Mixture

[Guest]

Two given mixtures of p% and q% respectively are of a certain ingredient. Determine how x units of the first and y units of the second are combined so the resulting mixture is r% of the ingredient.

My start:

px + qy = r

Unsure how to continue.

[Guest]

The equation should be
[tex]px+qy = r(x+y)[/tex]

Unfortunately your system of equations is under-constrained, i.e. there are not enough equations to uniquely determine [tex]x[/tex] and [tex]y[/tex].
For example suppose we are given [tex]p=0, q=100, r=50[/tex] one solution is [tex]x=1, y=1[/tex] (one unit without any ingredient, plus one unit with 100% ingredient, makes 2 units with an overall concentration of 50%). However, another solution is [tex]x=2, y=2[/tex], or [tex]x=10, y=10[/tex]. In fact as long as [tex]x=y[/tex] (and the numbers are positive) any solution will work.

In general if there exists a solution then there will be infinite solutions (just scale the quantities [tex]x[/tex] and [tex]y[/tex] by the same amount to get new solutions). However, there may be no solutions, it all depends on the values of [tex]p, q, r[/tex]. You need [tex]r[/tex] to lie between [tex]p[/tex] and [tex]q[/tex], otherwise there will be no solution. For example if [tex]p=75[/tex] and [tex]q=10[/tex], then there is no way to get [tex]r=80[/tex], because the most concentrated the mixture can be is [tex]75[/tex] (mixing up the first mixture with the second will only reduce the concentration of the first not increase it). Similarly [tex]r=5[/tex] is also impossible.

Assuming [tex]r[/tex] lies between [tex]p[/tex] and [tex]q[/tex]
If [tex]p=r[/tex], then [tex]x=1, y=0[/tex] is a solution,
if [tex]p\neq r[/tex], then [tex]x=\frac{r-q}{p-r}, y=1[/tex] is a solution.

Hope this helped,

R. Baber.

[Guest]

Ok, thanks.