Mixture and Alligation
Mixture and Alligation
Alligation is a quick way to find the ratio in which two ingredients at different prices, strengths or rates must be mixed to get a given mean. NTPC asks it in three forms: two varieties of rice or tea mixed to a mean price, two milk-water mixtures combined, and repeated removal and replacement of liquid.
1. The Alligation Rule
Cheaper : Dearer = (Dearer value − Mean) : (Mean − Cheaper value)
Write the two values on top, the mean in the middle, and cross-subtract:
| Cheaper (c) | Mean (m) | Dearer (d) |
|---|---|---|
| ₹42 per kg | ₹46 | ₹54 per kg |
| d − m = 8 | m − c = 4 | |
| Ratio cheaper : dearer = 8 : 4 = 2 : 1 |
The rule works for any quantity that averages by weight: price per kg, percentage of milk, percentage of copper, interest rate, marks, speed with equal times.
2. Mixture Formulas
| Situation | Formula |
|---|---|
| Two mixtures combined in quantities q₁ and q₂ | Milk = q₁ × (milk fraction 1) + q₂ × (milk fraction 2); same for water |
| Ratio to mix two mixtures to reach a target milk fraction f | (f₂ − f) : (f − f₁), using the MILK FRACTIONS (not the ratios) |
| x litres removed from V litres of pure liquid and replaced with water, n times | Liquid left = V × (1 − x/V)ⁿ |
| Mixture sold at a price for profit | Mean cost = selling price ÷ (1 + profit %/100), e.g. 162 ÷ 1.2 = 135; then alligate |
✗ Milk : water 7 : 3 and 2 : 3 mixed to get 1 : 1, alligate 7/3, 2/3 and 1 | ✓ Convert to milk fractions first: 0.7, 0.4 and 0.5; ratio = (0.5 − 0.4) : (0.7 − 0.5) = 1 : 2
✗ 6 litres out of 60 replaced twice leaves 60 − 12 = 48 litres of milk | ✓ The second removal takes some water too: 60 × 0.9² = 48.6 litres
हिंदी नोट: एकांतरण नियम: सस्ती : महँगी = (महँगी − माध्य) : (माध्य − सस्ती)। दो मिश्रणों को मिलाते समय अनुपात नहीं, दूध का भिन्न (fraction) लेकर एकांतरण कीजिए।
Exam Pointer: Verified NTPC patterns: two varieties of rice mixed, with total weight and cost given (June 2026 UG CBT-1), two milk-water mixtures combined (June 2022 CBT-2), and repeated removal and replacement of milk (January 2021). Mixing for profit and alligation with interest rates or alloy percentages are tagged syllabus-based.
Pariksha Pattern: Every Way NTPC Asks This Topic
Pattern 1: Two varieties mixed to a mean price
[PYQ: NTPC UG CBT-1 16-Jun-2026 Shift-1]
EXAM LEVEL
Q. In what ratio must rice at ₹42 per kg be mixed with rice at ₹54 per kg so that the mixture costs ₹46 per kg?
Cheaper : dearer = (54 − 46) : (46 − 42) = 8 : 4 = 2 : 1.
Answer: 2 : 1
EXAMATLAS LEVEL
Q. A shopkeeper mixes rice costing ₹32 per kg with rice costing ₹47 per kg to make 60 kg of a mixture worth ₹2,340. How many kg of each did he use?
Mean price = 2,340/60 = ₹39. Ratio cheaper : dearer = (47 − 39) : (39 − 32) = 8 : 7. Cheaper = 60 × 8/15 = 32 kg, dearer = 28 kg. Check: 32 × 32 + 28 × 47 = 1,024 + 1,316 = 2,340.
Answer: 32 kg at ₹32 and 28 kg at ₹47
Pattern 2: Combining two mixtures
[PYQ: NTPC CBT-2 12-Jun-2022 Shift-1]
EXAM LEVEL
Q. 3 litres of a mixture with milk and water in the ratio 3 : 1 is mixed with 4 litres of a mixture in the ratio 1 : 1. Find the ratio of milk to water in the new mixture.
First: milk 2.25 L, water 0.75 L. Second: milk 2 L, water 2 L. Total milk 4.25 L, water 2.75 L, ratio 17 : 11.
Answer: 17 : 11
EXAMATLAS LEVEL
Q. In what ratio should two mixtures with milk and water in the ratios 7 : 3 and 2 : 3 be mixed to get a mixture with milk and water in equal quantities?
Milk fractions: 7/10 = 0.7 and 2/5 = 0.4; target 0.5. Ratio = (0.5 − 0.4) : (0.7 − 0.5) = 0.1 : 0.2 = 1 : 2 (first mixture : second mixture). Check: 1 L of the first and 2 L of the second give milk 0.7 + 0.8 = 1.5 L out of 3 L.
Answer: 1 : 2
Pattern 3: Repeated removal and replacement
[PYQ: NTPC CBT-1 19-Jan-2021 Shift-1]
EXAM LEVEL
Q. A container holds 60 litres of milk. 6 litres are taken out and replaced with water, and this is done once more. How much milk is left?
Each operation keeps 54/60 = 0.9 of the milk. Milk left = 60 × 0.9² = 48.6 litres.
Answer: 48.6 litres
EXAMATLAS LEVEL
Q. A vessel contains 50 litres of pure milk. x litres are drawn out and replaced with water, and the operation is repeated twice more. The vessel now has 25.6 litres of milk. Find x.
50 × (1 − x/50)³ = 25.6, so (1 − x/50)³ = 0.512 = 0.8³. Then 1 − x/50 = 0.8 and x = 10 litres.
Answer: 10 litres
Pattern 4: Mixing for a target profit
[Pattern: syllabus-based, PYQ-style]
EXAM LEVEL
Q. Tea worth ₹120 per kg and ₹150 per kg are mixed in the ratio 2 : 3. The mixture is sold at ₹161 per kg. Find the profit percent.
Mean cost = (2 × 120 + 3 × 150)/5 = 690/5 = ₹138. Profit = 23/138 × 100 = 16 2/3%.
Answer: 16 2/3%
EXAMATLAS LEVEL
Q. In what ratio must tea at ₹126 per kg be mixed with tea at ₹156 per kg so that selling the mixture at ₹162 per kg gives a 20% profit?
Cost price of the mixture = 162/1.2 = ₹135. Ratio = (156 − 135) : (135 − 126) = 21 : 9 = 7 : 3. Alligating against 162 directly is the trap; the mean must be a cost price.
Answer: 7 : 3
Pattern 5: Alligation with percentages: interest rates and alloys
[Pattern: syllabus-based, PYQ-style]
EXAM LEVEL
Q. ₹10,000 is lent partly at 6% and partly at 10% simple interest. The total yearly interest is ₹820. How much is lent at 6%?
Mean rate = 8.2%. Ratio 6% part : 10% part = (10 − 8.2) : (8.2 − 6) = 1.8 : 2.2 = 9 : 11. At 6% = 10,000 × 9/20 = ₹4,500.
Answer: ₹4,500
EXAMATLAS LEVEL
Q. Alloy A is 60% copper and alloy B is 85% copper. How many kg of B must be mixed with 20 kg of A to get an alloy that is 75% copper?
A : B = (85 − 75) : (75 − 60) = 10 : 15 = 2 : 3. With 20 kg of A, B = 30 kg. Check: 12 + 25.5 = 37.5 kg copper in 50 kg = 75%.
Answer: 30 kg
60-Second Revision
- Cheaper : dearer = (dearer − mean) : (mean − cheaper).
- Combining mixtures: work with milk fractions, not ratios.
- Replacement: left = V(1 − x/V)ⁿ.
- For profit: mean cost = SP ÷ (1 + profit %/100), then alligate.
- Alligation works for prices, strengths, rates and any weight-averaged quantity.
Next Step: Average, Weighted Average and Mixture done. Practise these in the ExamAtlas RRB NTPC 2026 mock tests; the "work with totals" habit from this chapter also speeds up data interpretation.