Right, PuzzleScot. And all that's left is parentheses. They take precedence over everything else, which is how we force changes in precedence.

Full precedence list:
( ) highest: do the embedded computations first
multiplication & division: next highest, equal precedence
addition and subtraction: next highest, equal precedence

Examples:

4 + 3 + 2 = 9 is calculated as 4 plus 3 (7), then 7 plus 2.
Since plus is the same precedence as itself, we work left to right. But it would have worked the other way too, right? (3 plus 2 (5), then 4 plus 5 (9))

Consider:
10 - 4 - 2 = 4 is calculated as 10 minus 4 (6), then 6 minus 2 (4)
Suppose operators with equal precedence worked right to left. Then we would have:
10 - 4 - 2 = 8 (calculated as 4 minus 2 (2), then 10 minus 2)

So left to right can make a difference. (And this holds if we mix addition and subtraction, and also with multiplication and division.)

Now parentheses. Just do them first no matter what - they trump everything.
1 + 2 x 3 = 7 (multiplication first, standard rule)
(1 + 2) x 3 = 9 (parentheses first, overrides standard rule)

The comments about clarity are exactly right. Use parentheses to show (force) which operations to do first, then neither you nor your reader need worry about the math rules.

Class dismissed - go back to puzzling and thanks for playing math!

I've had a FB discussion about this.
PEMDAS is correct IF AND ONLY IF you write it PE(MD)(AS)
ie, M/D have equal precedence, and A/S have equal precedence.
When you have a tie, work from left to right.

MDAS -> Multiplication than Division etc...
This would mean 1/3*3 = 1/9, and 6-3+1=2, by that reasoning.

No-one on the planet knows mathematics better than the gurus at Wolfram.
http://www.wolframalpha.com/input/?i=1%2F3*3 (D before M!)
http://www.wolframalpha.com/input/?i=6-3%2B1 (S before A!)

Feel free to try anything else that helps you learn that ...MDAS... is wrong!

The answer is 20. PEMDAS: parentheses, exponents, multiply, divide, add, subtract; the order in which you solve it. My son had to learn this last year in algebra.

Still shaking my head that people are being led to think they're stupid by getting the 'wrong' answer.

Ask them what this is: 4 / 2 * 2 = ?

BOMDAS (and wiki) will tell you it's 1. It's actually 4. (and "bad maths")
Not convinced? What about 1/3*3? 1/9? No. You'd write that as 1/3 * 3, which of course is 1.

Anyone remember BOMDAS (or, more correctly, BODMSA) at school?
[Brackets over Mult/Div Add/Sub]
Perform all multiplications first, and the evaluate:
16 + 16 + 4 - 16 = 20.
In reality, this is bad notation. There should never be ambiguity in an expression, but standard operator precedence suffices.

"73% failed to answer this". Shows how 'clever' the author was - That actually means only 27% stuck around long enough to give an answer, but we don't know how many got it right.

It's amazing how much we forget! lol I had a vague idea but I'm glad I looked it up. My husband is an engineer, I'm sure he'd be embarresed that I forgot the rules. But I suppose he'd be proud that I looked it up. lol :)

In mathematics and computer programming, the order of operations (sometimes called operator precedence) is a rule used to clarify unambiguously which procedures should be performed first in a given mathematical expression.

For example, in mathematics and most computer languages multiplication is done before addition; in the expression 2 + 3 × 4, the answer is 14. Brackets, "( and ), { and }, or [ and ]", which have their own rules, may be used to avoid confusion, thus the preceding expression may also be rendered 2 + (3 × 4), but the brackets are unnecessary as multiplication still has precedence without them.

Since the introduction of modern algebraic notation, multiplication has taken precedence over addition.[1] Thus 3 + 4 × 5 = 4 × 5 + 3 = 23. When exponents were first introduced in the 16th and 17th centuries, exponents took precedence over both addition and multiplication and could be placed only as a superscript to the right of their base. Thus 3 + 52 = 28 and 3 × 52 = 75. To change the order of operations, originally a vinculum (an overline or underline) was used. Today, parentheses or brackets are used to explicitly denote precedence by grouping parts of an expression that should be evaluated first. Thus, to force addition to precede multiplication, we write (2 + 3) × 4 = 20, and to force addition to precede exponentiation, we write (3 + 5)2 = 64. Also BOMDAS stands for Brackets Of Multiplication Division Addition Subtraction.

The order of operations rule is to multiply and divide first and then do addition and subtraction, if I am remembering the rule correctly. However I do not know if there should be parentheses in place for the rule to hold. I remember having a hard time with this rule when I was in school...

It depends on how you read it! Is it (4 x 4) + (4 x 4) + (4 - 4) x 4? Or (4 x 4) + 4, then multiply that by 4, and so on? If the latter, I get 320 as well. If the former, I get 128.....

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