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I have to find a solution for this problem: given $N$ materials of density $\rho_k$, find the mixture of them giving a compound of density $\rho$. From a mathematical point of view, we have to find $q_k$ and $\rho_k$ for which: $$\rho=\frac{\sum_{k=1}^Nq_k\rho_k}{\sum_{k=1}^Nq_k}$$ with $\rho$ given. The constrains are: $\sum_{k=1}^Nq_k=Q$,$\forall k\le N$ $q_k\gt 0$. I suppose there are infinite solutions for this problem. How to find one of them? Thanks.

Riccardo.Alestra
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    Shouldn't it be $\rho=\frac{\sum_{k=1}^Nq_k\rho_k}{\sum_{k=1}^Nq_k}$? Otherwise, $\rho$ always equals $1$. – Wolf Aug 08 '13 at 08:38
  • @Wolf: Just an error. I corrected it. Thnks – Riccardo.Alestra Aug 08 '13 at 08:50
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    You might find your answer here if you are ok with a numerical solutation: http://stackoverflow.com/questions/16365723/find-positive-solutions-to-underdetermined-linear-system-of-equations (just multiply your equation with $Q$). – Wolf Aug 08 '13 at 08:53

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