60-second revision
- Alligation: cheaper quantity : dearer quantity = (dearer price - mean price) : (mean price - cheaper price).
- Mean price of a mixture = total cost / total quantity.
- Repeated replacement: pure liquid left = initial × (1 - x/V)^n, where x is removed and replaced n times from V.
- When only water is added, the quantity of milk stays fixed — build the new ratio around it.
- Alligation works for any weighted average: rates of interest, speeds, profit percentages, marks.
- A milkman who sells the mixture at the cost price of milk and gains x% has water : milk = x : 100.
What alligation does
Alligation is a quick way to find the ratio in which two ingredients at different prices (or strengths) must be mixed to get a mixture at a given mean price. It is just the weighted-average formula rearranged.
The rule
cheaper quantity : dearer quantity = (dearer - mean) : (mean - cheaper). Write the two prices on top, the mean in the middle, and cross-subtract diagonally.
Worked example: wheat at Rs 24/kg and Rs 32/kg is mixed to get Rs 27/kg. Ratio cheaper : dearer = (32 - 27) : (27 - 24) = 5:3. Check: (5 × 24 + 3 × 32)/8 = 216/8 = 27.
Mean price (the reverse question)
If the quantities are known, find the mean directly: 3 kg at Rs 40 and 5 kg at Rs 56 cost 120 + 280 = Rs 400 for 8 kg, so the mean is Rs 50/kg.
Adding one component
When water is added to milk, the milk does not change. Example: 45 litres of milk and water in the ratio 4:1 has 36 L milk and 9 L water. To make the ratio 3:2, water must be 36 × 2/3 = 24 L, so add 15 L.
Percentage version: keep fixed the component you are NOT adding. 30 litres with 20% water has 6 L of water. To bring water down to 10% by adding the other liquid, water stays 6 L, so the new total = 6/0.1 = 60 L and 30 L must be added.
Repeated removal and replacement
From a vessel of V litres of pure liquid, x litres are taken out and replaced with water, and this is done n times. Pure liquid left = V × (1 - x/V)^n. Example: 50 L of juice, 5 L replaced with water three times: 50 × (0.9)³ = 36.45 L juice left.
Mixing two mixtures
- Equal quantities: add the fractions of each component. Vessels with milk:water 3:1 and 5:3 mixed equally give milk 3/4 + 5/8 = 11/8 and water 1/4 + 3/8 = 5/8, so 11:5.
- Unknown proportion: use alligation on the fraction of one component. To get milk = 2/3 from vessel A (3/4 milk) and vessel B (5/8 milk): B : A = (3/4 - 2/3) : (2/3 - 5/8) = 1/12 : 1/24 = 2:1.
Alligation beyond mixtures
| Situation | The "prices" | Example |
|---|---|---|
| Part sold at one profit %, rest at another | Profit percentages | 50 kg sold partly at 8% and partly at 18% profit, overall 14%: (18 - 14):(14 - 8) = 2:3, so 20 kg at 8% |
| Money lent at two rates | Interest rates | Overall rate as the mean |
| Journey at two speeds | Speeds (alligate on time) | Average speed as the mean |
| Two classes with different average marks | Averages | Combined average as the mean |
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