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Math Help - Odd question

  1. #1
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    Odd question

    10 apples = 1cents
    5 oranges = 2cents
    2 pear = 5cents

    Using at least 1 of each, get 100 fruits and 1 dollar. WITHOUT GUESSING.
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  2. #2
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    Re: Odd question

    Here's a start

    Making A = number of apples, O = number of oranges and P = number of pears

    A+O+P=100

    and

    0.01A+0.02O+0.05P=$1
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  3. #3
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    Re: Odd question

    Quote Originally Posted by pickslides View Post
    Here's a start

    Making A = number of apples, O = number of oranges and P = number of pears

    A+O+P=100

    and

    0.01A+0.02O+0.05P=$1
    That won't work cause its 3 variables for 2 equations right?
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  4. #4
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    Re: Odd question

    Yep, you'll have to find a third.
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  5. #5
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    Re: Odd question

    Hello, pvpkillerx!

    \begin{array}{ccc}\text{10 apples} &=& \text{1 cent} \\ \text{5 oranges} &=& \text{2 cents} \\ \text{2 pears} &=& \text{5 cents} \end{array}

    Using at least one of each, get 100 fruits and 1 dollar ... without guessing.
    The unit prices are: . \begin{bmatrix}\text{apples} & 0.1\,\rlap{/}c \\ \text{oranges} & 0.4\,\rlap{/}c \\ \text{pears} & 2.5\,\rlap{/}c \end{bmatrix}

    Let: . \begin{Bmatrix}A &=& \text{no. apples} \\ \theta &=& \text{no. oranges} \\ P &=& \text{no. pears} \end{Bmatrix}


    There are 100 fruits: . A + \theta + P \;=\;100 \;\;[1]

    They cost 100 cents: . 0.1A + 0.4\theta + 2.5P \;=\;100 \;\;[2]


    Multiply [2] by 10: . A + 4\theta + 25P \;=\;1000
    . . . . Subtract [1]: . A +\; \theta +\; P \;\;\;=\;\;\;100

    . . . . And we have: . . . 3\theta + 24P \:=\:900

    . . P \:=\:\frac{900 - 3\theta}{24} \quad\Rightarrow\quad P \;=\;37 + \frac{4-\theta}{8}


    Since P is an integer, 4-\theta must be divisible by 8.
    . . Hence: . \theta \,=\,4 \quad\Rightarrow\quad P \,=\,37

    Substitute into [1] and solve for A.

    Last edited by Soroban; September 30th 2011 at 01:59 PM.
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