have to multiply 2X12. Basic aglebraic distributive property.

[W_HAMILTON]

55. Hmm, that makes sense, too.

But isn't 2(12) fundamentally the same as 2 x 12?

[lumberjack_jeff]

62. Yes.

The issue is the order in which you perform x/y(z). Is it x/y*z or is it x/(y*z)?

ournal of Algebra Vol 17: "We shall sometimes use • and sometimes juxtaposition to indicate ring multiplication. When both occur, juxtaposition takes precedence, ie, xy ■ z = (xy)z."

Tamkang Journal of Mathmatics: "Multiplication is indicated both by juxtaposition and by "x". When both forms occur, juxtaposition takes precedence. So (ab x c) x de means ((ab)c)(de) ."

In 1972, the answer was unambiguously x/(y*z)... or "2". There's a lot of evidence to show that modern calculators and software use x/y*z.

I guess the solution is to write equations in such a way that there's no doubt. Both x/(y(z)) and (x/y)*(z) are clearer.

[Motown_Johnny]

234. that just means that modern calculators and software are flawed

there can not possibly be some point in time when the unambiguous answer of "2" somehow became ambiguous and possibly "288".


The answer is 2

[lumberjack_jeff]

250. I agree

http://epsstore.ti.com/OA_HTML/csksxvm.jsp?nSetId=103110

[salin]

68. certainly if you do them both first before dividing the answer into 48.

if the quantity in the parenthesis had a variable - the only way to get rid of the parenthesis would be to use the distributive property (multiply 2 by both of the terms in the parenthesis.) The issue between the answers seems to be whether or not you do the multiplication an the bottom of the fraction of the equation, or whether you divide the top by 2 and then take that result and multiply by 12.

I think that if the equation was expressed with a variable in the parenthesis, in the original equation, more folks would come up with the same answer (as we has the distributive property drilled into us in algebra. When considering the distributive property the question is 48/<2*9 + 2*3>. Then there is no question to the answer.

Under what algebraic rule do we ignore the distributive property - a(x+b) = ax + bx ?

[Anser]

61. Both are right

Either answer is correct. The notation is ambiguous.
If it was meant as a fraction with the 48 on top then the answer is 2.
If not, and was thus a straight application of order of operations, then the answer is 288.

Needs more parenthesis :)

[demwing]

88. 288 is the correct answer /nt

[struggle4progress]

50. I don't think most people ever use these conventions for avoiding parentheses

According to what I was taught,

48 ÷ 2 x (9+3)

should be evaluated "from the right" as

(48 ÷ 2) x (9+3)

but your writing

48 ÷ 2(9+3)

naturally suggests a closer binding of 2 to (9 + 3) (with an implicit multiplication) than to 48 (with the explicit division), so I automatically read it as

48 ÷ (2(9+3))

The only time I seen a question like this, in the last 30 years or so, was on a test for a temp position, where none of the available answers matched any of the rules I'd ever been taught: one of the answers matched some nonstandard evaluation, so I gave that and was congratulated for my "perfect score," which means (I suppose) that they were trying to test whether I could do arithmetic or whether I was some sort of stickler for the rules

I say, use a few parentheses: write (48/2)(9+3) or 48/(2(9+3)), depending on what you mean

[Anser]

53. Ambiguous Notation

As a math teacher, and an under 40, I'm gonna call a muligan.

But here lies an important lesson: Parenthesis are your friends!

Rewrite as 48 ÷ (2(9+3))= If the answer you want is 2 OR

Rewrite as (48 ÷ 2)(9+3)= If you want 288

As originally written, either answer would be acceptable on one of my tests.

Side Note: The shorthand of dropping the multiply sign is fine; it does not add or detract from the ambiguity.

[ecstatic]

56. What would be your initial response to a question like that nt

[Anser]

58. My "Gut" reaction..

As soon as I looked at it, I saw: 2

I think it is because I read it as a fraction with the 48 on top.


48
_________

2(9+3)

[BlueMTexpat]

137. +2 nt

[RUMMYisFROSTED]

149. +288 nt

:evilgrin:

[Adsos Letter]

138. yep.

[Buzz Clik]

63. +1

[salin]

72. think algebra and the distributive property. Then this is unambiguous.

If he 3 was x ... 48/3(9+x)

In the next notation of this problem is it really ambiguous between 48/(2*9 +2x) vs 48/2 * (9+x) ?

To do the latter ignores the distributive property. The way the question is written verbally would be 48 divided by the quantity of 2 times the product of 9 times 3.

Per your sidenote - when did the distributive property cease to be valid? That multiplication sign is key.

[Anser]

85. Hmm

The distributive property is valid, but it is also a red herring.

The "distributive property" that you are referencing is shorthand for "The distributive property of multiplication over addition." Therefore, the question of whether one should do the division or the multiplication first still needs to be answered. Once answered, you will distribute. The question is then are you distributing 2 over the sum 9 + 3 (ending up with final answer 2), or are you distributing the quotient 48/2 = 24 over 9 + 3 (ending up with final answer 288.)

And on the sidenote discussion, my point was using the shorthand does not change the problem. You can distribute 2 over 9+3 in exactly the same way if written 2 * (9+3) or 2(9+3) because they mean exactly the same thing.

This is not an "alternative view" it is just a poorly constructed problem that does not follow conventions itself... thus all the discussion.

[high density]

66. 2 and I don't see anything ambiguous about it

Is it because I'm a computer guy or something? And I'll make sure to never buy a TI-86.

[progressoid]

71. lotta old people here...

:eyes:



Progressoid included :blush:

[IDemo]

73. Here's another one: 0.999... = 1, true or false?

[Paradoxical]

78. Logically, .9999... is not equal to 1. Statistically, the two are identical.

[IDemo]

87. Not sure what you mean by logically vs statistically

They can be shown to be equal arithmetically.

We can agree that 3/3 or 7/7 equal one.

1/3 = 0.333...
Multiply both sides by 3 and you get 3/3 = 0.999...

1/7 = 0.142857...(I don't know how to type the correct overhead bar to include all the digits in the series)
7/7 = 0.999...

[Paradoxical]

90. Never mind, you did not foul up.

[IDemo]

92. You multiply only the numerator, so it's 3 * 1/3 = 3/3

Both sides means both sides of the equals sign, not all factors inclusive.

I'm trying to show that using the two different forms of notation, 0.999... equals the decimal 1. It doesn't appear correct to the eye, but is.

[MindPilot]

197. You are looking at the difference between exact and approximate

A fraction is exact; 1/7 is exactly one seventh of the whole, seven of those makes exactly one. But if that 1/7 is expressed as a decimal, it's 0.14285714285714285714285714285714 and so on. At what point is it accurate? Four decimal places? Six? 100? 19,000?

[IDemo]

207. It's not just approximate

This will clear it right up. :)

http://en.wikipedia.org/wiki/0.999...

[MindPilot]

212. Now my brain hurts...

I'm still sticking with approximate since I can only barely begin to comprehend that esoteric dissertation. From now on I will simply end every sentence with ...

[Hosnon]

209. The "and so on" makes it mathematically equivalent to 1.

It is not an approximation, just two different ways to write the exact same number: 0.9_ and 1

[Hosnon]

195. Logically they are equivalent.

(a) 1/3 * 3 = 1

(b) 1/3 = 0.3_

(c) 0.3_ * 3 = 0.9_

Therefore, 0.9_ = 1

0.9_ and 1 are just two ways of writing the same number.

[boppers]

82. For certain values of 1, true.

However, that's also a very high value of 0. :evilgrin:

[Doctor_J]

290. This is true, since there is no real number between the two

[LibertyFox]

74. I really don't get it- oh wait.

I don't see any reasonable way that the answer is not 2. I was taught in college algebra that when you see a number next to parenthesis that you do the parenthasis first and then multiply the result by the number next to it.

That is unless you first do 48 / 2 = 24 * (9+3) = oh I get it now.

[krispos42]

77. Shit, now what was I taught in math? I thought it was multiplication before division.

I got "2".

Do the parenthesis first, so 9+3=12.

Then you have 48÷2x12.


So if you do multiplication before division, you get "2".


But if you do it from left-to-right, you get "288".

Shit.


THIS IS WHY YOU USE PARENTHESIS!!!!

[soleiri]

79. I picked 2 as my answer.

Then I check my answer by asking my 15 year old son, who happens to be great at math.
He agreed that the answer is 2.

[Bonobo]

80. 288. Over 40.

Complete the addition in the parentheses first. That is the way (I think) I was taught.

[Zorra]

81. Yes. nt

[underseasurveyor]

83. I'm over 40

And I hate my dear aunt Sally:dunce:

Math not my strong suit.

[Electric Monk]

84. For some reason this problem reminds me of the "airplane on a treadmill" thing

that was "viral" a couple years back.

The plane moves, and the answer is 2.

http://www.airplaneonatreadmill.com/

[nonconformist]

89. I'm under 40 (barely) and I got 2.

[deaniac21]

93. You're a lumberjack and you're ok....

[Renew Deal]

95. That was pretty east

Now what's this about?

[necso]

97. Mathematical notation must be precise (and precisely interpreted),

so that there's only one correct interpretation.

And the use of implied parenthesis can complicate interpretation.

For example, this form

48 ÷ (2(9+3)) =

or

(48 ÷ (2(9+3))) =

resolves unambiguously

48 ÷ (2(12)) =

48 ÷ (24) =

2

Notably, from a math point of view,

2(9+3) = 2 x (9 + 3)

but this isn't true from a notation point of view (as I was taught), where

2(9 + 3) = (2(9 + 3))

Which is illustrated by the form 48 ÷ 2(9 + 3) =

since 48 ÷ 2 x (9 + 3) <> 48 ÷ (2(9 + 3))

...

To illustrate the point a little better, I'll resolve the "(48 ÷ (2(9 + 3))) =" form step by step:

(48 ÷ (2(9 + 3))) =

(48 ÷ (2 x (9 + 3))) = ; a notation form change

(48 ÷ (2 x (12))) = ; a little arithmetic (innermost parenthesis)

(48 ÷ (2 x 12)) = ; a notation form change

(48 ÷ (24)) = ; a little arithmetic (innermost parenthesis)

(48 ÷ 24) = ; a notation form change

(2) = ; a little arithmetic

2 ; final notation form change

[demwing]

101. You're adding parentheses that weren't originally there

(48 ÷ (2(9 + 3))) =/= (48 ÷ 2(9 + 3))

The former equals 2, the latter equals 288.

Operation 1: Parentheses
Operation 2: Exponents
Operation 3: Multiplication and Division
Operation 4: Addition and Subtraction
Within each operation, work left to right...

[Electric Monk]

102. See post #52

http://www.democraticunderground.com/discuss/duboard.php?az=show_mesg&forum=439&topic_id=982937&mesg_id=983374

Implied parenthesis by the spacing in the OP.

[demwing]

112. gotcha /nt

[necso]

111. The full form is either:

(48 ÷ (2(9 + 3))) = (48 ÷ (2 x (9 + 3))) or

(48 ÷ 2 x (9 + 3)) =

But the outer parenthesis are often dropped, so (doing this) it's either:

48 ÷ (2(9 + 3)) = 48 ÷ (2 x (9 + 3)) or

48 ÷ 2 x (9 + 3) =

I was taught 2(9 + 3) as a "shorthand" for (2(9 + 3)). (Others may have been taught differently.)

But such "shorthands" can be subject to different interpretations.

[GoneOffShore]

98. There are 10 types of people in this world

Those who understand binary and those who don't.

[U4ikLefty]

99. lol

[Electric Monk]

100. 5?

:D

[cemaphonic]

115. You dig.

[lumberjack_jeff]

103. My 18 year old says 2. His girlfriend says 288. My 12 year old says 288.

I guess there's no clear generational consensus.

[dsc]

104. as a math teacher I am pretty saddened by this poll

it proves just how badly we have been doing our jobs.


First the correct answer is 288, not 2. The correct way of thinking of the order of operations is as follows. You do the most powerful operation first. Parenthesis are the most powerful because they alter the rules. Exponents are the second most powerful operation since they are repeated multiplication. Multiplication and division are equally powerful since they are merely inverses of each other, and are the next most powerful since they are repeated addition, and thus are worked from left to right. Similarly addition and subtraction, which are inverses of each other, are worked from left to right. You do NOT do all multiplication before doing all division.

So this problem 48 \ 2(9+3) is done in the following order. First you add 9 and 3 since they are in parenthesis. You then get 48 \ 2(12). Now you work left to right, you do not do the multiplication first. 48/2 = 24, 24 * 12 = 288. As to the distributive property, if you use that to do this problem you must divide before you distribute since the entirety of 48/2 must be distributed, not just the 2.

PENDAS, or please excuse my dear aunt sally is not the best way to think of how the order of operations works.

Again, this poll shows just how badly we math teachers have been doing our jobs.

[Electric Monk]

105. Implied parenthesis due to the spacing? See post #52. nt

http://www.democraticunderground.com/discuss/duboard.php?az=show_mesg&forum=439&topic_id=982937&mesg_id=983374

[dsc]

107. uh, no

no amount of space adds parenthesis which weren't there. Either you put parenthesis in the problem or you don't.

[ecstatic]

109. From another math teacher:

"This is a common and controversial topic actually that involved
(PE)(DM)(AS) Please Excuse My Dear Aunt Sally and some rules learned in
algebra. Both are correct but one is still incorrect. The reason being is
that thru pre algebra we teach you to separate and do everything as an
individual equation along the lines of PEDMAS which will yield the answer
288. However, once you get into algebra 1,2,3 and 4 you learn the rules
that go along with the PEDMAS and other basic math skills, such as in the
case the distribution rule. So while 288 is correct, it is only correct
for anyone in pre algebra of below. Once they learn the rules needed it is
an incorrect answer and is now the answer of 2. It is one of those
Building block in pre algebra that the first upon your first day of
algebra you walk in and your teacher goes 'remember all this stuff from
pre algebra. well forget it' type of things. It is a very good equation to
see how many people have beyond "basic" math skills but it is a horrible
problem and rather erroneous to begin with."

----

I tend to agree with that guy only because I've taken college level Calcs 1-3, DiffEQ, Statistics 1&2

[demwing]

118. Where did you find that quote?

curious :)

[ecstatic]

120. here

http://www.cherokeetech.com/VBull/showthread.php?1448-Math-Poll/page5

I can't believe how much attention that equation is getting! :crazy:

[demwing]

128. This is crazy, eh?

btw - that teacher is an 8th grade math teacher. He may be a good teacher, but 8th grade?

[PufPuf23]

119. You must not be a very good math teacher - no insult intended

I like the / rather than the divide. The notation is confusing.

The answer is 48 divided by two times twelve (3+9) and the result is two (2).

At best (or worst) this is a poorly written equation in text in that there is even a question.

All operations above the divide are done then all operations below the divide are done then do the division.

[dsc]

139. No since they used a division symbol in the OP

we know it isn't a fraction but a division problem followed by a multiplication problem. I couldn't put a division symbol in so used the slash. you would be correct if it had been a fraction but it isn't.

[MindPilot]

199. There is a difference between fractions and division?

Are you saying that one half, 1/2, and one divided by two are not the same expression?

[dsc]

225. No, what I am saying is that there is a difference bewteen

a fraction bar with a complicated expression under it and a division symbol which only encompasses one number by convention. 1 division symbol 2 + 3 is understood to mean 1 divided by 2 and then add 3 which is 3.5. On the other hand 1 fraction bar with 2 + 3 under it means 1 divided by 5 which is 0.2.

[lumberjack_jeff]

162. If us old timers are more likely to get the "wrong" answer, it's not your fault.

Can you please explain why

Journal of Algebra Vol 17 (1970): "We shall sometimes use • and sometimes juxtaposition to indicate ring multiplication. When both occur, juxtaposition takes precedence, ie, xy ■ z = (xy)z."

Tamkang Journal of Mathmatics: "Multiplication is indicated both by juxtaposition and by "x". When both forms occur, juxtaposition takes precedence. So (ab x c) x de means ((ab)c)(de) ."

No longer pertain?

Personally, I think the generational difference is between those who used a calculator in school and those who could not. Most consumer grade calculators will force you to put a multiplication sign into the formula where none actually exists.

The rules as stated in the JoA lead to the answer of 2.

[dsc]

192. when it is all multiplication then order doesn't matter

which means the rule there is irrelevent to this case. Order matters when you are doing division. I also don't think that rule gives you 2.

[lumberjack_jeff]

241. If it only pertains to multiplication, then it is irrelevant, period.

If it's only relevant to multiplication, the rule might as well say "when one of the values is printed in blue, it takes precedence over other multiplication".

I think this is a story problem.

"Aunt Sally has nosy neighbors. She has a $48 budget to make curtains. Curtains are $9 and curtain rods are $3. Each window requires two complete sets. How many windows can she cover?"

$48 / 2($9+$3) = 2 windows

It is not this one.
"Aunt Sally has a major grow operation. One of her customers will give her a 50% discount on the retail price of $48 to make curtains for each window. There are 9 windows on the south side and 3 on the north. What's the total cost?"

($48/2)*(9+3) = $288

[dsc]

263. You can add words till the cows come home

but the problem is, what the problem is. The only way the problem would say what you say it does is if you had brackets around the 2(9+3).

[fishwax]

295. so "2x divided by 2x" is x squared, but "2x divided by (2x)" is 1

That seems very weird to me, and doesn't match up with any math textbook I ever remember seeing.

2x / 2x = 2 * x / 2 * x = x * x = x (sq)

Also true, of course, of 2(x)/2(x)
2(x) / 2(x) = 2 * x / 2 * x = 2x/2 * x = x * x = x (sq)

I remain skeptical. :)

Of course, the truth of the matter is that the expression would rarely (if ever?) be presented in that way in, say, a textbook, where vertical fractions are used. (The obelus isn't really used beyond arithmetic.)

On edit: put a space between x and (sq) to avoid a x(

[dsc]

303. If you use a divsion symbol vs the slash

that example would be ambiguous which is why we use vertical fractions. once you have the parenthesis I think it becomes an unambiguous case. I can see thinking of 2x as one term in your example but no way can I see 2(9+3) as one term.

[fishwax]

312. so you would see "2(x) divided by 2(x)" as unambiguously "x squared"

"which is why we use vertical fractions"

Amen to that, which is why the question is frustrating.

There is a convention (I'm not sure how widespread it is, but it's the one I've always been familiar with) that treats, in a horizontal expression, everything (at least, everything involving multiplication) after the division line as the denominator. You can see this in the writing of common formulas such as the solving the radius of a circle from the circumference: r = c/2π. (There are a few more examples on the first page of this (PDF Warning) list of physics equations from the College Board for the AP Physics Exam.)

Another example is 1x/1x. That's x-squared according to the logic that we must treat 1x as 1 * x. But we know that x/x is 1, and we know that 1 is the unstated coefficient of x, so x = 1x.

I suppose that one could argue that in substituting 1x for x you would have to use parenthesis, making it (1x)/(1x) which would be (along with 1x/(1x)) still equal to 1. But (1x)/1x remains x-squared by that logic.

Ultimately, the logic would seem to say that, say 48 / 1x and 48 / x are not equivalent expressions.

Therefore, for 48/x = 288, x = 24, but for 48/1x = 288, x = 6.

Again--a hearty "exactly" to your statement about vertical fractions. I suppose one could also argue that "÷" and "/" are not actually the same thing, but ...

[dsc]

321. I think it is ambiguous when you have nothing but numbers

and/or variables but becomes less so when you add in operation symbols such as the plus sign. I would say that all of the fractions in your post are also ambiguous which is why I teach my students not to write horizontal fractions but to write vertical ones instead. I would also argue that there is a difference between a division symbol and a fraction bar in this regard. If you use a division symbol then you are implying that there isn't a vertical fraction to be written and thus you are implying an uncomplicated denominator.

[fishwax]

322. x ÷ 1x = x(sq)?

That still strikes me as very odd. But then again, nobody uses the "÷" symbol for algebra anyway, unless they're trying to start fights on the internet :rofl:

[mathematic]

224. Please stop quoting that

It's clear to me that you don't understand the bit you're quoting.

Ring multiplication is associative. That means (a*b)*c = a*(b*c). The outcome of the calculation does not change depending on which has precedence, * or juxtaposition. It does not refer to / and juxtaposition. Division is NOT associative with multiplication(as this whole 2-288 problem demonstrates).

Professional algebraists are notorious pedants and such a trivial note is not unexpected when reading their work. (I can think of one other reason why they might mention it but it would require additional context that's not present in the bit you quoted).

The point here is that you've construed a note about multiplication symbol conventions as having meaning for calculations involving division. Heck, division isn't even always defined on rings!

For the record, I have a degree in mathematics and the largest part of my undergraduate work was in algebra.

[lumberjack_jeff]

231. How about "no"? Does "no" work for you? n/t

[mathematic]

243. Seriously?

Why would you purposely continue to misuse that passage? Perhaps you didn't understand my explanation? I'm willing to address any questions you have.

[lumberjack_jeff]

248. If the rule doesn't apply to division then it's not just trivial and pedantic...

... it's gibberish.

There's no reason to admonish people to do implied multiplication before explicit multiplication if it doesn't also apply to division. It's equivalent to making a rule telling people to do multiplication in numeric order (eg "always multiply the smallest number next, but otherwise left-to right".)

[JonLP24]

171. .

[Hosnon]

196. The correct answer is that the equation is ambiguous, and therefore allows for more

than one correct answer.

In my opinion, the lack of a space between the 2 and the ( indicates that the multiplication must be done before the division to remove the parentheses. But I understand that this is a matter of interpretation.

[MindPilot]

200. It's not an equation and it's not ambigous

It's an expression and it simplifies to 2

[Hosnon]

210. I agree because of the lack of a space. With a space, it would be 288.

From a math website a math-tutor friend of mine sent me:

"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."

[Spider Jerusalem]

108. The answer is 288. Brackets first, then left to right (multiplication doesn't take precedence here).

Quite simple. 48/2*(9+3)=48/2*12=24*12=288. Treating the problem thusly: 48/2*(9+3)=48/2*12=48/24=2 is committing a fundamental error; multiplication and division take equal precedence and both take precedence before addition or subtraction (except in brackets). If the problem had been presented as 48*2/(9+3), the answer would be: 48*2/12=96/12=8 (since again, division and multiplication have equal precedence and one merely works from left to right).

[Throd]

114. CCLXXXVIII

[demwing]

116. Fine, but you only get full points when you show your work

IN Roman numerals

:P

[DevonRex]

152. Hahahaha. Best reply ever.

[thelordofhell]

121. 288

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

here is how it should be written to get 2

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

[Electric Monk]

122. The spaces in the OP weren't accidental. nt

[thelordofhell]

133. There was not an implied space *before* the 48 or the =

If there was implied space then the equation would be
48 / 2(9+3) =

Instead, the spacing was not there, the equation was
48 / 2(9+3)=

Translation

48(/)2(9+3)=48/2(9+3)=288

on edit----for some reason the first entry did not indent before the 48.....weird

[crickets]

125. 48 ÷ 2(9+3)=*waves cane*

This is how I learned it in the "show your work" era before pocket calculators, back when we used to sniff the mimeograph fumes before getting down to business:

48 ÷ 2(9+3)

= 48 ÷ 2(12)
= 48 ÷ 24
= 2

If you want a similar equation to equal 288 it might be written:

(48÷2)(9+3)

= 24 x 12
= 288

Now keep your eyes on your own paper!

[BillyJack]

129. Is there any useful way that I can apply this conundrum

when doing my taxes?

:shrug:

[MilesColtrane]

130. The answer is 2.

Elementary, my dear.

http://faculty.kutztown.edu/schaeffe/Mnemonics/MultRock/two.html

[thelordofhell]

135. Here ya go

BirthCertificate / Trump(Taitz+Birthers) = Crazy

[eridani]

136. The answer is 42

Do you know where your towel is?

[Township75]

142. Under 40 and 288.

I answered it this way, because I consider (9+3) to be in the numerator. If it were supposed to be in the denominator, it must be written as 48/(2(9+3)).

[CBGLuthier]

143. Proving once again that people will argue about anything

The answer is and has always been 2. Any other answer is a result of faulty practices. Math rules do not change.

[JoePhilly]

144. The answer you get depends on whether to interpret this to have 2 or 3 separate components.

Here is how I was taught ...

Starting with the original problem ... you have to answer the question is this 2 or 3 separate numbers.

48 ÷ 2(9+3)=?

The number 48 stands alone. But the string 2(9+3) is actually ONE number. The number 2 is PART OF the parenthetical (9+3), not separate from it. Thus this problem starts with 2 numbers to be divided, 48, by the product of 2(9+3).

And so 48 ÷ 2(9+3) =? becomes 48 ÷ 24 = 2

However, if the problem were written this way, 48 ÷ 2*(9+3)=?, you get 3 numbers to deal with. The 48, the 2, and the sum of 9+3. In this case, the * separates the number 2 from the (9+3). And you solve left to right.

And so 48 ÷ 2 * 12 = 288

That's how I was taught to decipher this kind of problem

And so as written, the answer would have been 2.

[intaglio]

145. Not generational just 2 different calculations

In the 1st calculation shown in the video he is using the calculator software to calculate

48/2(9+3) = 48/(2 x 9 + 2 x 3) = 48/(18+6) = 48/24 = 2 the order of the calculation brackets (multiplied and added), division.

In the second he gets the software to perform a different calculation

48/2 x (9+3) = 48/2 x 12 = 24 x 12 = 288 the order of this calculation is brackets (added), division, multiplication

Indeed you would get a third answer if you input

48/(9+3) x 2 = 48/12 x 2 = 4 x 2 = 8 order brackets (added), division, multiplication.

BODMAS (brackets, "over," division, multiplication, addition, subtraction)

BTW anyone else here remember "reverse polish notation"?

[Bonobo]

146. Two Japanese teachers today agreed it is 288 (and one is a math teacher)

Division gets done first.

Sorry, end of story.

[logosoco]

148. I am 40 or over....

totally hated math as a kid. When I started college in 1983, I had to take basic math!! Twice! Last May, I FINALLY conquered college algebra (okay, the teacher took it easy with me!). Couldn't have done it without the help of my kids (my middle child is just finishing up his freshman year of college and he took calculus in high school like it was nothing!).
Here is how i did it:
48/2(9+3)
48/2(12)
48/24
2
I don't have a regular division sign on my keyboard, but that wouldn't change it for me anyway.
This has been an interesting diversion , along with the royal wedding.

[demwing]

153. but why are you giving multiplication higher precedence than division?

shouldn't you:

1. work out the parentheses: (48/2(9+3) = 48/2(12) = 48/2*12
2. work out the exponents: none
3. work Multiplication AND Division, left to right: 48/2*12 = 24*12 = 288
4. work Addition and Subtraction, left to right: none

[logosoco]

176. One of the things i learned taking algebra in my late 40s...

never ask why!!!!!!
But I think I am looking at it as this also:

48
------
2(9+3)

Either way, I don't distribute the 2, except to what is in the parentheses.

Could also be as simple as I do multiplication first just because I like multiplication more than I like division!!!!

[DevonRex]

150. Over/288.

[NoGOPZone]

155. Damn, this thing is spreading all over! nt

[Kingofalldems]

156. Where is the division sign on the keybord?

[demwing]

161. if your keyboard doesn't have one, ue the ASCII code:

type the ampersand &
followed by #247
followed by a semicolon ;

There should be 6 characters with no spaces, formatted like this: &xxxx; (replace the xxxx with #247)

[Kingofalldems]

163. ÷

Yeah!

[slackmaster]

158. My brain wants to say 2 because of the proximity of the 2 and the parenthetical expression

My knowledge of the rules of precedence tells me that you have to work the problem from left to right, so the correct answer is 288.

[fifthoffive]

164. This is what I was taught

48 ÷ 2(9+3)=
48 ÷ 2(12)=
48 ÷ 24 =
2

Or

48 ÷ 2(a+b) =
48 ÷ (2a+2b) =
Where a=9 and b=2
48 ÷ (2*9 + 2*3) =
48 ÷ (18+6) =
48 ÷ 24 =
2

[Lindsey]

168. I took pre-algebra at my community college so I could get into

a required statistics class (I was in a program at a school where all of the J.C. classes transferred to Cal State where I got my Bachelor's). Anyway, I learned the Order of Operations (Please Excuse My Dear Aunt Sue = Parenthesis, Exponents, Multiplication, Division, Addition, Subtraction). Calculating the equation using this rule, my answer is 2.

[DevonRex]

181. It's multiplication AND division, addition AND subtraction. Four not six.

[ManiacJoe]

189. The key part there is "pre-algebra".

PEMDAS is an incomplete set of rules. See post #47.

[jayfish]

166. 2

48 ÷ 2(9+3)=x
48 ÷ 2(12)=x
48 ÷ 24=x
2=x

To get 288 the problem should be written...

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


ON EDIT: Over 40. ...barely.

[mrmpa]

172. Thank you!!!!

[jayfish]

193. Also,...

the problem seems to be written incorrecty. IMO, it should look like this:

48
2(9+3)

[HuckleB]

204. +1

Nicely done.

[Doctor_J]

265. Nope

48 ÷ 2(12)=x
48 ÷ 24=x


No. 2*12 is no in parentheses, so does not get performed before the division.

[jayfish]

278. Sorry, that is incorrect.

The 2 is in contact with the parenthetical which means it must be performed immediately following the solution of said parenthetical. if it was 48 ÷ 2*(12)=x then you would have something. Then again, it would never be written that way. It would be (48 ÷ 2)12=x or (48 ÷ 2)(12)=x. The parenthesis around 12 in the second example are superfluous though.

[L0oniX]

169. ***

[ecstatic]

170. Hilarious video about this equation:

(Hitler parody) http://www.youtube.com/watch?v=bwpWw-iVKHc

[Javaman]

173. Hell, if I knew there was going to be a quiz today, I would have stayed home! nt

[Iggo]

174. PEMDAS.

Parens, exponents, mult, div, add, subtr.

[InvisibleTouch]

175. I was taught to always do the parentheses first.

I was always decent at math. I didn't particularly LIKE it, didn't particularly find it easy, but was good at it. Good enough to study and get mostly A's, anyway, and then put it behind me. Oh, and in the "over 40" group here.

(edit: typo)

[NoGOPZone]

179. I'm on the 288 side. As for my age, I refer you to

the Miruts Yifter quote that is my signature.

[JHB]

180. In base 10 or 12?

[fishwax]

182. what is 4x ÷ 2x?

And is it the same as 4x/2x?

[MindPilot]

201. 2x and yes. n/t

[fishwax]

237. do you mean simply 2?

If one views 48/2(9+3) as 2 (and I think you're in that camp, right?), then 4x/2x should also be 2 (because the x's cancel each other out and the 4 over 2 reduces to 2 over 1).

If, on the other hand, one argues that 48/2(9+3) = (48/2)*(9+3), then 4x/2x = 4*x/2*x = 2x*x = 2xx. (I don't know if I can get superscripts on DU, but I mean 2x(sq).)

I'm sure that no math book I've ever seen presented 4x/2x as equal to 2x(sq).

[MindPilot]

269. No, because the "x" represents something.

4x/2x says you have 4 things divided by 2 things, which of course leaves you with 2 things, not just 2, but 2 things, so 4x/2x=2x.

1x/1x is x, not 1, because you don't know what x is, you just know you have one of them.

[fishwax]

275. the x represents a value, not a thing or things

it's not like 4 cows divided by 2 cows, it's (4 multiplied by a number represented by x) divided by (2 multiplied by a number represented by x).

Put any value (other, in this case, than 1) in for x and you will see that the final result can't be 2x.

(4*7)/(2*7) = 28/14 = 2
(4*5)/(2*5) = 20/10 = 2
(4*200)/(2*200) = 800/400 = 2

1x/1x is 1 because no matter what value of x is, dividing a number by itself gives you a value of 1.

[MindPilot]

277. You're stating the identity property of 1

I thought you were looking for some solution to X. I believe we are in agreement now.

[GreatCaesarsGhost]

183. 48 is the dividend, 2(9 + 3) is the divisor and 2 is the quotient

[TheKentuckian]

186. Slightly under and 288. I don't get the instinct to divide the result of the parentheses.

I never passed anything tougher than statistics though.

[JoePhilly]

203. Here is where that instinct comes from ...

If we look at the original equation ... 48 ÷ 2(9+3)=?

... the real question is how you interpret 2(9+3).

Most people know that you always "figure out" the "parenthetical phrase" first.

But as written above, what is the "parenthetical phrase"? Is it only (9+3), or is the 2 PART OF the parenthetical phrase?

When I was taught this (over 40), if the 2 is TOUCHING the parenthetical, it is PART OF the parenthetical. The result of which is 24.

If you want the 2 to be SEPARATE from the parenthetical, you would write it this way: 2*(9+3).

This latter notation makes it clear that the number 2 and the parenthetical (9+3) are SEPARATE elements. When you attached the number 2, to the parenthetical, as in the original example, the 2 becomes PART OF the parenthetical. And as such, you include the multiplication of the number 2 as part of resolving the parenthetical.

So for me, the answer to this question is driven by the presence (or lack there of) of a "*" between the 2 and the (9+3).

[ManiacJoe]

205. Right on the money!

[JoePhilly]

208. Thanks ... personally, I think the ability to use the 2 notations is useful.

It gives you some flexibility.

I mean, you could also go with this .... (2*(9+3)) = 2(9+3).

Those are (were), from what I learned, EQUAL.

And if I recall, the latter was also intended to reduce the number of compound or nested parentheticals in complex equations, in part because if you had multiple such entities, you end up with very asymmetric, nested, paren structures where it is almost impossible to figure out the order of simplification. Bolting a multiplier (or divisor) on to a parenthetical helped drop a pair of parens.

I suppose that the mathematical notation police could be changing the rules, but I think as you get into more complex math situations, you still need/want this flexibility in notation.

Of course, I am over 40 too. :-)

[Hosnon]

211. +1. The parenthetical is "2(9+3)" and we all agree that parentheticals are resolved first. nt.

[TheKentuckian]

213. I see. Thanks for the insight.

[DCBob]

215. Yes, there is a visual component to this conundrum..

to me 2(9+3) is the same as 2x where x=12.

48 / 2x where x=9+3 is of course 24.

[MindPilot]

219. Exactly and because an operator separates quantities

in the expression 48 ÷ 2(9+3) the quantity 48 is separated by the division operator from the quantity 2(9+3) which simplifies to 2. To get 288, the expression would have to have an additional operator, or
48 ÷ 2*(9+3) which really would be ambiguous and have to be expressed as (48÷2)(9+3) or 48/2 x (9+3)

[JoePhilly]

220. Great addition to the discussion.

[MindPilot]

194. In my fifties and I took an advanced algebra course this year.

It's 2...PEMDAS applies and multiplication (the additive operation) takes precedence over division the (subtractive operation)

Left to right has no bearing since the expression is actually a fraction expressed as 48 over 2 times the quantity 9 plus 3

Now, if you want to get really picky, there should be no equal sign because there is nothing to solve for...the exercise is merely to simplify the expression. It is not an equation, it is an expression.

[thelordofhell]

216. Please see reply #169

[MindPilot]

221. You have it exactly backward

You are introducing an operator where there isn't any. The expression as written is: quantity 48 divided by quantity 2(9+3). If there were another operator ie 48 ÷ 2*(9+3) then it would be quantity 48 divided by quantity 2 times quantity (9+3). Thats because of the commutative property, so we could look at the same expression this way: 48 ÷ 1(9+3)*2. The one in front of the parentheses is implied and it will not affect the calculation either way so regardless of if you distribute the one, or divide 48 by 1, subsequent application of the order of operations says add 9+3, then multiply that result by two, then divide into the 48.

[JoePhilly]

227. +1, 2(9+3) is ONE quanity, not two, unless you add the "*" in between

[thelordofhell]

229. No it's not

the ()=*

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

However, when you put the division part in, it changes the way you have to do the equation, because you have to do multiplication/division (or if it helps division/multiplication) in order from left to right.

48÷2(9+3)=48÷2(12)=48÷2*12=24*12=288

Sorry about the confusion, but that's how you do it.


:pals:

[JoePhilly]

236. The lack of the "*" makes 2(9+3) a single quantity.

And as I said earlier, this interpretation is not one I made up. It is the interpretation that many of us were taught so that we could differentiate the two possibilities without adding many new paren pairs.

The notation police may have changed that, but its not how many of us were taught this.

[fishwax]

239. by that logic, x = 1/x

Of course, the statement x = 1/x is true if x = 1 or x = -1, but it's obviously false for any other real number. 2 != 1/2.

Let's start with the assumption that 48/2(12) = (48/2)*(12).

Now let's apply the same logic to 1/x.

We know that if a variable doesn't have a coefficient, then the unstated coefficient for that variable is 1: x = 1x.

So we should be able to express that coefficient in the statement without changing its value: 1/x = 1/1x.

Based on the logic in our assumption, 1/1x = 1/1*x.

Going left to right with the order of operations, we get that 1/1*x = 1*x = x.

Therefore, if 48/2(12)=(48/2)*12, then 1/x = x.

:)

on edit: removed some wayward punctuation

[thelordofhell]

228. Sorry, but 288 is the answer

by the way......48÷1(9+3)*2=48÷1*12*2=48*12*2=576*2=1152.....so you can't look at the expression that way cause it's not the same.

Cool, huh

:pals:

[MindPilot]

270. Nope, addition does not clear the parentheses, multiplication does

48÷1(9+3)2
48÷1(12)2 -- still have the parenthetical quantity there so there is another pass of multiplication: 1*12*2
48÷24

Because of the commutative property of multiplication, the order doesn't matter; you just have to do all the parenthetical operations first.

[demwing]

240. Multiplication does NOT have precedence over division

They are within the same level, just as addition and subtraction have equal precedence.

[ceile]

198. My head is going to explode...

Awesome thread!

[DCBob]

217. What is 48 ÷ 2x where x = 9+3 ?

That equals 2 and is the exactly the same as saying 48 ÷ 2(9+3).

However if it was written 48 ÷ 2 * (9+3) then that's different. There is a visual component to this that some calculators cannot interpret.

[Hosnon]

230. THIS! nt.

[lumberjack_jeff]

232. Yes. n/t

[demwing]

244. That's not correct!

According to the Wiki page on order of operations:

An expression like 1/2x is interpreted as 1/(2x) by TI-82, but as (1/2)x by TI-83.<2> While the first interpretation may be expected by some users, only the latter is in agreement with the standard rules stated above.

http://en.wikipedia.org/wiki/Order_of_operations

and they cite the Texas Instruments Knowledge Base as their resource:

Solution 11773: Implied Multiplication Versus Explicit Multiplication on TI Graphing Calculators.

Type: Question and Answer Last Updated: 16-JAN-2011 23:45:41
Does implied multiplication and explicit multiplication have the same precedence on TI graphing calculators?

Implied multiplication has a higher priority than explicit multiplication to allow users to enter expressions, in the same manner as they would be written. For example, the TI-80, TI-81, TI-82, and TI-85 evaluate 1/2X as 1/(2*X), while other products may evaluate the same expression as 1/2*X from left to right. Without this feature, it would be necessary to group 2X in parentheses, something that is typically not done when writing the expression on paper.

This order of precedence was changed for the TI-83 family, TI-84 Plus family, TI-89 family, TI-92 Plus, Voyage™ 200 and the TI-Nspire™ Handheld in TI-84 Plus Mode. Implied and explicit multiplication is given the same priority

http://epsstore.ti.com/OA_HTML/csksxvm.jsp?nSetId=103110

Texas Instrument calculators formerly gave higher precedence to implied multiplication, but they stopped the practice with/after the TI-83 line..

So, if it were the correct method, why did the world's leading manufacturer of calculators stop?

[fishwax]

245. was the world's leading manufacturer of calculators doing it wrong before?

Implied multiplication has a higher priority than explicit multiplication to allow users to enter expressions, in the same manner as they would be written. For example, the TI-80, TI-81, TI-82, and TI-85 evaluate 1/2X as 1/(2*X), while other products may evaluate the same expression as 1/2*X from left to right. Without this feature, it would be necessary to group 2X in parentheses, something that is typically not done when writing the expression on paper.

So this basically says that, when written, 1/2x means 1/(2x). It's not normally written that way, but that is what it means.

I'm speculating, since I know relatively little about the manufacturing calculators or about the calculator industry, but I would guess that the change had more to do with ease of use than the realization that previous models had been incorrect.

[demwing]

247. Apparently so, and that's why they corrected their products

The quote says:

"Implied multiplication has a higher priority than explicit multiplication to allow users to enter expressions, in the same manner as they would be written"

In other words, they were trying to make things easier. But your guess is that they changed it AGAIN to make it easier? I thought it was easier the previous way? Well...at least that's what the quotes says...

My guess is that they want their calculators to accurately reflect accepted principles. So, when it was accepted to give implicit multiplication priority, that's what TI calulators did. Now, the practice is to give implied and explicit multiplication equal priority.

"Dr Math" agrees, lol:

In my research for another Dr. Math "patient," I found that some
calculators have experimented with this rule. Calculators have
somewhat different needs than mathematicians, since they have to take
input linearly, one character after another, so they are forced to
make a decision about it. On the TI Web site I learned that they
deliberately put this "feature" into the TI 82, and then took it out
of the TI 83, probably because they decided it was not a standard rule
and would confuse people.

http://mathforum.org/library/drmath/view/54341.html

[fishwax]

252. actually, the quote says the *old* way reflects standard convention

"... to allow users to enter expressions, in the same manner as they would be written"

So the *old* way reflected how expressions are actually written. Then they changed it.

All the Dr. Math quote demonstrates is that one shouldn't necessarily trust how calculators do things, since they apparently have an agenda of their own.

[lumberjack_jeff]

251. "So, if it were the correct method, why did the world's leading manufacturer of calculators stop?"

Because they're easier to make without "this feature"?
Because it's "a feature" that they felt people weren't using? (which is pretty much what the article says)
Because they screwed up on the TI-86 and expect people to live with the bugs.

The article you posted says: "Implied multiplication has a higher priority than explicit multiplication to allow users to enter expressions, in the same manner as they would be written."

Explain that sentence to me in a way that supports your contention.

[demwing]

257. It's a specific answer to a question about why the caculators responded in a given way

Q. Does implied multiplication and explicit multiplication have the same precedence on TI graphing calculators?

A. Implied multiplication has a higher priority than explicit multiplication to allow users to enter expressions, in the same manner as they would be written.

To me that seems pretty straightforward. TI does not say:

"Implied multiplication has a higher priority than explicit multiplication to allow users to enter expressions, in the same manner as they SHOULD be written"
-or-
"Implied multiplication has a higher priority than explicit multiplication, as is the standard in algebraic expressions."

No, their technical writer used the term "would," the past tense of will. TI is saying "This is how people would write it out, back in the day."

The proof is in the pudding - if TI thought implied multiplication should maintain its higher priority, they never would have changed the programming.

To address your individual points:

Why did the world's leading manufacturer of calculators stop?
Because they're easier to make without "this feature"?
Actually, any product change would require additional research and planning. The easiest thing to do would be nothing at all!

Because it's "a feature" that they felt people weren't using? (which is pretty much what the article says)
Well yeah, that's my point - it wasn't an accepted practice

Because they screwed up on the TI-86 and expect people to live with the bugs.
Which exaplains why the fix has been included on the caculaters since theb.

[DCBob]

260. TI does not set algebraic standards.

I would like to see a quote from a mathematics academic journal stating that the world should now look at equations like this in a different way.

[ZombieHorde]

218. I think the answer is two. Step one: 9+3=12. Step two: 2*12=24.

Step three: 48/24=2.

[Doctor_J]

266. No, step 2 is 48/2

There is no reason to compute 2*12 before 48/2

[NutmegYankee]

223. I'm under 40, and the answer is 2.

But I'm an engineer, and not representative of my generation.

[lumberjack_jeff]

233. These guys get it.

But being under 40, I'm sure they'll get the wrong answer. ;)

http://www.youtube.com/watch?v=B5XfCE_x6S4

[BillyJack]

238. Geeeeeezzzzz.....no WONDER we can't all get along when we can't even agree

on the answer to a MATH problem? :freak: :think:

Thanks OP. Point taken.

[lumberjack_jeff]

242. I find it fascinating that there's such a marked difference between age cohorts.

[Renew Deal]

281. DU gets along just fine

And this isn't a DU only argument. It started elsewhere.

[U4ikLefty]

253. You are all banned from my Excel files.

This thread scares me.

[demwing]

256. Ubuntu says having "48÷2(9+3) = 2" is a Bug, and incorrect

and that earlier versions of the calc tool had an incorrect evaluation order for implied multiplication.

No worries though, the bug was removed from the calc tool in version 6.0.1, and the problem now resolves correctly, with an answer of 288.


https://bugs.launchpad.net/ubuntu/+source/gcalctool/+bug/756277

[NuttyFluffers]

258. with no other decimals or fractions or integers present, whole numbers are assumed as presented

thus PEMBAS kicks in and the answer is two. if someone intended decimals/fractions in the equation then they would have written the equation with that in mind and used the already established protocol in order to be more clear and concise for those assumptions.

bad notation is killer, almost as bad as over-interpretation of what is written. write well. further, read as written and assume no greater complexity. do this and things become clear.

damn, bustin' out my grammarian and linguistic skills on a math problem. what has the post post-modern world come to...

[Blecht]

259. Of course it's unambiguously 288 no matter how old you are

Maybe it would help to think about the fact that dividing by something is the same as multiplying by that something's reciprocal. The equation becomes:

48(1/2)(9+3)= 288

And to those of you bringing in what a calculator does with the equation are definitely not reading the manual. Calculator input and mathematical rules are two different things.

On edit: It's even better if we write the 1/2 as 0.5:

48(0.5)(9+3)= 288

[Doctor_J]

264. Math prof checking in

the answer is 288. 2*(12) is not a parenthetical operation, and thus doesn't get preference. after performing 9+3, the problem becomes

48 / 2 * 12, performed left to right. Try this on any calculator, as written in the OP, and 288 will come out.

[DCBob]

267. Math Prof, If this question was on a math exam: What is 48 ÷ 2x where x = 9+3 ?

Clearly the answer to that is 2. Why is that not the same as the original equation??

[Doctor_J]

273. I would not have such an expression on an exam

It is deliberately misleading. But, since you asked, technically your first expression simplifies not to 48/(2x), but to 48/2*x, or (48/2)*x, = 24x. BTW where are all of these people getting the division sign? My keyboard has no such thing.

[DCBob]

283. Huh?? Looks like a very typical question for basic grade school algebra exam..

How about this one..

Solve for x: 48 ÷ 2x = 2

There is no doubt the answer is 12 or (9+3). Right?

[Doctor_J]

284. I just tried this on my variable-capable calculator

I set x = 9+3, and then typed in 48/2*x, and 288 came out.

Edit: As to this question, this is one of the reasons we seldom use the division sign in algebra or anything. Here's what you wrote

48 ÷ 2x = 2

I think what you mean, I would write as a fraction, with 48 in the numerator and 2x in the denominator. In this case 2x would be evaluated first and you answer would be correct. But the insertion of a variable x modifies the OP irrevocably. That one has a string of integers with some ops.

[DCBob]

302. Yes, of course when typing into calc that way you would get that answer..

because you separated the "2" and the "x" with a "*". That changes the result and the left to right rule would take effect. However, it is common sense that 2x is a separate entity when written that way and 99% of math students would calc that portion first. Seems to me that is a basic math convention.

[demwing]

311. watch here

http://www.youtube.com/watch?v=JHX8fBb8-ac

[demwing]

291. If your keyboard doesn't have one, use the ASCII code

Type the ampersand: &
followed by these four characters: #247
followed by a semicolon: ;

There should be a total 6 characters with no spaces between, formatted like this: &xxxx; (replace the xxxx with #247)

[demwing]

288. where x = (9+3)

http://www.youtube.com/watch?v=JHX8fBb8-ac

[DCBob]

304. Using an older TI and you get "2"... so a calulator proves nothing.

Forget about using a calculator. Just solve the equation on a piece of paper using basic common sense.

[demwing]

309. Proves you need a newer TI

And solving the equation on a piece of paper using basic common sense gives an answer of 288.

[DCBob]

314. So are you saying algebraic conventions have changed recently??

Please provide citing from a mathematical journal. FYI.. Texas Instruments help page is not a mathematical journal.

[demwing]

316. The thread has been over this several times

Order of operations dictates working inside the parentheses, then the exponents, then multiplication and division (l to r), then addition and subtraction (l to r).

The idea that the implied multiplication of 2(9+3) takes precedence over the order order of operations (in this case, or in general), has not been sufficiently established.

[DCBob]

318. In your mind it has been "sufficiently established"..

but not in mine or most other people commenting here. Anyway, I am done with this debate. Ciao.

[demwing]

319. I think you meant "not been sufficiently established"

the explicit "not" takes precedence over the implied "not"

Sorry, I just had to... :P

anyway, peace?

[DCBob]

325. HA! Yes, your are right..

Good one! Peace.

[MindPilot]

268. Prof, how is 2*(12) not parenthetical?

Addition doesn't clear parenthesis, multiplication does:

Simplify the expression: 48 ÷ 2(9+3)
48 ÷ 2(12) -- still has parentheses
48 ÷ 24 -- NOW the left-to-right sequence can apply

[Doctor_J]

271. The + is INSIDE the (). The * is not

[lumberjack_jeff]

274. With all due respect.

You're saying that you're right because
a) you're a math professor
b) when you push those buttons on a calculator your result is what comes out?

I can't speak to point A, but Point b is demonstrably false.
http://www.youtube.com/watch?v=U706g7_m_78&

http://www.democraticunderground.com/discuss/duboard.php?az=show_mesg&forum=439&topic_id=982937&mesg_id=983325
http://www.democraticunderground.com/discuss/duboard.php?az=show_mesg&forum=439&topic_id=982937&mesg_id=991783

[demwing]

286. Use a better calculator

:)

http://www.youtube.com/watch?v=JHX8fBb8-ac

[lumberjack_jeff]

296. Even the best calculator in the world suffers when you push the wrong buttons.

The guy in the video asked the calculator a different question from the one posed.

[demwing]

310. No, he assigned a value to x of (9+3)

then ran the problem as 48÷2x - which is exacly the same as 48÷2(9+3)...

and got an answer of 288

[Doctor_J]

289. No, what I said was,

a. I am a math professor
b. The answer to your original question is 288, according to the order of operations that I and every other math teacher in the world put forth
c. If you type your expression from the OP into any calculator, 288 will be reported as the answer

as an aside. So whether or not someone on youtube was able to push buttons on a calculator to make something unusual come out, the answer to your original question is 288, and any calculator, including the MS Windows one, will get it right.

Whether or not this garners your respect won't change much in this case.

[lumberjack_jeff]

300. It's been amply demonstrated that about half of the calculators will get it wrong.

The question is which half.

[demwing]

306. As far as TI is concerned, its the older half

and as time progresses and new models hit the market, that "half" will become an ever dwindling fraction

[Renew Deal]

279. I've talked to lots of other smart people about this and I get a mix of answers

And the funniest thing is that people are adamant they are correct. I'm still not convinced it's 288.

[demwing]

308. Yeah, its odd

I just saw a youtube video of a Berkely Ring Theory Professor who had never hear of PEMDAS, or of the Order of Operations, and when presented with the concept, gave his opinion that it must not be a very well known principle:

http://www.youtube.com/watch?v=SB3Ekgt1pHw&feature=fvsr

He also said that as written: 48÷2(9+3) is ambiguous, and that either answer could be correct.

Which all come together to make me suspect that maybe this Ring Theorist needs to get out a little more :)

[lumberjack_jeff]

276. The results are in.

Those over 40 understand the answer to be 2 by a 4:1 margin.
Those under 40 understand the answer to be 2 by a less than 2:1 margin.

My takeaway;
a) the rules for math drift in the same way that the rules for language do.
b) in practice, the tools dictate the rules, not the reverse. (eg "I'm a math professor and my calculator says 288")

[Renew Deal]

280. Rules of math don't drift

They are fixed, at least in the simpler calculations. There is something severely wrong when people and calculators cannot agree on an answer.

[lumberjack_jeff]

285. The written rules in 1972 said that multiplication by juxtaposition takes precedence.

If that is still the case today, the answer is 2.

If the answer is 288, then the rules (conventions?) have drifted.

I prefer the former, but I can live with the latter if we could at least acknowledge *that*.

[Renew Deal]

297. That's my problem too.

I can accept being wrong, but it bothers me that no one really knows (even if they say they do). There's too much disagreement.

[rug]

287. I'm over 40 and the answer is 143.

[Renew Deal]

298. Finally someone gets it right.

:D

[lumberjack_jeff]

299. Eureka! The compromise solution!

You'll get a nobel for this, you know.

[Dappleganger]

293. Mittens

[BillyJack]

324. LOL - tossels hair, says "you goof"....NOW

BACK TO THE ARGUMENT!