Alligation and Mixture Formulas
| Concept / Rule | Description | Formula / Representation |
| Alligation | A rule to find the ratio in which ingredients at given prices must be mixed to produce a desired price. | — |
| Mean Price | The cost of a unit quantity of the mixture. | — |
| Rule of Alligation | The relationship between the quantities and costs of cheaper and dearer ingredients. | $\frac{\text{Quantity of cheaper}}{\text{Quantity of dearer}} = \frac{d – m}{m – c}$ |
| Repeated Dilution | Formula for pure liquid remaining after $n$ operations of replacing $y$ units from $x$ total units. | $\text{Pure liquid} = x \left(1 – \frac{y}{x}\right)^n$ |
Alligation Graphical Representation
To find the ratio of Cheaper Quantity : Dearer Quantity, use the following cross-method:
- $c$ = C.P. of unit quantity of cheaper
- $d$ = C.P. of unit quantity of dearer
- $m$ = Mean Price
| Ingredient | Cost | Difference (Ratio Part) |
| Cheaper ($c$) | $c$ | $(d – m)$ |
| Dearer ($d$) | $d$ | $(m – c)$ |
Resulting Ratio: $(d – m) : (m – c)$
