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

1. ## 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?

2. 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.

3. 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

5. 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: $x_2 = 95 - x_1 = 95 - \frac{95 \cdot 0.375 - 95 \cdot 0.27}{0.375 - 0.22}$

6. 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

7. ## still not got it

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