Mixtures & Alligation
Scenario
Swiggy Instamart blends two varieties of coffee beans (Rs400/kg and Rs600/kg) to create a 'house blend' that costs exactly Rs450/kg. What's the exact proportion?
Mental model
Alligation is just a visual seesaw finding the center of gravity between two different prices or concentrations.
Instead of solving simultaneous equations to find proportions, we use the differences from the target average to instantly get the mixing ratio. The farther one ingredient is from the target, the less of it you use.
Formula
Alligation ratio: Quantity(Cheaper) : Quantity(Dearer) = (Dearer Price - Mean Price) : (Mean Price - Cheaper Price). Repeated dilution: Final pure quantity = Initial x (1 - Replaced/Total)^n.
Set up a cross-diagram: write Cheaper on top-left, Dearer on top-right, Mean in middle. Cross-subtract diagonally. The left-cross gives the quantity of cheaper, the right-cross gives the quantity of dearer. Repeated dilution applies when a fixed volume is removed and replaced with another liquid n times.
Pattern discovery
Mix items priced at Rs10/kg and Rs20/kg to hit various target prices.
- Mix Rs10/kg & Rs20/kg 1:1 -> Avg Rs15/kg (equidistant from both)
- Mix Rs10/kg & Rs20/kg 1:3 -> Avg Rs17.5/kg (closer to Rs20 side)
- Mix Rs10/kg & Rs20/kg 3:1 -> Avg Rs12.5/kg (closer to Rs10 side)
What is the relationship between the ratio of quantities and their distances from the target price?
Rule: The ratio of quantities is inversely proportional to their distances from the target. Closer ingredient gets more quantity.
Worked example
Problem: In what ratio must Rs60/kg rice be mixed with Rs90/kg rice to get a mixture worth Rs70/kg?
- Set up the alligation cross: Cheaper = 60, Dearer = 90, Mean = 70.
- Cross-subtract: Dearer - Mean = 90 - 70 = 20 (quantity of cheaper).
- Mean - Cheaper = 70 - 60 = 10 (quantity of dearer).
- Ratio Cheaper : Dearer = 20 : 10 = 2 : 1.
- Sanity-check: (2 parts x 60 + 1 part x 90) / 3 = (120 + 90) / 3 = 210 / 3 = 70. Correct.
Answer: Mix in ratio 2:1 (Rs60/kg : Rs90/kg).
Problem: A 100-litre cask is full of wine. 20 litres are drawn out and replaced with water. This is done twice more (3 times total). How much wine remains?
- Use repeated dilution formula: Final wine = Initial x (1 - Replaced/Total)^n.
- = 100 x (1 - 20/100)^3 = 100 x (0.8)^3.
- (0.8)^3 = 0.8 x 0.8 x 0.8 = 0.512.
- Final wine = 100 x 0.512 = 51.2 litres.
- Sanity-check: After 1st draw: 80L wine. After 2nd: 80 x 0.8 = 64L. After 3rd: 64 x 0.8 = 51.2L. Matches.
Answer: 51.2 litres of wine remain.
Shortcuts
- Repeated Dilution: Final Quantity = Initial x (1 - Replenish/Total)^n (A 100L cask of wine has 10L drawn and replaced with water, done 3 times. Final wine = 100 x (1 - 10/100)^3 = 100 x 0.9^3 = 72.9L)
Common mistakes
- Confusing Cost and Selling Price: Always apply alligation on the Cost Price per unit. If given Selling Price and profit %, calculate CP first before alligating.
Glossary
- alligation
- A quick visual math trick to find the right mix between two different things.
- dearer
- An older word for more expensive or costing more money.
- dilution
- Making a liquid weaker or less concentrated by adding water or another fluid.
Recall questions
- In what ratio must Rs60/kg rice be mixed with Rs80/kg rice to get Rs75/kg rice?
- If 10L is removed from 50L milk and replaced with water twice, how much milk remains?
- Can you alligate volumes directly?
Questions & answers
A merchant mixes two types of tea costing Rs40/kg and Rs60/kg. He sells the mixture at Rs54/kg, making a 20% profit. What is the ratio of the mix?
3:1
Approach: SP = 54, Profit = 20%, so CP of mixture = 54 / 1.20 = 45. Alligation: (60 - 45) : (45 - 40) = 15 : 5 = 3:1.
Two vessels contain milk and water in the ratio 3:2 and 4:1. In what ratio must they be mixed to get a mixture with milk and water in the ratio 11:4?
1:2
Approach: Milk fractions: 3/5 (V1) and 4/5 (V2). Target milk fraction: 11/15. Alligation: (4/5 - 11/15) : (11/15 - 3/5) = 1/15 : 2/15 = 1:2.
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