If you follow the order of operations, why are you multiplying 2 by twelve before dividing 48 by two? Multiplication doesn't take precedence over division, they're performed from left to right: [tex]\frac{48}{2}(9+3)[/tex]288? Following PEMDAS, I get 2:
48÷2*(9+3) = 48÷2*(12) = 48÷24 = 2
The reason why some people are getting 288 is that they're forgetting that there's an invisible multiply sign in the expression.
Honestly though, if I ever saw this I would apply a facepalm. It's not good notation and like you're seeing, it's ambiguous without proper use of parentheses and will confuse people. It's better notation to say [tex]\frac{48}{2(9+3)}[/tex]
That's simply the order in which they're stated. Multiplication and division are equal, so are addition and subtraction. I could just as easily say that PEDMSA represents the order of operations. One way to look at it is to say that division is simply multiplication of the reciprocal, and subtraction to be addition of the opposite.PEMDAS = Parenthesis -> exponents > multiply -> divide -> add -> subtract
I think it comes from viewing division as multiplication by the reciprocal. I wasn't sure, either.You go left to right? I didn't know that.
Even I'm getting confused. :tongue: It's just bad notation.And this is why you never see ÷ used for any serious purpose...the notation is just awful.
MATLAB and Python say 288.
So does C#. I'm positive that C and C++ would report the same result.MATLAB and Python say 288.
They do. So do Wolfram Alpha, Google, PHP, Perl, and Ruby.So does C#. I'm positive that C and C++ would report the same result.
And this is why you never see ÷ used for any serious purpose...the notation is just awful.
No, the notation is fine - 48/2*12 really isn't ambiguous if you understand that arithmetic operators at the same precedence level are evaluated left to right. It seems clear from this thread that not everyone is taught this fine point.Even I'm getting confused. :tongue: It's just bad notation.
I agree it's not ambiguous, but I just hate the ÷ symbol.No, the notation is fine - 48/2*12 really isn't ambiguous if you understand that arithmetic operators at the same precedence level are evaluated left to right. It seems clear from this thread that not everyone is taught this fine point.
But if you follow the rules, you'll always get the correct answer - which I've just ran in python (as above have) to get 288.i will say that the question problem because it haven't make that clear for the
Your entire argument is based on the point that multiplication by juxtaposition takes precedence over multiplication by operator, despite the fact that they're just two ways of expressing one operation. It's like saying that ÷ takes precedence over / or vice versa.Looking at it like this though:
48 ÷ x(9 + 3) = 288
48 ÷ 9x + 3x = 288
48/12x = 288
4/x = 288
4 = 288x
4/288 = x
1/72 = x
Would suggest that 288 is wrong, and that 2 is correct.
Also, with the order of operations aren't we using the distributive property which states as an example:
Simplify 16 ÷ 2[8 – 3(4 – 2)] + 1.
16 ÷ 2[8 – 3(4 – 2)] + 1
= 16 ÷ 2[8 – 3(2)] + 1
= 16 ÷ 2[8 – 6] + 1
= 16 ÷ 2[2] + 1 (**)
= 16 ÷ 4 + 1
= 4 + 1
= 5
Then, looking at what I have bolded below:
The confusing part in the above calculation is how "16 divided by 2[2] + 1" (in the line marked with the double-star) becomes "16 divided by 4 + 1", instead of "8 times by 2 + 1". That's because, even though multiplication and division are at the same level (so the left-to-right rule should apply), parentheses outrank division, so the first 2 goes with the [2], rather than with the "16 divided by". That is, multiplication that is indicated by placement against parentheses (or brackets, etc) is "stronger" than "regular" multiplication. Typesetting the entire problem in a graphing calculator verifies this hierarchy.
Note that different software will process this differently; even different models of Texas Instruments graphing calculators will process this differently. In cases of ambiguity, be very careful of your parentheses, and make your meaning clear. The general consensus among math people is that "multiplication by juxtaposition" (that is, multiplying by just putting things next to each other, rather than using the "×" sign) indicates that the juxtaposed values must be multiplied together before processing other operations. But not all software is programmed this way, and sometimes teachers view things differently. If in doubt, ask!
This all seems to point toward 2 being the correct answer.
You are tacitly assuming that 48 ÷ x(9 + 3) means 48 ÷ [x(12)]. I maintain that it means (48 ÷ x)* 12. Writing 12 as 9 + 3 needlessly complicates things.Looking at it like this though:
48 ÷ x(9 + 3) = 288
48 ÷ 9x + 3x = 288
48/12x = 288
4/x = 288
4 = 288x
4/288 = x
1/72 = x
Would suggest that 288 is wrong, and that 2 is correct.
Also, with the order of operations aren't we using the distributive property which states as an example:
Simplify 16 ÷ 2[8 – 3(4 – 2)] + 1.
16 ÷ 2[8 – 3(4 – 2)] + 1
= 16 ÷ 2[8 – 3(2)] + 1
= 16 ÷ 2[8 – 6] + 1
= 16 ÷ 2[2] + 1 (**)
= 16 ÷ 4 + 1
= 4 + 1
= 5
Then, looking at what I have bolded below:
The confusing part in the above calculation is how "16 divided by 2[2] + 1" (in the line marked with the double-star) becomes "16 divided by 4 + 1", instead of "8 times by 2 + 1". That's because, even though multiplication and division are at the same level (so the left-to-right rule should apply), parentheses outrank division, so the first 2 goes with the [2], rather than with the "16 divided by". That is, multiplication that is indicated by placement against parentheses (or brackets, etc) is "stronger" than "regular" multiplication. Typesetting the entire problem in a graphing calculator verifies this hierarchy.
Note that different software will process this differently; even different models of Texas Instruments graphing calculators will process this differently. In cases of ambiguity, be very careful of your parentheses, and make your meaning clear. The general consensus among math people is that "multiplication by juxtaposition" (that is, multiplying by just putting things next to each other, rather than using the "×" sign) indicates that the juxtaposed values must be multiplied together before processing other operations. But not all software is programmed this way, and sometimes teachers view things differently. If in doubt, ask!
This all seems to point toward 2 being the correct answer.
I agree, but in this particular case, doesn't juxtaposition take precedence?Your entire argument is based on the point that multiplication by juxtaposition takes precedence over multiplication by operator, despite the fact that they're just two ways of expressing one operation. It's like saying that ÷ takes precedence over / or vice versa.