Mixture and Alligation for Bank Exams: The One Rule You Need
Mixture and alligation questions for bank exams are solved with one rule: cheaper quantity : dearer quantity = (dearer price − mean price) : (mean price − cheaper price). If rice at ₹40/kg is mixed with rice at ₹55/kg to get a mixture worth ₹45/kg, the ratio is (55 − 45) : (45 − 40) = 10 : 5 = 2 : 1.
Key takeaway: Write the two values on top, the mean in the middle, and cross-subtract. The same cross works for prices, percentages of milk or alcohol, average marks and average salaries. For repeated "draw out and replace with water" questions use Final = Initial × (1 − x/V)ⁿ.
IBPS PO, SBI PO and the clerk prelims usually carry one or two direct mixture questions, and the alligation idea also hides inside profit-and-loss, average and data-interpretation sets. Every answer below was checked in Python before publishing.
What Is Alligation? (Rule of Alligation Explained)
Alligation is a quick way to find the ratio in which two ingredients must be mixed to reach a given mean value. It is simply the weighted-average formula written as a cross.
| Cheaper (c) | Dearer (d) | |
|---|---|---|
| Mean (m) | ||
| d − m | m − c |
Ratio of cheaper : dearer = (d − m) : (m − c).
Why it works: if the cheaper item is mixed in quantity x and the dearer in quantity y, then c·x + d·y = m(x + y), which rearranges to x : y = (d − m) : (m − c).
Mixture and Alligation Formulas for Bank Exams
| Situation | Formula / method |
|---|---|
| Ratio of two ingredients | (d − m) : (m − c) |
| Mean price of a mixture | (c₁q₁ + c₂q₂) ÷ (q₁ + q₂) |
| Mixture sold at a profit | First find mean CP = SP × 100 ÷ (100 + gain%), then use alligation |
| Repeated replacement | Final pure liquid = V × (1 − x/V)ⁿ |
| Two mixtures of the same liquids | Use the fraction of one liquid in each mixture as c and d |
| Adding water to change a ratio | Keep the milk constant and rebuild the new ratio |
| Groups with different averages | Alligation on the averages gives the ratio of group sizes |
Mixture and Alligation Questions with Solutions (Bank Exam Level)
Q1. Two varieties of rice cost ₹40 and ₹55 per kg. In what ratio should they be mixed to get rice worth ₹45 per kg? Ratio = (55 − 45) : (45 − 40) = 10 : 5 = 2 : 1.
Q2. Tea at ₹36/kg and ₹46/kg is mixed and the mixture is sold at ₹50/kg for a 25% profit. Find the mixing ratio. Mean CP = 50 × 100 ÷ 125 = ₹40. Ratio = (46 − 40) : (40 − 36) = 6 : 4 = 3 : 2.
Q3. 60 litres of milk and water are in the ratio 7 : 5. How much water must be added to make the ratio 1 : 1? Milk = 60 × 7/12 = 35 L, water = 25 L. For 1 : 1, water must be 35 L, so add 10 litres.
Q4. A can has 40 litres of pure milk. 8 litres are taken out and replaced with water. This is done 3 times in all. How much milk is left? Milk = 40 × (1 − 8/40)³ = 40 × (4/5)³ = 40 × 64/125 = 20.48 litres.
Q5. Vessel A has milk and water in the ratio 5 : 3, vessel B in the ratio 3 : 1. In what ratio should A and B be mixed to get milk and water in the ratio 7 : 3? Milk fraction: A = 5/8, B = 3/4, target = 7/10. A : B = (3/4 − 7/10) : (7/10 − 5/8) = (1/20) : (3/40) = 2 : 3.
Q6. The average salary of 60 employees is ₹18,000. Officers earn ₹30,000 on average and the rest ₹15,000. How many officers are there? Others : officers = (30,000 − 18,000) : (18,000 − 15,000) = 12,000 : 3,000 = 4 : 1. Officers = 60 × 1/5 = 12.
Q7. How many litres of a 20% alcohol solution must be mixed with a 50% solution to get 30 litres of a 30% solution? 20% : 50% = (50 − 30) : (30 − 20) = 2 : 1. Litres of 20% solution = 30 × 2/3 = 20 litres.
Q8. A vessel holds 80 litres of wine. Some wine is drawn out and replaced with water, and this is repeated once more. Wine and water are now 9 : 16. How much was drawn each time? Wine left = 9/25 of 80. So (1 − x/80)² = 9/25, 1 − x/80 = 3/5, x = 32 litres.
Q9. 2 kg of sugar at ₹120/kg is mixed with 3 kg at ₹145/kg and sold at ₹150/kg. Find the profit percent. Mean CP = (240 + 435) ÷ 5 = ₹135. Profit % = 15/135 × 100 = 11 1/9% (about 11.11%).
Q10. Three varieties at ₹25, ₹30 and ₹40 per kg are mixed in the ratio 1 : 1 : 2. Find the price of the mixture. (25 + 30 + 80) ÷ 4 = ₹33.75 per kg.
Replacement Formula in Mixture Questions (Step-by-Step)
The replacement formula trips up many candidates, so learn it as a habit:
- Write the capacity V and the amount removed each time x.
- Each operation keeps (1 − x/V) of the original liquid.
- After n operations: pure liquid = V × (1 − x/V)ⁿ.
- If the question gives the final ratio, convert it to a fraction of the whole and take the n-th root (see Q8).
This works only when the removed amount is replaced so the volume stays constant. If liquid is removed without replacement, just use ordinary ratios.
Mixture and Alligation Short Tricks
- Profit hidden in the question? Convert SP to mean CP first (Q2). Applying alligation on the selling price is the most common error.
- Two mixtures? Use the fraction of one liquid (milk) in each, never the raw ratio numbers (Q5).
- Adding pure water or pure milk? Water is a "mixture" with 0% milk; pure milk is 100% milk. Alligation then works directly.
- Averages: groups with averages a and b and an overall average m are in the ratio (b − m) : (m − a), the same cross (Q6).
- Check: the mean must lie between the two values. If it does not, recheck the question.
For more speed tricks that combine with these, revise simplification tricks for bank exams and the ratio work in SI and CI difference formulas.
Common Mistakes in Mixture and Alligation Questions
| Mistake | Fix |
|---|---|
| Using SP as the mean value | Mean must be the cost price of the mixture |
| Writing the ratio as dearer : cheaper | The difference opposite each value belongs to that value |
| Using ratio numbers (5 : 3) instead of fractions (5/8) | Convert to the fraction of one liquid first |
| Applying (1 − x/V)ⁿ when liquid is not replaced | The formula needs constant volume |
| Forgetting that the result is a ratio, not a quantity | Multiply by the total to get quantities |
Mixture and Alligation Quick Revision Table
| Question type | First step |
|---|---|
| Two prices, find ratio | Alligation cross on prices |
| Sold at profit | Convert SP to mean CP |
| Two milk-water mixtures | Use milk fractions |
| Add water to change a ratio | Keep milk fixed |
| Draw and replace n times | V × (1 − x/V)ⁿ |
| Group averages | Alligation on the averages |
Mixture and Alligation Practice Questions (with Answers)
- Sugar at ₹20/kg and ₹32/kg is mixed to get ₹24/kg. Ratio? Answer: 2 : 1 (8 : 4).
- 30 litres of milk and water are in the ratio 2 : 1. How much water must be added to make it 1 : 1? Answer: 10 litres (milk 20 L, water 10 L → needs 20 L water).
- From 50 litres of pure acid, 5 litres are drawn and replaced with water twice. Acid left? Answer: 40.5 litres (50 × 0.9²).
- In what ratio must 15% and 40% solutions be mixed to get 25%? Answer: 3 : 2 (15 : 10).
- A mixture of ₹50/kg and ₹70/kg items is sold at ₹66/kg with 10% profit. Ratio? Answer: 1 : 1 (mean CP = ₹60).
- Class average is 72; boys average 70, girls 75. Boys : girls? Answer: 3 : 2.
- Mixtures with milk : water 3 : 1 and 1 : 1 are mixed in equal volumes. New ratio? Answer: 5 : 3 (milk fraction 5/8).
- A 64-litre can of milk: 16 litres replaced with water, twice. Milk : water? Answer: 9 : 7 (36 L milk, 28 L water).
- 10 kg of an alloy has copper : zinc = 3 : 2. How much copper must be added to make it 4 : 1? Answer: 10 kg (zinc 4 kg stays, copper must be 16 kg).
- 40 students average 50 and 60 students average 60. Overall average? Answer: 56.
Time yourself in the IBPS PO Quantitative Aptitude Sectional Mock Test 02 and the SBI Clerk Numerical Ability Sectional Mock Test 03.
Mixture and Alligation FAQs
What is the rule of alligation?
Cheaper : dearer = (dearer value − mean) : (mean − cheaper value). It gives the ratio in which two items must be mixed to get the mean value.
How many mixture and alligation questions come in bank exams?
Usually one or two direct questions in IBPS and SBI prelims, plus the idea shows up inside arithmetic word problems and caselet DI in mains.
What is the replacement formula?
If x litres are drawn from V litres and replaced with water n times, the original liquid left is V × (1 − x/V)ⁿ.
Can alligation be used for averages?
Yes. Two groups with averages a and b and a combined average m are in the ratio (b − m) : (m − a).
Is mixture and alligation asked in SSC CGL?
Yes, mostly as one arithmetic question in Tier 1 and a few in Tier 2, so this method also helps SSC candidates.
Practice Next
- Arithmetic tests with mixture and alligation questions: SBI Clerk Arithmetic Mock Test 02 and IBPS RRB Clerk Arithmetic Mock Test 02.
- Mock tests: IBPS Clerk Numerical Ability Sectional Mock Test 02 and SBI PO Quantitative Aptitude Sectional Mock Test 01.
- New SSC topic test: SSC CGL Ratio, Proportion and Mixture Mock Test 01: Shares, Changed Ratios and Alligation (15 Questions).
- Mixed arithmetic practice: SBI Clerk Arithmetic Word Problems Test 01 and IBPS RRB Clerk Numerical Ability Sectional Mock Test 01.
- New SSC topic test: SSC CHSL Mixture and Alligation Mock Test 01.
- Related notes: boats and streams tricks for bank exams, quadratic equation tricks for bank exams, probability questions for bank exams, approximation questions for bank exams, age problems for bank exams and partnership questions for bank exams; also see ratio and proportion for SSC CGL, plus average tricks for SSC CGL.
- Exam details: SBI Clerk 2026 notification.
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