Assumptions in Mathematical Word Problems

  1. Hi there,

    Consider a related rates problem,

    Gas is being added into a balloon at a rate of dE/dt = x ft^3/sec Find the rate at which the radius is changing at time h.

    This is a completely false problem; I am just giving an example. My question is,

    How would you know if there is a HOLE in the balloon (from which air LEAVES)?

    I mean, what assumptions should you make? This is a word problem (keep that in mind).

    How do you know if a hole (from which air leaves) EXISTS or DOES NOT EXIST ???

    Thanks guys =)
     
  2. jcsd
  3. mathman

    mathman 6,436
    Science Advisor
    Gold Member

    You are being very picky. The word problem is sensible only on the assumption that there is no leakage.
     

  4. What do you mean by "sensible"? I would REALLY appreciate it if you could describe this a LITTLE more? PLEASE?

    Thank you so much =)
     
  5. bump
     
  6. HallsofIvy

    HallsofIvy 40,310
    Staff Emeritus
    Science Advisor

    So you "bump" but don't answer the question you were asked?

    Generally speaking, if you want to solve a "word problem", you should assume it can be answered. That is, that there is an answer and all necessary information has been given. In the problem you state, you must assume that there is NO hole through which air is escaping because, if there were, since you have not been told the rate at which air is escaping, you could not do the problem.
     
  7. jbunniii

    jbunniii 3,353
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    It depends on the context. If it's a math homework problem, then in general, don't read anything into it which isn't stated. If you are actually trying to design a balloon that someone is going to fly in, then by all means worry about holes, power lines, aggressive birds, etc.
     
  8. symbolipoint

    symbolipoint 3,076
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    Gold Member

    Amad27, look for all the information in the problem which is needed for answering the problem. Identify any information that is unnecessary. Your example contains no indication about any leaks in the balloon. You have no reason to wonder about this detail.
     
  9. Hi there! Thanks a bunch for the reply.

    I suppose you are right. But this concept requires the concept of "solvability," meaning you must assume the problem is solvable.

    What about the problems with no solution or no sufficient data provided?
     
  10. So you should take all the facts you are given, and not worry about other things, which exist? Tell me one thing.

    In a mathematical word problem, do you always take the information literally and ignore what isnt mentioned? Do you Assume what ISNT mentioned doesn't exist??

    Thanks a bunch ;)
     
  11. Basically, when presented a word problem, the technique is to not be a pedantic goob.

    And like HOI said, you interpret the problem so that it makes sense and is solvable. There are no extra points for turning the problem into one about language. I remember someone asking what day a cube would be half the size of a room, if the cube started 1/50th the size of the room and it doubled in size every day. They "smartly" pointed out that the question "had no answer" because it was never specified (it was, but by "it") which of the black and room were doubling. Their nit pick gets them exactly zero points because it is obvious that if the room was doubling, there is no solution, and so it is not a reasonable way to interpret the question.
     
  12. Yes; I do understand that. What do you mean by "reasonable way to interpret the question"?

    What you mean is that you are basing this off of the concept of "solvability"

    What about the problems, which have the answer: Insufficient data to solve the problem?
     
  13. Mentallic

    Mentallic 3,684
    Homework Helper

    Does this really need to be explained?

    The balloon problem you presented had sufficient data to be solved, but once you included the possibility of air escaping (that the problem does not mention in any way) then it becomes unsolvable.

    I'll leave the rest to you to figure out how this is different to questions that actually have insufficient data.
     
    1 person likes this.
  14. If there isn't sufficient data to solve the problem, then there isn't sufficient data.

    That's wildly different from there being sufficient data, and then inventing some technicality (hole in the balloon) that makes the problem unsolvable.
     
    1 person likes this.
  15. Hey; Thanks for the reply; I appreciate it.

    So really the bottom-line is that, your assumptions are based on making a problem (w/ sufficient data) solvable?

    Think about it for a second, I want your opinion; before this question, did you ever wonder about these assumptions (you blindly made) yourself? Did you ever think WHY you wouldn't assume there to be a hole?

    Thanks
     
  16. Astronuc

    Staff: Mentor

    Most homework/textbook problems are solvable. The goal is to take the knowledge gained/learned and use it to solve a problem. Usually, when something is unstated, the problem is built upon an underlying understanding which does not need repeating.

    In the original post of this thread, one asks about a hole in a balloon. Well, most balloon have at least one hole through which the balloon is filled. That hole is then sealed/closed in order to retain the contents. The membrane of the balloon is the physical barriers which retains the contents, usually under a greater pressure than the environment (atmosphere) in which the balloon exists.

    There could be a second hole in the balloon through which the contents escapes, but that would have to be explicitly mentioned. There is no mention of fluid exiting the balloon.

    The statement "Gas is being added into a balloon at a rate of dE/dt = x ft^3/sec" implies that there is at least one hole through which the gas enters the balloon. One also needs to know about the geometry of the balloon, e.g., although one might assume it is spherical, as opposed to cylindrical or toroidal. Also, one would need to know about the compressibility of the gas, and the tension (elasticity) of the balloon material.

    It is important to understand if the volumetric rate relates to the gas (at a given pressure) or if it refers to the volumetric rate of change of the balloon. There is a difference.

    Ideally, homework problems are carefully worded to avoid confusion or misinterpretation. Sometimes, it might be necessary to state assumptions in solving a problem in order to clarify the solvers interpretation of the problem. However, I believe most textbook problems have a unique solution.
     
  17. For the same reason that you blindly made the assumption that there isn't a green hobgoblin trying to pop the balloon by throwing darts across the room once per ten second interval.
     
  18. What is the "same reason" you are pointing out here?
     
  19. Hi, thanks (very much).

    So you are suggesting that the reason is that a textbook problem is solvable. What about the problems with the wording,

    find the rate at which the radius changes IF ANY...?

    How would you do this? You cant assume its solvable; there is a chance it ISNT because of IF ANY.
     
  20. Vanadium 50

    Vanadium 50 17,442
    Staff Emeritus
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    Amad, it's very simple. You are free to invent assumptions that are not in the problem that make it unsolvable. And your instructor is free to mark this wrong. Which she will.
     
  21. I want anyone's honest opinion. What was the reason you never assumed something exists BEFORE I asked this question?

    Did you all really not assume the existence of leaks because they would make the problem unsolvable or did you blindly assume that nothing besides the "said" exists?

    Anyone ? Opinions?
     
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