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Math Help - abv percentage help please I cant work this out

  1. #1
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    abv percentage help please I cant work this out

    Hi Did not really know where to post this but can anyone explain how to do this I am trying to work out the following and cant come up with an answer and explain it please help.

    Vodka
    I would like to make a final product of 27% abv from 22% abv spirit and 37.5% abv spirit. What % of each strength would I need?

    How many litres of each would I need to make a batch size of 95 litres?
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  2. #2
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    There are two constraints to take care off. First, your final product must have 0.22 abv, such that:

    \frac{x_1}{x_1 + x_2} \cdot 0.22 + \frac{x_2}{x_1 + x_2} \cdot 0.375 = 0.27

    Where x_1 is the total amout of your 0.22 abv liquid and x_2 is your 0.375 abv liquid. Also, the total batch size must be 95 litres. So:

    x_1 + x_2 = 95

    Just solve the last equation for one x_i, i \in \{1,2\} and fill this solution into the first equation. Solve for the left variable and compute the second one afterwards.
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  3. #3
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    Sorry I still dont get it whats the answer and then I might be able to work backwards to find out how it was done. Thanks I am being a bit thick
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  4. #4
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    Please help me I am lost with this one
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  5. #5
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    Since x_1 + x_2 = 95 we know that: x_2 = 95 - x_1

    Thus:

     \frac{x_1}{x_1 + x_2} \cdot 0.22 + \frac{x_2}{x_1 + x_2} \cdot 0.375 = 0.27
    \Rightarrow \frac{x_1}{x_1 + 95 - x_1} \cdot 0.22 + \frac{95 - x_1}{x_1 + 95 - x_1} \cdot 0.375 = 0.27
    \Rightarrow \frac{x_1}{95} \cdot 0.22 + \frac{95 - x_1}{95} \cdot 0.375 = 0.27
    \Rightarrow x_1 \cdot 0.22 + (95 - x_1) \cdot 0.375 = 95 \cdot 0.27
    \Rightarrow 95 \cdot 0.375 - 95 \cdot 0.27 = (0.375 - 0.22) \cdot x_1
    \Rightarrow x_1 = \frac{95 \cdot 0.375 - 95 \cdot 0.27}{0.375 - 0.22}

    Since x_2 = 95 - x_1 you know: x_2 = 95 - x_1 =  95 - \frac{95 \cdot 0.375 - 95 \cdot 0.27}{0.375 - 0.22}
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  6. #6
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    Easier if you look at these this way:
    Code:
       x @ 22
    95-x @ 37.5
    ===========
     95  @ 27
    So:
    [22x + 37.5(95 - x)] / 95 = 27

    Solve for x
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  7. #7
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    still not got it

    So whats the answer how much of each will I need to make 95 litres?
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