Alligation Rule and Mixtures and Replacements Formulas

Alligation Rule

Alligation rule helps us to find, in what ratio two mixtures with different concentrations are to be mixed to get a target concentration.

For example, we have two mixtures of alcohol concentrations with 40% and 85% and we need to find in what ratio these two are to be mixed to get 50% concentration.   To know this we have to learn about weighted average.

Alligation Rule Derivation:
Let $m$ quantities have an average $x$ and $n$ quantities have an average $y$ are mixed together.To get the final Average $A$, the following weighted average formula to be used.
$$\displaystyle\frac{{{m} \times {x} + {n} \times {y}}}{{{m} + {n}}} = {A}$$
Let us re-arrange the terms above,
$\Rightarrow m \times x + n \times y = A(m + n)$
$\Rightarrow m \times x + n \times y = A \times m + A \times n$
$$\Rightarrow n \times y - A \times n = A \times m - m \times x$$
$$\Rightarrow n\left( {y - A} \right) = m\left( {A - x} \right)$$
$$\Rightarrow \dfrac{m}{n} = \dfrac{{y - A}}{{A - x}}$$
To easily apply the alligation rule, the following diagram is very useful. Here, $m,\,n$ are weights or units to be taken, and $x$, $y$ are initial averages or concentrations, $A$ is the target average or concentration.
Note: Alligation rule gives us only the ratio in which the initial mixtures are to be mixed to get desired concentration but never gives the actual quantities.

Mixtures and Replacements

The problems related to mixtures based on two important concepts.  Alligation rule and Inverse proportionality rule are the two.

In these problems we are asked to find the resultant concentration after mixing two or three components or the final concentration when one component of the mixture is being replaced by another component which is mostly one the components of the mixture.

Replacement formula:
The general formula for replacements is as follows: $FC = IC \times {\left( {1 - \dfrac{x}{V}} \right)^n}$

Here,
FC = Final concentration
IC = Initital concentration
x = replacement quantity
V = Final volume after replacement
n = number of replacements

Note:  Always remember FC and IC are the concentrations of the second component in the mixture.  "x" is the concentration of the first component.