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Math Help - solving for proportions

  1. #1
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    solving for proportions

    Seasons greetings!

    I apologize for the format of this post. The FAQ link on how to format equations seems to be broken.

    I have a "compound" (C1) that is to be made from two other compounds and two pure elements (C2, C3, L, T)

    Compound 2 is composed of elements in this proportion: C2 = 30X + 70L
    Compound 3 is composed of elements in this proportion: C3 = 4X + Z + 95L

    The final compound must have the following proportion of elements:
    C1 = 8X + 2T + .2Z +89.8L

    So, to find the proportions needed to make compound C1, I would need to solve the following for a, b, c, and d?
    C1 = aC2 + bC3 + cL + dT

    I this makes no sense, I'm sorry. My head is already hurting...
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  2. #2
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    Hello, scunning!

    I have a compound (C_1) to be made from two other compounds
    . . and two pure elements: . C_2,\:C_3,\:L,\: T.

    Compound 2 has elements in this proportion: . C_2 \:=\: 30X + 70L
    Compound 3 has elements in this proportion: . C_3 \:=\: 4X + Z + 95L

    The final compound must have the following proportion of elements:
    . . C_1 \:=\: 8X + 2T + 0.2Z +89.8L

    So, to find the proportions needed to make compound C_1
    . . . I would need to solve the following for a, b, c,\text{ and }d\,?
    . . . . . C_1 \:=\: aC_2 + bC_3 + cL + dT . Yes!

    We have: . . . . . . . . . . . aC_2 + bC_3 + cL + dT \;=\;8X + 2T + 0.2Z + 89.8L

    . a(30X + 70L) + b(4X + Z + 95L) + cL + dT \;=\;8X + 2T + 0.2Z + 89.8L

    . . 30aX + 70aL + 4bX + bZ + 95bL + cL + dT \;=\;8X + 2T + 0.2Z + 89.8L

    . . . . (30a + 4b)X + dT + bZ + (70a + 95b + c)L \;=\;8X + 2T + 0.2Z + 89.8L


    Equate coefficients: . \begin{bmatrix}30a + 4b \:=\: 8 \\ d \:=\: 2 \\ b \:=\: 0.2 \\ 70a + 95b + c \:=\: 89.8 \end{bmatrix}


    And solve the system of equations: . \begin{bmatrix} a &=& \frac{6}{25} \\ b &=& 0.2 \\ c &=& 54 \\ d &=& 2 \end{bmatrix}


    Therefore, we will use: . \begin{Bmatrix} \frac{6}{25}\text{ units} & \text{compoind 2} \\<br />
0.2\text{ units} & \text{compound 3} \\<br />
54\text{ units} & \text{element L} \\<br />
2\text{ units} & \text{element T} \end{Bmatrix}

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  3. #3
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    Soroban, you are a lifesaver! Thank you so much for the explanation. This forum is a wonderful resource.

    Have a great Christmas!

    Edit: It didn't seem like those proportions worked, but then I realized that since I expressed the compounds in terms of 100 parts, the elements needed to be expressed the same way. I change it to the following, and your logic worked perfectly! Thanks again.

    C1 = aC2 + bC3 + c100L + d100T
    Last edited by scunning; December 24th 2010 at 05:58 PM.
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