Alligation is a see-saw: the mix price sits between the cheap and dear prices, and the quantities are in the inverse of the distances.
Mix tea at Rs 200 a kg with tea at Rs 300 a kg and sell the mix at Rs 240. How much of each did you use? Alligation answers that with one cross.
Mixture questions (milk and water, two kinds of rice, alloys, even a class's average) all reduce to the alligation cross or the removal formula.
Write the cheaper and dearer prices at the top, the mixture's price in the middle. Cross-subtract: cheaper : dearer = (dearer - mean) : (mean - cheaper).
It works for any weighted average: prices, percentages of milk, speeds over equal times, marks of two classes.
In what ratio must tea at Rs 200/kg be mixed with tea at Rs 300/kg so the mixture is worth Rs 240/kg?
Answer: 3 : 2
A vessel has × litres of milk. y litres are taken out and replaced with water, n times. Milk left = x (1 - y/x)n.
Each time, the same fraction of whatever milk is left goes out.
A 40-litre can is full of milk. 4 litres are removed and replaced with water; this is done 3 times. How much milk is left?
Answer: 29.16
Add up each ingredient separately. Mixture 1 has milk : water = 3 : 1 and mixture 2 has 5 : 3; take equal amounts, say 8 litres each: milk 6 + 5 = 11, water 2 + 3 = 5. New ratio 11 : 5.
Two vessels have milk and water in ratios 3 : 1 and 5 : 3. Equal quantities are mixed. New ratio?
Answer: 11 : 5
Use it when: Two prices, one mean.
Rice at 60 and 75 mixed to sell at 65. Ratio?
Answer: 2:1
A 40 L can of milk: 4 L removed and replaced with water, three times. Milk left (L)?
Answer: 29.16
60 L mixture of milk:water = 2:1. Water to add to make it 1:2?
Answer: 60 L
Alloys gold:copper 7:2 and 7:11 are mixed in equal amounts. Gold:copper in the new alloy?
Answer: 7:5
Teas at Rs 126 and 135 per kg are mixed with a third variety in 1:1:2. The mix is worth Rs 153/kg. Price of the third?
Answer: Rs 175.5
10 questions with full solutions.