Alligation or Mixture
Mixing substances of different rates
Alligation is a rule that enables us to find the ratio in which two or more ingredients at given prices must be mixed to produce a mixture of a desired price. It is the reverse of weighted average and is highly useful in solving mixture problems, especially those involving different concentrations, costs, or qualities being blended together.
Key formulas and rules
Key concepts
Understanding the Alligation Cross
The alligation cross is a visual method where the cheaper price (c) is placed at the bottom left, dearer price (d) at the top left, and mean price (m) in the centre. The differences (d - m) and (m - c) are calculated diagonally to give the mixing ratio. This method is faster than algebraic equations for simple two-ingredient mixtures.
Types of Mixture Problems
1. Mixing two different priced items (grains, liquids)
2. Dilution problems (adding water to milk, acid solutions)
3. Successive replacement (removing and replacing liquid multiple times)
4. Mixing two different concentration solutions (acid-water mixtures)
5. Profit/loss in mixtures when selling at certain rates
The Replacement Formula
When a container has 'T' litres of pure liquid and 'x' litres is taken out and replaced with water, this process repeated 'n' times: Final pure liquid = T x (1 - x/T)^n. This exponential decay formula is crucial for repeated dilution scenarios. For example, if 10 litres is removed from 100 litres and replaced 3 times, pure liquid remaining = 100 x (0.9)^3 = 72.9 litres.