Mixtures and alligation questions, with answers

The questions, with answers

1. 1.What is the rule of alligation?

   In short: Two ingredients mixed to a mean value are in the inverse ratio of their distances from it: cheaper : dearer = (dearer − mean) : (mean − cheaper).

   Rice at Rs 42 a kg is mixed with rice at Rs 57 a kg to make a blend worth Rs 48 a kg. The distances from the mean are 48 − 42 = 6 for the cheaper rice and 57 − 48 = 9 for the dearer, and the quantities are in the inverse ratio: cheaper : dearer = 9 : 6 = 3 : 2. Check it forwards: 3 kg at 42 and 2 kg at 57 cost 126 + 114 = Rs 240 for 5 kg, which is Rs 48 a kg. The mean must lie between the two values; if it does not, there is no mixture answer and the question has been misread. The rule gives a ratio — multiply it out only if the question fixes one of the quantities.

2. 2.How is alligation related to a weighted average?

   In short: It is the same equation solved the other way: a weighted average finds the mean from the quantities, and alligation finds the quantities from the mean.

   Forwards: 3 kg of tea at Rs 40 and 5 kg at Rs 56 cost 120 + 280 = Rs 400 for 8 kg, a mean of Rs 50. Backwards, alligation on the same numbers gives (56 − 50) : (50 − 40) = 6 : 10 = 3 : 5 — the quantities we started with. Because it is only a weighted average, it works for anything that averages. In a class the boys average 58 marks, the girls 67 and the whole class 62, so boys : girls = (67 − 62) : (62 − 58) = 5 : 4; in a class of 36, that is 20 boys and 16 girls. Whenever a question gives two groups and one combined average, the rule applies.

3. 3.How do you solve questions that mix solutions of different strengths?

   In short: Use each strength as the value in the alligation rule, or track the quantity of the pure substance, which simply adds across the two solutions.

   How many litres of a 20% acid solution must be mixed with 12 litres of a 45% solution to make a 30% solution? Alligation gives 20% : 45% = (45 − 30) : (30 − 20) = 15 : 10 = 3 : 2. The 45% solution is 12 litres, which is 2 parts, so a part is 6 litres and 18 litres of the 20% solution are needed. Check with the acid itself: 0.2 × 18 + 0.45 × 12 = 3.6 + 5.4 = 9 litres of acid in 30 litres, which is 30%. The acid check is worth doing every time, because it cannot be applied the wrong way round and alligation can.

4. 4.How much water must be added or removed to change a solution's strength?

   In short: The pure substance does not change when only water moves, so the new volume is the pure quantity divided by the target strength.

   24 litres of a 25% alcohol solution contain 6 litres of alcohol. To dilute it to 15%, the new volume must be 6 / 0.15 = 40 litres, so 40 − 24 = 16 litres of water are added. The reverse is evaporation: 40 litres of a 10% salt solution contain 4 litres of salt, and to make it 16% the volume must fall to 4 / 0.16 = 25 litres, so 15 litres of water must evaporate. Treating water as an ingredient at 0% gives the same answers through alligation: for the dilution, water : solution = (25 − 15) : (15 − 0) = 10 : 15 = 2 : 3, and two-thirds of 24 is 16.

5. 5.How do you solve the repeated drawing-out and replacing question?

   In short: Each replacement keeps the same fraction of the pure liquid, so after n rounds the pure quantity is original × (1 − removed/total)^n.

   A tank holds 80 litres of pure juice. 8 litres are drawn out and replaced with water, and this is done three times in all. Each round removes one-tenth of whatever is in the tank, so nine-tenths of the juice survives each round: 80 × 0.9³ = 80 × 0.729 = 58.32 litres of juice remain. The tempting wrong answer, 80 − 3 × 8 = 56, assumes every round removes 8 litres of pure juice; after the first round, the 8 litres drawn out are partly water. This is compound depreciation told as a different story — the same (1 − r)^n that values a machine losing a fixed percentage of its value each year.

6. 6.How do you combine two mixtures to reach a target ratio?

   In short: Turn each ratio into the fraction of one component, then apply alligation to those fractions: the result is the proportion in which to combine the mixtures.

   Vessel A holds milk and water in the ratio 5 : 1, and vessel B in the ratio 2 : 3. In what proportion should they be combined to get equal amounts of milk and water? Milk is 5/6 of A, 2/5 of B and 1/2 of the target. Alligation: A : B = (1/2 − 2/5) : (5/6 − 1/2) = 1/10 : 1/3 = 3 : 10. Check: 3 litres of A contain 2.5 litres of milk and 10 litres of B contain 4, so 13 litres hold 6.5 litres of milk — exactly half. Working with the fraction of one component is what makes this work; alligating the ratios 5 : 1 and 2 : 3 directly means nothing.

7. 7.How do you use alligation when the mixture is sold at a profit?

   In short: Convert the selling price back to the mixture's cost price first — alligation works on cost, never on a price that includes profit.

   A trader mixes tea costing Rs 150 a kg with tea costing Rs 200 a kg and sells the mixture at Rs 198 a kg, making a 10% profit. The cost price of the mixture is 198 / 1.1 = Rs 180 a kg. Alligation on cost: cheaper : dearer = (200 − 180) : (180 − 150) = 20 : 30 = 2 : 3. Check: 2 kg at 150 and 3 kg at 200 cost Rs 900 for 5 kg, which is Rs 180 a kg, and 10% on 180 is 198. Alligating with 198 as the mean gives (200 − 198) : (198 − 150) = 2 : 48, an implausible ratio that is the sign the profit was never removed.

8. 8.Can alligation solve questions that are not about mixtures at all?

   In short: Yes — any two groups combining into one average, provided the ratio you find is read as the ratio of whatever the average is weighted by.

   A man covers 60 km in 5 hours, partly on foot at 6 km/h and partly by bicycle at 16 km/h. His average speed is 60 / 5 = 12 km/h. Speed is distance over time, so it averages over time, and alligation gives the ratio of times: walking : cycling = (16 − 12) : (12 − 6) = 4 : 6 = 2 : 3, or 2 hours on foot and 3 by bicycle. That is 12 km walked and 48 km cycled, 60 in all. Alligating the speeds and calling the result a ratio of distances is the classic error here. The simple interest page uses the same rule for a sum split between two rates.

How the diagnostic asks it

One question from the Aptitude bank, exactly as a sitting would show it. The bank has 7 on mixtures & alligation and 84 across Aptitude.