May 8, 2011

Ending the 48÷2(9+3) debate.

An issue of mathematical grammar circulating the internet lately has been how to read a mathematical expression like
LaTeX Code: 48 \\div 2 (9 + 3)
that involves a combination of division and implied multiplication.

The standard way to read arithmetic expression (i.e. order of operations) involves dealing with parentheses first, then you do all division and multiplication operations from left to right, then all addition and subtraction operations from left to right.


http://en.wikipedia.org/wiki/Order_of_operations

So, this is computed as:

48÷2(12)
=24(12)
=288

It doesn't matter whether the multiplication sign is shown, say:

48÷2 x (9+3)

or whether they are in different forms: (representing divisions by a slash):


48/2(9+3)

they all mean the same thing.

This problem arises due to the ambiguous nature of the equation, and it can represent different things for people who follow different conventions. So, the next time you write an equation, try to avoid writing like the above, write as unambiguous as possible, say:

(48/2)(9+3)


Note: Someone suggested that the (9+3) integer be represented by a variable y i.e. 48÷2(y), where y=9+3. Now, it is understood that the order of operations proceeds like usual, and:


48÷2(y) , y = (9+3)
=(48/2)y   , y = 12
=24y
=288.

Perhaps this wolfram page will clear any remaining doubts.
http://www.wolframalpha.com/input/?i=48%C3%B72%28x%29%2C+x%3D+9%2B3



edit: 48÷2y =/= 48÷2(y)
In reference to the above, one cannot simply remove the parenthesis, as when y=9+3 is applied to the equation, the only solution must be 48÷2(y) as you cannot remove the brackets through substitution of an unknown.


In other words, you cannot remove the brackets by substituting y = 9+3 into ...(9+3). Substitution will only bring about the general form of 48÷2(y), nothing else.

57 comments:

  1. long time ago since i had math in school but ok i belive you :D

    ReplyDelete
  2. Unfortunately people will still disagree after seeing this.

    ReplyDelete
  3. Actually the answer is 1 :)

    ReplyDelete
  4. The key here is that you do the operations left to right. Some will say that multiplication is first in order of operation because they have learned a mnemonic for order of operations that puts multiplication before division PEMDAS (parenthesis, exponents, multiplication, division, addition, subtraction). Unfortunately, the math does not workout in this particular case.

    ReplyDelete
    Replies
    1. AnonymousMay 07, 2012

      PEMDAS works fine as long as you remember that multiplication and divison, along with addition and subtraction are paired together (meaning just work left to right, not worrying about which one to do first). Unfortunately, most forget this

      Delete
  5. Nice, usually don't much care for learning about math, but i enjoyed that.

    ReplyDelete
  6. Thats a hard one, im that stupid that i dont know =(

    ReplyDelete
  7. Nice man now my day is complete now I'm off to bed lol

    ReplyDelete
  8. Definitely 288. I asked my teacher too.

    ReplyDelete
  9. For those that need an explanation, I am a little frustrated. But, great post!

    ReplyDelete
  10. AnonymousMay 18, 2011

    @metaphysicalfarms

    It is not PEMDAS but PEDMAS, or BEDMAS, or BODMAS -> all simply different ways of saying the same thing, but the mnemonic always puts Division BEFORE Multiplication. So, the math does indeed work out.

    P=Parenthesis / B=Brackets
    E=Exponents / O=Orders
    D=Division
    M=Multiplication
    A=Addition
    S=Subtraction.

    So the original post as stated works here.

    ReplyDelete
    Replies
    1. Divison before multiplication isn't a rule. It's
      Brackets
      Exponents
      Divison and Multiplication (left to right)
      Addition and Subtraction (left to right)

      Delete
  11. http://www.wolframalpha.com/input/?i=48%2F2%289%2B3%29
    Just believing Wolfram Alpha in such things.

    ReplyDelete
    Replies
    1. AnonymousJuly 24, 2012

      http://www.wolframalpha.com/input/?i=48%3A2%289%2B3%29

      At least copy it properly.

      Delete
  12. Fuck, finally someone who gets this. I'll share this article next time someone is doing it wrong

    ReplyDelete
  13. AnonymousJune 25, 2011

    Sorry, but the answer is 2 and it does not have anything to do with order of operations. The reason is Distributive Properties. The 2 must be distributed to completely resolve the () before moving on. 2*(12) does not equal 2(12); and that is the major issue.

    ReplyDelete
    Replies
    1. AnonymousJune 13, 2012

      This is absolutly correct. 2(9+3)=2(12) Notice there is still a parenthesis. To remove the parenthesis you multiply 2 into it resulting in 24. This follows PEMDAS perfectly. Parenthesis, exponent, multi, div,add, sub. Problem people have is that they are not completing the parenthesis part of the order

      Delete
    2. AnonymousJuly 25, 2012

      http://www.wolframalpha.com/input/?i=2%2812%29
      http://www.wolframalpha.com/input/?i=2*%2812%29
      http://www.wolframalpha.com/input/?i=2%289%2B3%29
      http://www.wolframalpha.com/input/?i=2*%289%2B3%29

      When you work through the parentheses 2(9+3), no matter which way you do it, or if it is indicated with a multiplication sign or not, and the numbers involved end up the same after, they are equal essentially equal operations. Easier to see by working it out by hand.

      Delete
    3. But if you used the Distributive Law properly, the answer would be 8.67!

      48 / 2(9 + 3)
      48 / 18 + 6
      2.67 + 6
      8.67

      Delete
  14. @Anonymous. Unfortunately the definition of implied multiplication and the Distributive Law of algebra itself disagrees with you.

    Read more at:
    http://en.wikipedia.org/wiki/Distributivity

    Also, the major issue here is the division symbol, not the 2*(12). 2(12) is clearly 24, but division takes place before the implied multiplication.

    ReplyDelete
  15. @Corridor

    Division doesn't take place before multiplication. They are the same, and take place left to right. If you have 2 * 4 / 5* 6 you don't divide 4 by 5 and then multiply across, you multiply 2 by 4, then divide by 5, then multiply again by 6.

    ReplyDelete
  16. @Anonymous

    Yes, division is of the same order as multiplication. They do take place from left to right. I never did say otherwise.

    2(12) equals 2*(12). The law of distribution does not take place before the division.

    ReplyDelete
  17. Wabun:
    @ Note: Someone suggested that the (9+3) integer be represented by a variable y i.e. 48÷2(y), where y=9+3.

    if you mean me by this: i was referring to calculator syntax and that calculators treat two factors written next to each other without the multiplication sign as if they were set into brackets which is obvious if you write it like 2y (everyone knows the calculator treats this as (2*y) ) but not so obvious if you write 2(y+x) (which in calculator syntax is (2*(y+x)) ).

    I simply pointed this out because it is the reason why some ppl got 2 as result when typing it in calculators.

    ReplyDelete
    Replies
    1. AnonymousJuly 26, 2012

      Don't always believe your calculator. It won't always give you the right answer.

      Delete
  18. sorry for the anonymous posters saying it's 2... they are from greece.

    ReplyDelete
    Replies
    1. AnonymousJune 13, 2012

      step by step:
      48/2(9+3)
      Parenthesis:
      48/2(12)
      48/24 (it isnt until this step, that the parenthesis portion of PEMDAS is satisfied. Once 2 is multiplied into the parenthesis, the parenthesis are then removed. Now we are free to move on to the next steps of PEDMAS.)
      Exponents
      There are none
      Multiplication/Division
      There is no multiplication, but there is division.
      48/24= 2
      Addition/Subtraction
      None

      This follows all the orders of operation perfectly...Don't forget to remove the parenthesis before moving on to next step.

      Delete
    2. AnonymousJuly 23, 2012

      This is wrong because you are multiplying 2*12 before you are dividing 48/2...

      Delete
    3. AnonymousJuly 24, 2012

      The step from 48/2(12) to 48/24 is wrong wrong wrong wrong wrong.

      Delete
    4. AnonymousJuly 25, 2012

      Whatever gives people the idea that the 2 is part of the parenthesis? What's the answer here then - 4(3-1)^2?

      Delete
    5. AnonymousJuly 25, 2012

      In mathematics, a multiplication can be represented by one value being next to another in parenthesis. For example, 2*3 can be written also as 2(3). Each give the value of 6. The first step is mathematically correct, simplifying the quantity of 9 plus 3. However, you fail to realize that 2(12) is multiplication. If you had realized that 2(12) is multiplication, you would not have made the error of stating the answer is 2. That is assuming that you know to complete a math problem from left to right.

      Delete
    6. AnonymousJuly 30, 2012

      This is correct, and the way he wrote the equation on wolfram in this article is wrong.
      If you are to change anything into X it would be the complete parenthesis resulting in 48/2x with x being 9+3 or (9+3) if u prefer (means the same thing) and what would wolfram say about that?
      http://www.wolframalpha.com/input/?i=48%C3%B72x%2C+x%3D+9%2B3
      Simply, the answer is 2.

      Delete
    7. When you removed the parentheses, you changed the problem.

      http://www.wolframalpha.com/input/?i=48%C3%B72%28x%29%2C+x%3D+9%2B3

      Delete
    8. When you removed the parentheses, you changed the problem.

      http://www.wolframalpha.com/input/?i=48%C3%B72%28x%29%2C+x%3D+9%2B3

      Delete
  19. AnonymousJune 06, 2012

    The answer is 2....even if you work it out long ways...."old math"....you conclude 48/( 2*9 +2*3 ).... Equals 2....

    ReplyDelete
    Replies
    1. AnonymousJuly 24, 2012

      Wrong wrong wrong... I wouldn't fly in one of your plane.

      Delete
    2. AnonymousJuly 24, 2012

      correct its 2, even if u do it how theyre describing. its 48/2 times 1/12 which is still 2. case closed everyone stfu and get a life cuz ur wrong and u fail at being nerds lol

      Delete
    3. AnonymousJuly 25, 2012

      You've added parantheses in there that werent there before. There is no bracket after the division sign. You've changed the math to fit your delusion. This is the correct way in terms of algebra:

      If you had:

      48/2*(X+Y)

      it would be

      24*(X+Y)

      Now if X is 9 and Y is 3.

      288.

      Try typing the equation (correctly without adding brackets) into any programming language and see if you can get 2.

      Delete
  20. AnonymousJuly 23, 2012

    The is 2, now stop fking arguing and move on with your life.

    ReplyDelete
  21. AnonymousJuly 24, 2012

    48/2(9+3) ohk so now BEDMAS or BODMAS tells us that Brackets are first right? and because there is no * next to the 2 you * whats in the brackets so 48/(2*9+2*3) now when you done with the brackets you got 24 sp NOW only you work left to right 48/24=2 thank you very much

    ReplyDelete
  22. AnonymousJuly 24, 2012

    Learn algebra you retard! you never learned this 2(x+y)= (2x+2y) so in this case it would be 2(9+3)= (2*9+2*3) and you have to solve the braquest first! damm! I cant believe you even had the balls to make a blog about it, you should feel ashamed sir

    ReplyDelete
  23. http://www.youtube.com/watch?v=a-e8fzqv3CE#t=153

    This is pretty good in explaining it.

    ReplyDelete
  24. AnonymousJuly 25, 2012

    The original equation is not ambiguous unless one doesn't know that a(b+c) is a mathematical shorthand for a*(b+c), along with the other precedence rules. If we ignore those precedence rules we'll have numerous equations that have different answers depending on the whim of whomever is doing the calculation. That's why mathematics has a convention regarding the order of precedence.

    ReplyDelete
  25. AnonymousJuly 25, 2012

    In order to frustrate the intellectual progress of the human race, allow me to propose a third solution:

    Between two possible solutions, the more elegant is the correct solution. Hence, by the principle of mathematical elegance, the answer is 2.

    ReplyDelete
  26. AnonymousJuly 25, 2012

    To all those who say it is 2, you are incorrect. The math problem 48/2(9+3) is read as "48 divided by 2 multiplied by the quantity of 9 plus 3." You would simplify this problem by reading it from left to right. This means you must divide 48 by 2 first before multiplying any value by the quantity of 9 plus 3. This means it would be as such

    48/2(9+3)
    24(9+3)
    24(12)
    288

    To all of those who say I should add 9 and 3 first, I am aware of order of operations but this was simply another way to explain the math problem.

    Another way to explain it would be to change everything into improper fractions. 48/1 divided by 2/1 multiplied by the quantity of 9/1 plus 3/1. As you may know from the 5th grade, you may divide fractions by multiplying by the reciprocal of the divisor. This will make it 48/1 multiplied by 1/2 multiplied by the the quantity of 9/1 plus 3/1. If you use this method, as they are the same math problem, it will be done as such:

    48/2(9+3)
    (48/1)(1/2)(9+3)
    24(12)
    288

    The common error is that many people multiply the divisor 2 and the quantity of 9 plus 3 together first. The problem is that the 2 serves as a denominator and to put the quantity of 9 plus 3 in the denominator too would make your entire work mathematically incorrect.

    As a rule of thumb, work your orders of operations from left to right.

    ReplyDelete
  27. AnonymousJuly 26, 2012

    I think the answer is 2.
    When you type 48/2(9+3), the answer is 288. However, this is ONLY because they spread the equation further to (48/2)*(9+3).
    The answer would be 288 if and ONLY IF the parentheses came before 48 and after 2.
    If there is no parentheses, the equation ACTUALLY looks like this: 48/(2(9+3)). The key problem of this equation is not the division sign, but the parentheses.
    This also goes for the variable concept. Changing (9+3) to y actually makes the equation 48/2y, making the answer 2.
    I don't know if I'm correct, but this is what I think. I think that the placing of the parentheses is so important and is the problem to this controversy.

    ReplyDelete
  28. Why do you say, "...the equation ACTUALLY looks like this: 48/(2(9+3))." Why do you add the extra set of parentheses? Is it because that's the only way to make the answer come out as 2? Leave the extra parens out, evaluate the problem as it's written, and the answer is 288. 48 divided by 2 (24) times 9+3 (12). 24x12 = 288.

    ReplyDelete
    Replies
    1. Just as 2x or 2y represents a number 2() also represents a number

      Delete
  29. OK guys... here's my demonstration....

    48:2(9+3) =
    = 4(9+3):2(9+3) = [pongo (9+3)=x]
    = 4x : 2x =
    = 2

    ReplyDelete
  30. http://www.studygs.net/pemdas/pemdas2.htm

    ReplyDelete
  31. The reason the answer can't be 2 is this

    2(9+3) = 24
    24/2 = 12
    So (2(9+3)/2 = 12

    If we set the equation equal to 2

    48/2(9+3) = 2

    If this is true we can divide both sides by 2 and the left and right sides will still be equal

    for those saying you distribute the two first consider this

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

    If you consider 2 as the coefficient of (9+3)

    you would end up with this

    24/12 =1 this is not true, the coefficient of (9+3) was actually 48/2 or 24

    the only to get 2 is if you thought the / was a fraction bar

    ReplyDelete
    Replies
    1. The / is a fraction bar. 50/3 is the same as 50 over 3.

      Delete
  32. 2y = y + y
    2y equal the PRODUCT of 2 and y, ie, the product of 2 * y.
    2y = (2y) = (2(y)) = y + y = (y+y)

    48/2y = 24/y
    What is 1a/1a ? It is 1.

    The algebraic notation invoke algebraic thinking:
    2(9+3) = [(9+3) + (9+3)]

    The 2 is a factor of the 2 terms inside the parentheses.
    research "factorizing" or "factorization".

    Also look up how a group of coefficients are communtative.
    abc/xyz is (abc)/(xyz)
    Google around. There are many math and algebra sites which show how this is done and why, and there are NONE that say "Insert a multiplication sign between all factors, then solve"
    Have you ever been taught this or seen it is a math text??

    http://cstl.syr.edu/fipse/algebra/unit2/parenth.htm

    This shows inline division. There are OTHER SITES TOO!! See for youself.

    What is cos2x ? Trust me, it isn't cos2 * x, or xcos2... 2x is (2x) and cos2x is cos(2x)
    What is 2/3² ? It is not (2/3)².
    If you want to use fractions or expressions, you need to contain them in parentheses.
    48/2(9+3) is not (48/2)(9+3).
    2 is clearly the coefficient of (9+3), as coefficients are directly adjecacent to the parentheses without other operators. If you want to factor our (48/2) from 216 + 72, you get: 216 + 72 = (48/2)9 + (48/2)3 = (48/2)(9+3)

    ReplyDelete
  33. I have been trying to tell people this for YEARS

    ReplyDelete
  34. It all depends on which order of operations you use.

    If using Division before Multiplication:
    48/2(9+3)
    48/2(12)
    24(12)
    288

    However, if you're using Multiplication first:
    48/2(9+3)
    48/2(12)
    48/(24)
    2

    It all depends on the interpretation and what mathematical state of operations you use. For example: My calculator says 2, yet Wolfram says 288. It all depends on the order.

    ReplyDelete

Yes, Cerebrate?