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Math Help - Problem with apples driving me crazy!

  1. #1
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    Problem with apples driving me crazy!

    Hi
    I hope someone can shed some light on this question in my workbook. It is a holiday week and I can ask my teacher until tuesday - I'm going mad.

    You buy 12 dollars worth of apples. They are kind of small so the cashier gives you two extra for free. This reduces the price per dozen by exactly 1 dollar. How many apples did you get for 12 dollars?

    ok, I know the answer in the back of the book says 18 apples, but how??

    any help much appreciated
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  2. #2
    MHF Contributor
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    You buy 12 dollars worth of apples. They are kind of small so the cashier gives you two extra for free. This reduces the price per dozen by exactly 1 dollar. How many apples did you get for 12 dollars?
    "You buy 12 dollars worth of apples."
    Let x = number of apples you buy
    And y = cost per apple, in dollars
    So,
    x*y = 12 ----------(1)

    "...the cashier gives you two extra for free."
    So you have now (x+2) apples. ------***

    "This reduces the price per dozen by exactly 1 dollar."
    New price of apples = (12*y -1) per dozen.
    In per apple, that is (12y -1)/12 = y -(1/12)
    So,
    (x+2)(y -1/12) = 12 ------------(2)

    12 in (1) = 12 in (2),
    x*y = (x+2)(y -1/12)
    xy = xy -x/12 +2y -2/12
    0 = -x/12 +2y -2/12
    Clear the fractions, multiply both sides by 12,
    0 = -x +24y -2
    x = 24y -2 ------------(3)

    From (1), y = 12/x
    Substitute that into (3),
    x = 24(12/x) -2
    x = 288/x -2
    Clear the fraction, multiply both sides by x,
    x^2 = 288 -2x
    x^2 +2x -288 = 0
    Use the Quadratic Formula to solve for x.
    x = {-2 +,-sqrt[(2)^2 -4(1)(-288)]} /2*1
    x = {-2 +,-sqrt(1156)}/2
    x = {-2 +,-34}/2

    x = (-2 +34)/2 = 16 ---***
    x = (-2 -34)/2 = -18 --------reject, because there are no negative number of apples.

    So, x = 16 apples -----you bought.
    And you were given 2 extra apples for free.
    Therefore, for 12 dollars, you got 16+2 = 18 apples altogether. ----answer.
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  3. #3
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    Nov 2005
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    wow - I was so stuck and would have spent all weekend just going insane. I couldnt see how to set it up at the beginning.

    Thankyou!!
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  4. #4
    Junior Member
    Joined
    Oct 2005
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    Here's an eq in terms of cost/apple:

    $12 / x apples - $12 / (x+2) apples = $1.00 / 12 apples
    Multiply by common denom -->
    12(12)(x+2) - 12(12)(x) = x(x+2)
    144x + 288 - 144x = x^2 + 2x
    0 = x^2 + 2x -288
    0 = (x-16)(x+18)

    16 purchased + 2 for free = 18 apples
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