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Discussion in 'Mathematics' started by Hawkwind66, Jul 8, 2011.

  1. Hi all,
    Need some help, came across this problem from my brother-in-law who is studying to be a nurse and needs to brush up his maths. The book he is using had this problem:
    90 ÷ 9(8-3x2)
    He got an answer of 20, which I agreed with, but the book gave the answer as 5
    I used a scientific calculator to check and got these results:
    90 ÷ 9(8-3x2) = 5, but if you insert a multiply sign as follows:
    90 ÷ 9 x (8-3x2) = 20
  2. 90 ÷ 9(8-3x2) means 90 divided by the answer to 9(8-3x2) whereas 90 ÷ 9 x (8-3x2) means 90 ÷ 9 and then multiply the answer to that by the answer to (8-3x2).
    Expressed as quotients, 90 ÷ 9(8-3x2) would have 90 on the top and everything else on the bottom but 90 ÷ 9 x (8-3x2) would have 90 x (8-3x2) on the top and only 9 on the bottom (IYSWIM).
  3. PaulDG

    PaulDG Occasional commenter

    Because BIDMAS is a convention that has developed over the centuries and still has some remaining ambiguities that are dealt with by "experience" and "writing it another way".
    With the multiplication sign inserted, the expression becomes ambiguous - does it mean "divide by nine then multiply by the contents of the brackets" or does it mean "do the multipy first then divide 90 by the result"?
    Without the multiply sign, the expression isn't ambiguous as is clear if you write it as algebra.
    Just as it would also have been unambiguous if it had been written as a fraction.
  4. Please not another BIDMAS thread. The previous ones went on for ever.
  5. afterdark

    afterdark Occasional commenter

    This is a misprint. Or just simply wrong.
    Juxtaposition with numbers along means place value.
    Juxtaposition of number before a letter or bracketed algebraic expression means multiplication.
    BODMAS itself is an attempt to give a rule of priority, it flawed due to the multiple ambiguities of what we mean in different circumstances, for instance juxtaposition.
    BODMAS fails at the simple level of 30 - 12 + 5
    I say this because when we write 30 - 12 + 5 we mean:
    thirty subtract twelve then add five = (30-12) + 5 = 18 + 5 = 23
    Blindly following BODMAS would have us add first then subtract
    30 - 17 = 13
    as to your question
    90 ÷ 9 x (8-3x2) = 20
    follows BODMAS strictly
    90 ÷ 9(8-3x2) = 5
    here you are applying 'common sense' to say to say 9(8-3x2) is completed first
    to give 18
    so 90 ÷ 9(8-3x2) becomes 90 ÷ 18 =5
    take a look at this link
    notice that different makes of calculator have different algorithms for calculations
    Personally my stance on this matter is twofold
    1 that 90 ÷ 9(8-3x2) is not a valid expression [to qualify as a calculation]
    2 It is unwise to blindly follow rules without thought for the conditions under which they apply.
    Good luck helping you bro.
  6. Maths_Mike

    Maths_Mike New commenter

    BODMAS is a memory tool - thats it.
    It does not mean that division is done before mulitiplication (or addition before subtraction) - these operations are of equal precedence and should be done from left to right.
    Brackets first
    Other things second
    then division and multiplication
    then addition or subtraction
    So BoDMAS does not fail at the simple level of 30-12+5 If it is taught and applied correctly.
    However as I posted on the BODMAS thread all of these contrived examples are totally pointless.
    Put some context on the problem and it will be clear what order the operations should be done in.
  7. afterdark

    afterdark Occasional commenter

    What do <u>you</u> mean by this?

  8. Isn't part of the problem due to the fact that + and - are used as both operations and to indicate 'directed' numbers. This is, presumably, why some texts now have negative numbers with the negative sign placed at the upper left hand side of the number. Hence -12, in the given example, can be thought of as +(-12) or - (+12). (No wonder our students get confused!)
  9. Maths_Mike

    Maths_Mike New commenter

    I mean it doesnt fail.
    Its just how it is sometimes taught that can be a problem.
  10. I understand that they use PEMDAS in the USA
    so I suppose that Americans who don't understand how to apply the MD and AS parts will insist that multiplication takes precedent over division......
  11. Colleen_Young

    Colleen_Young Occasional commenter

    I have some sympathy with Guzintas here!
    WolframAlpha says 20.
    Casy says - why not just pop some brackets in to avoid ambiguity - or write it differently as someone has said.
  12. TBH I think there is an element of "***** wanging" beyond a certain point with the order of operations. It seems people just want to bend the rules when they get bored or realistic examples where we would use the order of operations.
    TBH at GCSE level is there that much depth required?
    Going back to the Americans and PEMDAS...that goes with a natty little sing song but they are more aware of the idea of operatins being 'on the same level'
  13. afterdark

    afterdark Occasional commenter

    Thank you for that.
    I think you are confusing your more complete understanding with how BODMAS is presented generally speaking.
    I note your use of O instead of I which is another point of potential confusion.
    Why do you object the phrase that 'the rule fails'?
    Do you mean taught as interchangeable pairs of things in 3 tiers?
    B & O
    D & M
    A & S
    Or 4 tiers?
    I am curious.

  14. PaulDG

    PaulDG Occasional commenter

    FWIW, I think the problem is that it's taught at all.
    When I was at school, my class was never taught "BODMAS", we were taught "you do multiplicaiton before addition". Then as we learnt about indices, there wasn't an issue about doing indices before multiplicaiton as it was obvious things would go wrong if you did the multiplicaition first.
    Division is simply multiplication by the recriprocal, so no confusion there and subtraction is adding the number after multiplying it by -1 so again, no confusion.
    No one in my class was in any doubt, no one had a funny word they needed to remember to use when they were facing a BODMAS question. No one had to spot BODMAS questions in the exams because we just knew which order to do everything in all the time.
    That still seems the best approach to me. Trouble is, it's always spoilt now because even the year 7s have heard of BODMAS and know they need to spot BODMAS questions... (Even if they then do them incorrectly, and even if they often fail to spot BODMAS questions because they thought they were doing an "area question" and not a BODMAS one.)
  15. Maths_Mike

    Maths_Mike New commenter

    thanks Paul I agree and you described it better than me. Its not BODMAS that fails anymore than SOHCAHTOA fails to give them the ability to do trig.

    Waht is failing is the students understanding of the order in which mathematical operations should be performed.

    If I understood this and I happen to use BODMAS as a memory tool all would be well - knowing BODMAS would not cause me to make a mistake.
  16. Maths_Mike

    Maths_Mike New commenter

    <font size="2">For some reason it did not let me edit my previous post so i try again! </font>
    <font size="2">thanks Paul I agree and you described it better than me. Its not BODMAS that fails anymore than SOHCAHTOA fails to give them the ability to do trig.</font><font size="2"> </font><font size="2">What is failing is the students understanding of the order in which mathematical operations should be performed.</font><font size="2"> </font><font size="2">If I understood this and I happen to use BODMAS as a memory tool all would be well - knowing BODMAS would not cause me to make a mistake.</font><font size="2"> </font><font size="2">However when teachers simply issue it and expect students to apply it blindly then of course ultimately, with anything other than the very straight forward questions, they make mistakes due to lack of understanding.</font> Anyway I will not be debating this further as I already indicated I don&rsquo;t like BODMAS and silly made up out of context questions to test students ability to apply it are pointless.

  17. So no real reason then as to why I get two different answers by either using the multiplication sign or not? Just a bunch of bitching about BIDMAS / BODMAS?
  18. pwc9000

    pwc9000 New commenter

    Surely the first reply gave a perfect explanation
  19. Thank you [​IMG]
  20. Nazard

    Nazard New commenter

    Shazgo did explain this well. Guzintas was right too about this thread. I almost posted links to the three previous threads about this exact same thing ... but was worried that someone would resurrect those threads and the forum would become an utter nightmare!Deep breath - here is my stab at an explanation.The reason you get two different answers is that the question is ambiguously stated. That's all. Where different mathematical conventions collide confusion occurs.Convention number 1: we have an order of precedence for operations in which brackets are evaluated first, followed by indices, then {multiplication and division} and lastly {addition and subtraction} (with the curly braces indicating that multiplication and division have equal precedence and are evaluated from left to right).Convention number 2: 4t mean "four lots of t" - we usually see 4t as a single mathematical object.Convention number 3: In some cases brackets are implied. For example, if we have a fraction:4 + 5--------5 + 13(the numerator of the fraction is 4+5 and the denominator is 5+13) then we work out the two additions first, before dividing one answer by the other.The issue arises when you write this last example in a different way. As it is stated currently (as a fraction) you work out the numerator and denominator separately and then divide, so the answer is 0.5The fraction line means the same as divide, but if we just substitute a divide sign for the fraction line we get this: 4 + 5 &divide; 5 + 13 In this case the first convention kicks in and the division is calculated first, lead to an answer of 4 + 1 + 13, which is 18. So there is clearly a difference.
    When we have 4t we usually treat this as a single entity and are used to working with it. We still remember, though, that it means 4 multiplied by t.
    If we have the fraction
    then this is clearly the same as 2/t
    A lazy way of writing this is: 8 &divide; 4t, which some people interpret as 8 &divide; (4t) and some interpret as 8 &divide; 4 x t. The former version is the same as the fraction version. The latter one should be evaluated left-to-right and gives 2xt.
    The problem, then, is that the statement is ill-defined (brackets would help, as would writing it as a fraction rather than using the divide symbol), and not that there is any confusion over BODMAS/BIDMAS/PEMDAS/PEDMSA/etc.

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