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Math Help - system of equations

  1. #1
    EPZ
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    system of equations

    Q: A doctor's prescription calls for a daily intake containing 40 mg of vitamin C and 30 mg of vitamin D. Your pharmacy stocks two liquids that can be used: one contains 20% vitamin C and 30% vitamin D, the other 40% vitamin C and 20% vitamin D. How many milligrams of each compound should be mixed to fill the prescription?

    The system I came up with is

    .2C+.3D=40
    .4C+.2D=30, which means we need 75 mg of C and 50 mg of D.

    The book says the solution is 50 mg of C and 75 mg of D. Simply switching the outputs from equation 1 and 2 doesn't change anything. Moreover, I am not entirely sure I am seeing the problem correctly. Can anyone explain?

    Thanks
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  2. #2
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    .2C+.3D=40
    .4C+.2D=30
    From what I see you decided to label the amount of liquids C and D. Lets call them A and B instead to avoid confusion with Vitamin C and D.
    So from now on A = Volume of liquid A and B = Volume of Liquid B.

    We need 40mg of Vitamin C. Liquid A has 20% Vitamin C and Liquid B has 40% Vitamin C. So

    0.2A + 0.4B = 40

    We need 30mg of Vitamin D. Liquid A has 30% Vitamin C and Liquid B has 20% Vitamin C. So

    0.3A + 0.2B = 30

    With these two equations, you should be getting the answer given in your book.
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  3. #3
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    Hello, EPZ!

    You're using the wrong variables.


    A doctor's prescription calls for a daily intake
    . . containing 40 mg of vitamin C and 30 mg of vitamin D.

    Your pharmacy stocks two liquids that can be used:
    . . one liquid contains 20% vitamin C and 30% vitamin D,
    . . the other liquid 40% vitamin C and 20% vitamin D.

    How many milligrams of each liquid should be mixed to fill the prescription?
    Let: . \begin{array}{ccc} x &=& \text{mg of Liquid \#1} \\<br />
y &=& \text{mg of Liquid \#2}\end{array}


    Liquid #1 has x mg which is 20% vitamin C: . 0.2x mg of C
    Liquid #2 has y mg which is 40% vitamin C: . 0.4y mg of C

    . . Together, they must total 40 mg: . 0.2x + 0.4y \:=\:40 .[1]


    Liquid #1 has x mg which is 30% vitamin D: . 0.3x mg of D
    Liquid #2 has y mg which is 20% vitamin D: . 0.2y mg of D

    . . Together, they must total 30 mg: . 0.3x + 0.2y \:=\:30 .[2]


    \begin{array}{cccccc}\text{Multiply {\color{blue}[1]} by 5:} & x + 2y &=& 200 & {\color{blue}[3]} \\<br />
\text{Multiply {\color{blue}[2]} by 10:} & 3x + 2y &=& 300 & {\color{blue}[4]}\end{array}

    Subtract [3] from [4]: . 2x \:=\:100 \quad\Rightarrow\quad x \:=\:50

    Substitute into [3]: . 50 + 2y \:=\:200 \quad\Rightarrow\quad y \:=\:75


    Therefore, use: . \begin{Bmatrix}50\text{ mg of Liquid \#1} \\ 75\text{ mg of Liquid \#2} \end{Bmatrix}

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