Alligation rule
When two ingredients of prices (or concentrations) c (cheaper) and d (dearer) are mixed to get a mean m:
Quantity of cheaper : Quantity of dearer = (d - m) : (m - c)
This is the cross method: write c and d on top, m in the middle, and subtract crosswise. Target: 40 seconds.
| Cheaper c | Mean m | Dearer d |
|---|---|---|
| 30 | 36 | 45 |
| d - m = 9 | m - c = 6 |
So the ratio is 9 : 6 = 3 : 2.
Where alligation works
- Prices per kg, concentrations (% milk, % alcohol), profit percentages, interest rates, speeds (for equal times), average marks of groups.
- Any weighted average with two groups.
Replacement formula
From a vessel of V litres of pure liquid, x litres are removed and replaced with water, n times. Pure liquid left:
V x (1 - x/V)^n
Example: 40 L milk, 4 L replaced twice: 40 x (0.9)² = 32.4 L.
Mixing two mixtures
Convert each mixture to the fraction of one component (milk = 3/5 and 7/10), then alligate around the target fraction (2/3). Ratio = (7/10 - 2/3) : (2/3 - 3/5) = 1/30 : 1/15 = 1 : 2.
Adding water / fixed component
When only water is added, the amount of milk stays fixed. If milk = 48 L and must become 75% of the mixture, total = 48/0.75 = 64 L.
Worked example
A trader mixes tea at Rs. 180/kg and Rs. 240/kg in ratio 2 : 3 and sells at Rs. 270/kg. Cost of mixture = (360 + 720)/5 = 216, profit = 54/216 = 25%.
Worked example 2 (alligation on profit)
50 kg sugar sold partly at 8% profit and partly at 18% profit gives 14% overall. Ratio (18 - 14) : (14 - 8) = 4 : 6, so 30 kg at 18%.
Common traps
- Writing the ratio the wrong way round: the cheaper quantity goes with (d - m).
- Replacement: subtracting x litres n times (40 - 4 - 4 = 32) instead of using the formula.
- Using ratios of different components (milk in one, water in another).
Worked example 3 (water as the cheaper item)
In what ratio should water be mixed with milk costing Rs. 60 per litre to get a mixture worth Rs. 45 per litre? Water costs 0. Water : milk = (60 - 45) : (45 - 0) = 15 : 45 = 1 : 3.