CAT 2025 Lesson : Mixtures & Alligations - Concepts & Cheatsheet

$1)$ $x_{4} = \dfrac{x_{1}w_{1} + x_{2}w_{2} + ... + x_{n}w_{n}}{w_{1} + w_{2} + ... + w_{n}}$.

$2)$ $\dfrac{w_{1}}{w_{2}} = \dfrac{(x_{2} - x_{A})}{(x_{A} - x_{2})}$



$3$) Where $\bm{I}$ is the initial quantity of non-replaced liquid,
$\bm{r\%}$ is the replacement percentage,
$\bm{n}$ is the number of replacements,
$\bm{F}$ is the final quantity of the non-replaced liquid,

$F = I \times \left(1 - \dfrac{r}{100} \right)^{n}$

$4$) If the quantity replaced is provided (for example, in litres) instead of percentage,

$F = I \times \left(1 - \dfrac{\text{Replacement Quantity}}{\text{Total Quantity}} \right)^{n}$

$5$) Where the quantities of $2$ mixtures of different unknown concentrations are $\bm{a}$ and $\bm{b}$ , and a quantity of $\bm{x}$ is removed from each of the mixtures and placed in the other mixtures, and this results in both the mixtures having the same concentrations, then

$x = \dfrac{ab}{a + b}$