Mixtures & Alligation
🔒 Log in to trackAlligation is the highest-leverage tool in arithmetic — it solves mixtures, dilutions, adulteration profits and even average-group questions in one crossing. Replacement chains and two-mixture blending complete the set. Expect one sure question per Tier 1 shift and more in Tier 2.
One page per subtopic: detailed notes, every question type, formulas, tricks and practice sets.
Every formula on one printable page, grouped by subtopic.
5 exam-level questions worked step by step.
66 questions — untimed practice or a timed test with analysis.
Track record in the exam
Questions per shift in recent SSC CGL papers.
Test difficulty mix (66 questions)
Question patterns exams keep repeating
Taken from previous-year papers. If a pattern is marked "very common", expect to see it in your exam.
Two-ingredient mixing (alligation)
very commonTwo unit prices with a target mean price — the mixing ratio is asked.
How to solve: Alligate: dearer : cheaper = (M − C) : (D − M); the mean must sit between the prices. Scale the arms by any known quantity.
Example: In what ratio must tea at ₹320 per kg be mixed with tea at ₹250 per kg so that the mixture costs ₹300 per kg?
Dearer : cheaper = (300 − 250) : (320 − 300) = 50 : 20 = 5 : 2.
Repeated replacement
very commonx litres drawn from C litres and replaced with water, n times — find what remains or the final ratio.
How to solve: Milk left = C(1 − x/C)^n; water = C − milk. Reverse problems take the n-th root of the final fraction.
Example: A vessel contains 80 litres of pure milk. 8 litres are drawn and replaced with water; the operation is repeated once more. Find the milk left.
80 × (9/10)² = 64.8 litres.
Add water / add ingredient
very common'How much water (or milk) must be added to reach a given strength or ratio?'
How to solve: Freeze the unchanged component, grow the total, set one linear equation. Water's 'price' is 0, so alligation also works.
Example: How much water must be added to 60 litres of milk so that milk forms 75% of the new mixture?
Milk stays 60 L and must be 75% → total 80 L → add 20 L.
Milk-water ratio bookkeeping
commonA ratio plus a total (or an operation) is given — extract component amounts or the new ratio.
How to solve: Parts × total/parts-sum = litres per component. Adding one ingredient leaves the other column untouched; a draw preserves the internal ratio.
Example: A 40-litre mixture has milk and water in the ratio 3:1. How much milk must be added to make the ratio 4:1?
Water 10 L fixed → milk must reach 40 L → add 10 L.
Adulteration profit
commonA costly item is cut with a cheap (or free) one and sold at cost or marked price — find gain % or the ratio.
How to solve: Water sold at milk's price gains water/milk × 100; 'p% of the mixture' is p/(100−p) per unit milk. For a target gain, alligate to the blend CP = SP ÷ (1 + g/100).
Example: A milkman adds water equal to one-fifth of the milk and sells the mixture at cost price. His gain per cent is:
(1/5) × 100 = 20%.
Mixing two mixtures
commonTwo ready mixtures (given ratios or strengths) are combined — final ratio or strength is asked.
How to solve: Reduce each to the key-component fraction, pool the columns. For a target strength, alligate the fractions; the target must lie between them.
Example: Vessels of milk:water 4:1 and 3:2 are mixed in equal volumes. Find the milk : water ratio of the blend.
Milk fraction = (4/5 + 3/5)/2 = 7/10 → 7 : 3.
Alligation on averages
commonTwo group averages with a known overall — class marks, speeds over equal distances, wages.
How to solve: Alligate on the averaged quantity: sizes' ratio = (A₂ − A) : (A − A₁). Equal distances alligate the times, not the speeds.
Example: The average marks of boys is 70 and of girls is 75. If the class average is 72, the ratio of boys to girls is:
(75 − 72) : (72 − 70) = 3 : 2.
Dishonest stock replacement / watered stock
commonA trader replaces sold goods with inferior stock, or sells watered milk — quantitative gain is asked.
How to solve: Run the replacement multiplier C(1 − x/C)^n for stock; for watered milk sold at cost, the gain is the water : milk ratio in per cent.
Example: A milkman mixes water equal to 25% of the milk in his can and sells the mixture at the cost price of milk. His gain per cent is:
Water : milk = 1 : 4 → gain 25%.