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Thread: WOrd problems

  1. #1
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    WOrd problems

    Q: A party of boys purchased a boat and upon payment for the same discovered that if thay had numbered 3 more, they would have paid a dollar apiece less; but if they had numbered 2 less,they would have paid apiece more. How many boys were there, and what did the boat cost?

    let x= number of boys and y = number of dollars each paid. let xy = represents the number of dollars the boat cost


    i created these two sim. equations but i am wrong. This is due to my interpretation of the question. How do i interpret in probably to get the right equations

    $x + 3 + y = xy $
    $x-2 = x(y+1)$
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  2. #2
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    Re: WOrd problems

    Quote Originally Posted by bigmansouf View Post
    Q: A party of boys purchased a boat and upon payment for the same discovered that if thay had numbered 3 more, they would have paid a dollar apiece less; but if they had numbered 2 less,they would have paid apiece more. How many boys were there, and what did the boat cost?

    let x= number of boys and y = number of dollars each paid originally
    using your variables with one addition ...

    let $P$ be the price paid

    $xy = P$

    $(x+3)(y-1) = P$

    $xy+3y-x-3 = P$

    $P + 3y-x-3 = P$

    $x = 3(y-1) \implies y > 1 \text{ and } y \in \mathbb{Z}$ ... also, $x$ is a multiple of $3$

    so, what value of $y$ satisfies $xy=P$ and $(x+3)(y-1) = P$ ? (guess & check)

    ----------------------------------------------------------------------------

    method 2 ...

    $P = \text{ price paid}$

    $x = \text{ number of boys}$

    $\dfrac{P}{x}$ is the price each paid

    3 more in the party, each boy pays 1 dollar less

    $\dfrac{P}{x} - 1 = \dfrac{P}{x+3}$

    $\dfrac{P(x+3)}{x(x+3)} - \dfrac{x(x+3)}{x(x+3)} = \dfrac{Px}{x(x+3)}$

    $P(x+3) - x(x+3) = Px$

    $Px + 3P - x^2 - 3x = Px \implies x^2 + 3x - 3P = 0$

    $x = \dfrac{-3 \pm \sqrt{9 + 12P}}{2}$

    since $x$ is a positive integer, the value $(9+12P)$ must be a perfect square number ...

    $x = \dfrac{-3 + \sqrt{9 + 12P}}{2} \implies P = 6 \implies x = 3$
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    Re: WOrd problems

    Quote Originally Posted by bigmansouf View Post
    A party of boys purchased a boat and upon payment for the same
    discovered that if thay had numbered 3 more, they would have
    paid a dollar apiece less; but if they had numbered 2 less,they
    would have paid apiece more.
    How many boys were there, and what did the boat cost?
    Unclear: you omitted "how much more" apiece.
    Looks like you used 1, therefore do did Skeeter.
    But that doesn't seem to work (unless I need another coffee!).
    Using Skeeter's results (p=6, x=3) has this result:

    p x y
    6 3 2 (original)
    6 6 1 (3 more boys)
    6 1 6 (2 less boys) : only 1 boy left, who pays 4 more...
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  4. #4
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    Re: WOrd problems

    Here is how it can work out:

    Quote Originally Posted by bigmansouf View Post
    Q: A party of boys purchased a boat and upon payment for the same discovered that if thay had numbered 3 more, they would have paid
    a dollar apiece less; but if they had numbered 2 less,they would have paid [a dollar] apiece more. How many boys were there,
    and what did the boat cost?

    let x= number of boys and y = number of dollars each paid. let xy = represents the number of dollars the boat cost
    From the first hypothetical:


    xy = (x + 3)(y - 1)

    xy = xy - x + 3y - 3

    0 = -x + 3y - 3

    x = 3y - 3


    and


    From the second hypothetical:


    xy = (x - 2)(y + 1)

    xy = xy + x - 2y - 2

    0 = x - 2y - 2

    x = 2y + 2

    ____________________________

    Solve simultaneously:

    x = 3y - 3
    x = 2y + 2
    -------------

    3y - 3 = 2y + 2

    y - 3 = 2

    y = 5


    Substitute the y-value back into one of those equations to find x:

    x = 2(5) + 2

    x = 10 + 2

    x = 12


    There \ \ were \ \ 12 \ \ boys, \ \ and \ \ the \ \ boat \ \ cost \ \ 60 \ \ dollars.


    And you can check what happens with the cost apiece per boy when there are three more boys or two fewer boys.
    Last edited by greg1313; Dec 30th 2016 at 10:09 AM.
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  5. #5
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    Re: WOrd problems

    Quote Originally Posted by bigmansouf View Post
    A party of boys purchased a boat and upon payment for the same
    discovered that if thay had numbered 3 more, they would have
    paid a dollar apiece less; but if they had numbered 2 less,they
    would have paid apiece more.
    How many boys were there, and what did the boat cost?
    Geezzzzz......SO:
    x boys bought a boat: each paid y dollars.
    If instead (x+3) boys bought the boat, each would pay (y-1) dollars.
    If instead (x-2) boys bought the boat, each would pay (y+1) dollars.
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