Read this...Mathematics is supposed to provide you with single, definitive answers. However, in this case, it does not. I really scoured the internet and countless textbooks for the proper answer and have made inquiries with few people I know whom I venerate as some of the greatest minds that I have had the luxury of meeting and the answer isn't as definitive as you would hope. Sadly, the real answer is that both sides are correct to a point. (…and it pains me to even remotely admit that anything outside of 288 could be the answer).
Position 1: The Order of Operations: This stance states that since the standard order of operations puts multiplication and division on the same rank, the equation can be read as (48÷2)*(9+3). This reigns with truth as 48÷2(9+3) is the same as 48÷2*(9+3). Using the standardized left-to-right notation, the answer can be nothing outside of 288. This left-to-right concept is indicative of “PEMDAS” that we all learned in grade school.
Position 2: The Distributive Property: This stance states that multiplication through juxtaposition, being a commonplace concept, naturally makes a parenthetical implication around grouped numbers. Thus, the equation appears as 48÷[2(9+3)]. This technique is correct as well. However, this makes the answer 2.
If you are bored enough to read [reference 1], you’ll see that the writer has noted multiplication by juxtaposition as being used in algebraic nomenclature dating as far back as the fifteenth century. That being said, the standard left-to-right order of operations predates even that. The real problem is that both techniques are taught in our school systems, depending on the chosen literature. Both forms can be found dating far enough back that there’s a solid argument for either chosen notation. Sadly, there is no current ‘authority’ to standardize which is appropriate. As Doctor Peterson mentions in [reference 2] “When algebraic notation was first being developed, it was common for each writer to begin by explaining his own notation.” That seems to still be true as students are still being taught the distributive property and multiplication by juxtaposition as well as the standardized left-to-right order of operations. The correct answer breaks down to who is asking the question.
As much as you’d like it to be a single definitive answer, alas, it is not. In my opinion, that needs to change. Someone should standardize it. But the real answer here, to end all further questions on this topic, is the ambiguity of the formula makes it faulty to begin with. No self respecting mathematician would have used this notation. Formula ambiguity can destroy the outcome. This should have been written as 48 / (2 * (9+3)) or (48 / 2)*(9+3), thus guaranteeing the desired result.
To summarize, as much as one would like to claim one answer or the other, the sad reality is; being as there is no current standardization in this case, the answer is relative to the inquirer. In the future, if an “ab/cd” –type equation is proposed, you’re better off inquiring as to the proposer’s notation preference.
Source(s):
[Reference 1]:
http://jeff560.tripod.com/operation.html[Reference 2]:
http://mathforum.org/library/drmath/view/57021.html
