# Order of Operations

Discussion in 'Math' started by thatoneguy, Apr 11, 2011.

1. ### thatoneguy Thread Starter AAC Fanatic!

Feb 19, 2009
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This has been around on the net, I'm interested in how Math people interpret this:

48/2(9+3) = ?

2. ### Georacer Moderator

Nov 25, 2009
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This is a bit of a violation, since the parenthesis doesn't necessarily imply multiplication. It should be written explicitly.

Other than that, I think it is a global convention that operations are done from left to right, when on the same level of priority, so the right answer is right.

If you wanted the left result you should have written 48/(2*(9+3))

3. ### hgmjr Moderator

Jan 28, 2005
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I was taught that calculations are carried out in the order of (My Dear Aunt Sally) Multiplication followed by Division followed by Addition followed by subtraction. Parentheses are to be resolved from the inner-most to the outer-most. Based on this convention, I find the answer to be 2 not 288.

hgmjr

4. ### magnet18 Senior Member

Dec 22, 2010
1,232
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I was taught Please Excuse My Dear Aunt Sally, parentheses, exponents, multiplication&division, addition&subtraction, left to right, so I arrive at 288.

I actually ran into this issue in chemistry this morning before school when comparing my TI89's answer with my teachers TI84's answer.

5. ### Georacer Moderator

Nov 25, 2009
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I took it for granted, but it seems I had to add that multiplication/division and addition/subtraction have the same priority. At least this is what I was taught.

Jul 7, 2009
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It's ambiguous unless you define the precedence of the operations. For example, the usual programming languages that use infix notation will interpret it (48/2)*(9+3). Results will be different for prefix or postfix notations. Personally, I'd interpret it like the infix programming languages. However, there's one case where I am occasionally sloppy and write units like W/m*K and I mean it as W/(m*K), not (W/m)*K, especially when I'm working with the GNU units program where such notation is not ambiguous.

Frankly, the best practice is to use parentheses or define your notation so there is no ambiguity -- then everyone gets the same results.

7. ### yegonjapheth New Member

Apr 4, 2011
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Try B.O.D.M.A.S, it's easier that way! Brackets, Of, Division, Multiplication, Addition, Subtraction.

8. ### thatoneguy Thread Starter AAC Fanatic!

Feb 19, 2009
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I am used to RPN calculators, where you start from the parenthesis and work outwards. I guess this supports "implied multiplication" at a higher precedence then explicit multiplication, but it is how I was learned.

The formula I posted has flaws, near as I can tell, leaving the answer ambiguous. At the same time, there are thousands of people arguing To The Pain that their interpretation is correct.

9. ### Wendy Moderator

Mar 24, 2008
21,012
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I have the first calculator. I've long gotten in the habit of using paranthesis.

10. ### thatoneguy Thread Starter AAC Fanatic!

Feb 19, 2009
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It was driven into me in college, regarding programming, "If in doubt, add another set of parenthesis".

It seems that pre-1990 Ti and SVPAM Casio result in an answer of 2
Post 1990 Ti and HP 48 generation that give truly visual equations give the answer as 288.

288 is the answer, and this is an argument that's been going on for 20 years, it seems not everybody has gotten the memo.

11. ### victorhugo289 Member

Aug 24, 2010
49
3
I thought it was supposed to be:

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

The way it is written 48/2(9+3) is indeed 3 factors:

48 * 2^-1 * 12 = 288

Multiplication is a continued operation meaning all calculators will solve 288.

12. ### magnet18 Senior Member

Dec 22, 2010
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Georacer likes this.
13. ### DerStrom8 Well-Known Member

Feb 20, 2011
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I was taught with PEMDAS (or Please Excuse My Dear Aunt Sally--parentheses, Exponents, Multiplication, division, addition, and subtraction). The way I would do this problem is this:
48/2(9+3)
1: Paretheses--48/2(12)
2: Multiplication or Division left-to-right--24(12)
3: Multiply: 24x12=288

14. ### Wendy Moderator

Mar 24, 2008
21,012
2,744
If Multiplication is first, you did it out of sequence. My Dear Aunt Sadie is the one I learned. I interpret the 12 as under the division symbol, especially since the problem started with the ÷ symbol.

1. 48÷2(9+3)
2. 48÷2(12)
3. 48÷24
4. 2

Consistency is important.

6÷2(1+2)
6÷2(3)
6÷6
1

I also disagree with post #12.

Contrary to what I keep reading, from 1st grade to college I was taught they do not have the same precedence, which is why the MDAS exists. It is to prevent this exact ambiguity. As illustrated by all the examples, an order of precedence must exist.

Last edited: May 2, 2011
15. ### BillO Distinguished Member

Nov 24, 2008
993
137
The ambiguity seems to stem from the 2(9+3) term. It should read 2*(9+3) or (2(9+3)), then there would be no issue and the desired result (either 2 or 288) would be obvious.

I think most of us would interpret 48/2x as:

48
---
2x

So if we write 48/2x, where x=(9+3), we would come up with the answer 2, not 288.

For me, it has always been that implied multiplication is done before explicit multiplication or division. For this reason I would say the answer is 2.

I too disagree with post 12.

16. ### Wendy Moderator

Mar 24, 2008
21,012
2,744
I like the last argument best.

So does anyone dispute that 48÷2X is not

48
2X

?

17. ### Georacer Moderator

Nov 25, 2009
5,151
1,266
I agree with post #12.

Officially, however, the expression 48/2X is wrong, because there is no operator between 2 and X.
If I were a calculator I would display an error on my screen.

If you re-write it as 48/2*X, then I would interpret it as $\frac{48}2\cdot X$

I 'm quite sure that was the case when we examined things a bit deeper in highschool, but in my EE university school I 'm in this is definitely the case. Mostly because programming languages (those that I know of, at least) have multiplication and division in the exact same priority.

18. ### DerStrom8 Well-Known Member

Feb 20, 2011
2,428
1,333
Multiplication and Division are of equal importance--whichever comes first is done first. In that case, as was mentioned in another post, 48/2X would be READ as 48/(2X), but would actually be calculated as 24X.
The same goes for addition and subtraction--equal in importance, so just go left to right.
Der Strom

19. ### DerStrom8 Well-Known Member

Feb 20, 2011
2,428
1,333
If this was true, the problem would have to have been 48/2(1/x) in order to get the x in the denominator.

20. ### BillO Distinguished Member

Nov 24, 2008
993
137
I'm quite amazed at the last two responses. Things must have changed drastically in the way mathematics is being taught in school these days.

Do you not do functions of variables anymore?

Back in ancient times things like this could be written:

y = 3/2x + c

Wherein everyone in the class would see that 1st term as 3 over 2x

If we wanted it differently we would just write:

y = 3x/2 + c

But I guess that no longer makes sense. Oh well.

Like I said, we all knew and understood that implied multiplication was done first. Always. Made life easier when we were all on the same page.