Mixture and Alligation
What you must know
Alligation is a shortcut for combining two ingredients of different values. Arrange the cheaper price and the dearer price in a column, put the mean price in the middle, and cross-subtract: the ratio of quantities is (dearer − mean) : (mean − cheaper). This is equivalent to the weighted-average equation but far faster.
Two other tools matter. Repeated replacement: if a vessel contains V litres of liquid and x litres are removed and replaced by water, repeated n times, the remaining pure liquid is V(1 − x/V)ⁿ. And mixing with water: if milk and water are in ratio a:b, adding water changes only the water term; keep the milk quantity fixed and rescale.
Traps
- Applying alligation to prices when the quantities are given in different units.
- In replacement problems, forgetting that each removal takes out the current mixture, not pure liquid.
- Adding water to a mixture and assuming the milk quantity changes.
- Three-component mixtures: alligation works pairwise only; combine two first, then the result with the third.
- Mean price must lie between the two component prices; if it does not, the question is inconsistent.
Rule of thumb and example
Cross-subtract and simplify; always check the ratio direction (cheaper quantity corresponds to the dearer-minus-mean difference).
Worked example: In what ratio must tea at ₹240/kg be mixed with tea at ₹300/kg to get a mixture worth ₹260/kg? Differences: 300 − 260 = 40 and 260 − 240 = 20. Ratio = 40:20 = 2:1, i.e., 2 kg of the cheaper tea to 1 kg of the dearer.