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Math Help - Solving for Mass, Given Mass as Constant + Infinite # of Discrete Unknown variables)

  1. #1
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    Solving for Mass, Given Mass as Constant + Infinite # of Discrete Unknown variables)

    So today my teacher decided to challenge us with a "word problem".

    He placed an unknown amount of marbles, all of constant mass, in 8 boxes. The mass of each of these boxes are:

    Box 1: 36.03 g
    Box 2: 30.41 g
    Box 3 = 24.93 g
    Box 4 = 24.87 g
    Box 5 = 20.63 g
    Box 6 = 15.98 g
    Box 7 = 10.43 g
    Box 8 = 6.28 g

    Total mass = 169.56 g

    These are the values of just the MASS of the marbles. I have already subtracted the mass of the box from each one already. Now, how would you solve for the mass?

    Here's what I started with:

    Let's say the number of marbles is represented by 'n' (the subscripts being the # of marbles relative to a box), and 'm' being the mass of the marble (which is constant).

    For example:
    Mass of Marbles in Box 1 = m(n1)
    Mass of Marbles in Box 2 = m(n2)
    etc.

    Using elimination method for each of the boxes, we isolate until we get a single variable solved. The problem is, the variable n is a set of whole numbers (in other words, discrete) where n greater than or equal to 0 [of course, having 0 for n is useless I think, so I'd say it was actually greater than or equal to 1]. I have no idea how to approach this problem properly. It seems like every time I try to isolate, there is always an unknown variable that cannot be solved.

    Any ideas? I would like for you guys to use the same variables I've used, but if it helps to use other variables instead, that'll be fine.
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  2. #2
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    You want the mass of one marble? There is not enough information to determine absolutely conclusively the mass of one marble. However, you can get a number that is very likely the mass of one marble, but in the unlucky case it could be an integer multiple of the mass. Also, I'm assuming the measurements are extremely accurate (i.e. you did not round up significant digits).

    The idea is similar to that of Millikan's oil-drop experiment to determine the charge of a single electron.

    To make things simple, lets get rid of decimal points and work in units of cg.
    I.e. the masses are 3603g, 3041cg, 2493cg, ...

    All of these numbers must be an integer multiple of the mass. This means that an integer multiple of the mass is the greatest common divisor of these numbers.
    What is the greatest common divisor for 3603, 3041, 2493, ...? I get 1 because 3602 and 3041 are already relatively prime.

    So the greatest common factor is 1cg=0.01g

    This means that the mass m = (0.01 g)/(some integer). It is likely that "some integer" is 1 since you have many samples.

    Why can't we know the exact mass? If we said m = 0.01 g, then we can also create an indistinguishable situation where m=0.001g and put 10 times as many marbles in each box to get the same masses.
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