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Solve the following problem taken from Treviso Arithmetic, a math textbook published in 1478 by an unknown Italian author:

A man finds a purse with an unknown number of ducats in it. After he spends 1/4, 1/5 and 1/6 of the amount, 9 ducats remain. It is required to find out how much money was in the purse.

I'm a little bit vpnfused by the problem... am I spending 1/5 of the remaining 3/4? I just don't know where to even begin.

  • Math -

    Model the situation by making an equation.

    Let d be the number of ducats in the purse.

    d - 1/4 - 1/5 - 1/6 = 9
    Get a common denominator with the fractions. There may be a number lower than 60, but that's what I'm going to use. (I multiplied 4, 5, and 6 to get 120. Then, I divided by 2.)

    d - 15/60 - 12/60 - 10/60 = 9
    d - 37/60 = 9
    d = 9 + 37/60, or about 9.6167

    This doesn't seem right to me, so I'm going to do what you suggested second (spending 1/5 of the remaining 3/4). That will probably work out better.

    (3/4)d - 1/5(3/4)d - 1/6(1/5)(3/4)d = 9
    (3/4)d - (3/20)d - (1/40)d = 9

    Common denominator... 40.

    (30/40)d - (6/40)d - (1/40)d = 9
    (23/40)d = 9
    d = 9(40/23) = 15.65 ducats

    That doesn't seem right either because you have partial ducats, which doesn't seem possible.

    I'm not sure. Sorry. Maybe my work can spark some thought, though.

  • Math -

    Michael, your equation

    d - 1/4 - 1/5 - 1/6 = 9 should have been

    d - (1/4)d - (1/5)d - (1/6)d = 9 , this times 60
    60d - 15d - 12d - 10d = 540
    23d = 540
    d = 23.478
    he had appr. 23.5 ducats

    I don't know if a ducat has smaller denominations, but I'm 100% sure of my math

    1/4 of 23.478 = 5.87
    1/5 of 23.478 = 4.70
    1/6 of 23.478 = 3.91

    He spent 14.48 (add them up)
    leaving 23.48-23.48 = 9

  • Math -

    thanls for the correction :)

  • Math -

    I worked on this awhile and ended up with 23.48 ducats (coins), too. I looked on google and ducats don't come in pieces. "It" was so many troy ounces of gold and was "a" coin used in several European countries.

  • Math -

    hmmm, that's weird; I'll tell my teacher about it on Monday because, to me, it sort of makes things confusing, since one would ususally figure you can't exactly have only fractions of a coin... thanks for the info!

  • Math -

    This may also be it...

    If I had 100 ducats, I would end up with 50 ducats.
    100(3/4) = 75
    75(4/5) = 60
    60(5/6) = 50

    Trying similar numbers, the result of all the spending is always 1/2 of what you start with.

    That would lead to an answer of 18 ducats. Not sure if this is right again, but it's an idea.

  • Math -

    well, the equations make sense and I see no other way of doing so thank you so very much for taking the time to helping me with this math Problem.

  • Math -

    Glad to help. :)

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