..

Mixtures and Allegations

Why the weird X technique works?

  • The main problem we are trying to solve involves a ratio - a numerator and a denominator.
  • Generalizing all the case we have 2 things
    • A ratio
    • A quantity of something that has the said ratio
  • Suppose I have $A$ amount of a substance with a ratio $x$ and $B$ amount of something with the ratio $y$. Then when I mix them together I have $A+B$ amount with some ratio.
  • That ratio is given by $\frac{Ax+By}{A+B}$
  • But the question is usually not to find the resultant ratio, though it may be. The question are often to find $A:B$
  • Let us set some standard variables
    • $A$ be the amount that is taken in the ratio $x$
    • $B$ be the amount that is taken in the ratio $y$
  • Then, the following results are true
    • $\frac{Ax + By}{A + B}$ is the resultant ratio i.e the thing to put in the middle
    • To get the final ratio of $z$, $A:B$ must be $y-z:x-z$
  • Generalizing to 3 variables we get $\frac{Ax + By + Cz}{A + B + C}$