Order of Operations - Need Help! Huge Debate!

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sanchito9999

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Apr 7, 2011
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I am having a discussion with others as to what the answer to this is:
48÷2(9+3) = ?

I am finding that one class of people will first do:
48 / 2(12)
48 / 24
=2

and the other class:
48 / 2( 12)
24(12)
=288

Every website and calculator seems to agree with the latter method, yielding 288. However, certain individuals are determined that the 2 MUST be multiplied through the (9+3) before moving on in the order of operations. I see it as 48 / 2 * (12) in which you would divide first and then multiply since those two functions go from left to right with equal precedence. Can someone please clarify the actual law of math behind parentheses? Is the 2 in the 2*(9+3) actually considered part of the parentheses when doing order of operations? Distribution is an act of multiplication and should come AFTER doing the first division in my opinion.

Thank you.
 
#1 - Juxtaposition implying multiplication is a bad idea. Always use some sort of operator. It won't necessarily change anything, but it will help.

48 / 2 * (9+3)

#2 - Parentheses rule.

48 / 2 * 12

#3 - Multiplication and Division are the SAME precedence. Given confusion, convention says work left-to-right

(48/2) * 12 = 24 * 12

#4 - I just corrected a large issue in statistics. I saw books and popwer points and hand-writen papers, ALL stating 1/n, 1/2n, 1/3n, etc. where they MEANT 1/(n), 1/(2n), 1/(3n), etc. Even authors and editors can get it wrong. However, given sufficient contextual clues, it can be decoded.

#5 - Be very careful un what environment you are work. My primary programming language decodes right-to left. This would give, besides the syntax error for no operator...

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

In order to get it NOT to do that, add parentheses

(48 / 2)*(9+3)

So, you see, everyone can be right, but without special circumstances, let's go with 288. Of course, if there are special circumstances you are not sharing with us, you had better not quote us out of context. :)
 
Well a lot of the confusion has arisen because there was no context behind the question. Some people have assumed that everything after the "/" sign was intened to be a denominator, while it seems that is generally not a safe conclusion to draw when you're strictly following the order of operations.

The primary argument as to why it was "2" is that the 2 next to the (9+3) was somehow "attached" to the (9+3). My argument is that the "48/" is equally attached to the (9+3) since there is no +, -, or () to separate it, and that you cannot distribute the 2 through the (9+3) without distributing it as (48/2)*(9+3).

Your answer makes sense to me, and I can see why it may be confusing. But I think if you strictly follow what the order of operations dictate, then 288 would be correct.
 
sanchito9999 said:
I am having a discussion with others as to what the answer to this is:
48÷2(9+3) = ?

I am finding that one class of people will first do:
48 / 2(12)
48 / 24
=2

and the other class:
48 / 2( 12)
24(12)
=288

Every website and calculator seems to agree with the latter method, yielding 288. However, certain individuals are determined that the 2 MUST be multiplied through the (9+3) before moving on in the order of operations. I see it as 48 / 2 * (12) in which you would divide first and then multiply since those two functions go from left to right with equal precedence. Can someone please clarify the actual law of math behind parentheses? Is the 2 in the 2*(9+3) actually considered part of the parentheses when doing order of operations? Distribution is an act of multiplication and should come AFTER doing the first division in my opinion.

Thank you.
Click to expand...

Before calculators, back in the paleolithic days when people wore untanned hides, ate their food raw, and communicated by drums and bullroarers, people would have just said that the expression is sloppy presentation. Back then, you avoided ambiguity by using parentheses and, if necessary, nesting parentheses. There is no theoretical basis for the sequence of operations; left-to-right is used by many systems of writing natural languages, but the universe does not require it. Sequence of operations is a convention for permitting communication. When calculators were first introduced (in the neolithic about the same time as dogs were domesticated), some used Polish notation and some used reverse Polish notation.
 
sanchito9999 said:
strictly following
you cannot
if you strictly follow
Click to expand...

I obviously agree with your conclusion but personally, I prefer to write so that people can and will understand my intent, rather than demanding that they understand what I write. If it takes a little extra effort, because I know someone in my audience wil be confused, the extra effort will be worth it.
 
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