Calculate the ratio of two drinks in order to have a specific ratio of two common ingredients

Question

Every oil contains different quantities of omega3 and omega6. I would like to mix two oils so that the ratio omega6/omega3 is the one I want.

For example, this are two oils:

In 100 grams
           Omega6         Omega3 
---------------------------------
FLAXSEED    51 grams       7 grams
SOYBEAN     14 grams      57 grams

I want for example a ratio omega6/omega3 of 2.

What formula should I apply that tell me the flaxseed/soybean ratio to use, in order to eat omega6/omega3 in the proper ratio?

Answer 1

Let the numbers in the first column of your table be called $p_1= 51$ and $p_2 = 7$.

Let $k$ be the ratio you want.

For a mix that's $s$ of the first item and $1-s$ of the second (if $s = .25$, that means 25% flaxseed and 75% soybean), you want to have $$ (1-s) p_1 + s p_2 = k ((1-s) q_1 + s q_2) $$ So we solve this for $s$:

\begin{align} (1-s) p_1 + s p_2 &= k ((1-s) q_1 + s q_2)\\ p_1-s p_1 + s p_2 &= k (q_1 -sq_1 + s q_2)\\ p_1- kq_1 - s p_1 + s p_2 &= -skq_1 + s k q_2\\ p_1- kq_1 - s(p_1 - p_2) &= -sk(q_1 - q_2)\\ p_1- kq_1 &= s(p_1 - p_2) -sk(q_1 - q_2)\\ p_1- kq_1 &= s[(p_1 - p_2) -k(q_1 - q_2)]\\ \frac{p_1- kq_1}{(p_1 - p_2) -k(q_1 - q_2)} &= s \end{align}

And that's your formula.

Answer 2

Say you eat one unit of soybean and $x$ units of flaxseed. You get $14+51x$ Omega6's and $57+7x$ Omega3's. You are asking that $\frac {14+51x}{57+7x}=2$

Answer 3

This is a system of two equations in three unknowns:

$$51F+14S=2C \\ 7F+57S=C $$

For each fixed $ C $ this is two linear equations in two unknowns, which can be solved for instance by Gaussian elimination. Then pick out your $ C$ of choice.