I think I have to make this my last comment, since it seems like the "opinion poll" is more or less 50/50.For seem reason you (Math Guy #288) seem to think that "48÷2*(9+3)" is somehow different than "48÷2(9+3)".For me, they are exactly the same in every respect, except for one missing "multiplication" character. Let's call one expression using "explicit multiplication", and the other "implicit multiplication".You believe, and would have everyone believe, that "implicit multiplication" has a higher precedence, than "explicit multiplication". Please provide me some kind of authoritative reference for this.Not to let you off the hook -- I really want to see a reference to a rule or established convention, or some sort of authoritative guideline, that goes beyond your own anonymous opinion, or "mplatt"'s highly credentialed yet jovial anonymous New England math professor -- but here are my final thoughts:First, in the real world, such ambiguities never arise, because the right answer is defined by the real world situation, attempting to guess how much water it takes to fill a swimming pool, or how long it will take, rather than how some mathematician misrepresented it in a single line, infixed notation, in an e-mail. Real mathematicians have access to equation editors.I did a little google research and here I find that orders of precedence were "established at least as far back as the 1500s" (before any calculators or computers):http://www.purplemath.com/modules/orderops.htmIn my limited research, I couldn't find that any special treatment or change of priorities or precedences for "implied multiplication" versus "explicit multiplication" that was established by any sort of universal or de facto convention.I found this opinion relevant and indicative of all opinions. If you want to believe the answer is "2", you must put "2(9+3)" in the denominator, because of some strong yet implicit bonding. But here "http://www.purplemath.com/modules/orderops3.htm""purplemath" opines that:When the vertical fraction above is reformatted horizontally (say, for typing it into an e-mail or a forum posting), you must convert the (vertically) implicit grouping into an (horizontally) explicit grouping, or this grouping could be "lost" or at least misunderstood.Here, in our case, in the subject line, no explicit grouping exists, i.e. no parentheses or brackets were added to enforce and ensure the implicit grouping.Here I found that the one company of Texas Instruments was also of two minds, sometimes giving special priority to implicit multiplication, and sometimes not:http://mathforum.org/library/drmath/view/72166.htmlhttp://epsstore.ti.com/OA_HTML/csksxvm.jsp?nSetId=103110But TI said that did it as a matter of convenience for their users, not to have to enter extra parentheses, rather than some established or de facto convention by 16th century mathematicians.(This explains the amusing "juxtaposition" result of the TI-85 versus TI-86).So for me, the lack of consensus clearly demonstrates no convention exists that supports the priority of "implicit multiplication". In the face of a lack of consensus, and a lack of a rule, "implicit multiplication" should deserve no de facto special status or exemption.I will leave with these final two contradictory thoughts:1) Although Dr. Math also indicated a strong preference for explicit notation, and parentheses, he did concede that some felt implicit multiplication deserves special treatment:"The problem is that some calculators that allow multiplications to be implied as in the former case treat that as a higher precedence operation (as some algebra texts say)"http://mathforum.org/library/drmath/view/72166.html2) How would I evaluate "3x/2y"? I find myself jumping the fence and saying that "3x" and "2y" are tightly bounded by implicit multiplication, without being able to explain why 2(9+3) is not.So I will leave some of you with the benefit of the doubt that you were taught that "implicit multiplication" deserves special treatment, because the confusion existed in the authors of your algebra text book.
Math Guy #288 wrote:
You are wrong. The question is not written: 48÷2*(9+3)
It is clearly written: 48÷2(9+3)
You must multiply out the 2 first. Either 2x9+2x3 or 2x(12). Either way, the answer is 2.