Last Updated: May 2026
Mixtures and Alligation is a high-yield IPMAT 2027 quant topic, contributing 1–2 questions in nearly every IIM Indore IPMAT and JIPMAT paper. The chapter is a classic example of where the alligation rule converts a tedious word problem into a 30-second mental calculation. Once you internalise the cross-multiplication trick, mixture problems become free marks.
The Alligation Rule — One Sentence
When two ingredients with prices/qualities a and b are mixed to produce a mixture of mean price/quality m, the ratio in which they should be mixed is:
(b − m) : (m − a)
where a is the cheaper ingredient and b the dearer. Alternatively, draw a cross diagram with a top-left, b top-right, m in the middle, and the two diagonals give (b − m) and (m − a).
Quick Reference Table — Mixture Templates
| Scenario | Approach |
|---|---|
| Two-ingredient mixture for given mean price | Direct alligation rule |
| Replacement (remove x L, add x L pure water repeatedly) | Final = Initial × (1 − x/V)ⁿ |
| Mixing two mixtures of different ratios | Convert each to component fraction, weight by quantity |
| Three or more ingredients | Apply alligation pairwise |
| Profit-loss with mixture | SP = CP × (100 + profit%)/100; back-solve mean price |
The Replacement Formula
If a vessel contains V litres of liquid X and you remove x litres and replace it with water (or another liquid), repeat n times. The amount of original X remaining is:
Final X = V × (1 − x/V)ⁿ
This is a workhorse formula. IPMAT problems often combine it with finding a percentage or ratio at a specified step.
Worked Example 1 — Pure Alligation
Problem: A trader has tea worth ₹70/kg and ₹95/kg. In what ratio should he mix them to get a blend worth ₹85/kg?
Solution: Cheaper a = 70, dearer b = 95, mean m = 85. Ratio = (b − m) : (m − a) = (95 − 85) : (85 − 70) = 10 : 15 = 2 : 3.
Worked Example 2 — Replacement
Problem: A 20-litre vessel contains pure milk. 2 litres are removed and replaced with water. This is repeated 3 times. What fraction of the final mixture is milk?
Solution: Final milk = 20 × (1 − 2/20)³ = 20 × (9/10)³ = 20 × 0.729 = 14.58 L. Fraction of milk = 14.58/20 = 72.9%.
Worked Example 3 — Profit Mixed In
Problem: A grocer mixes rice worth ₹40/kg and ₹50/kg in ratio 3:2. He sells the mixture at ₹50/kg. Find his profit %.
Solution: Cost price of mixture = (3 × 40 + 2 × 50)/(3+2) = (120+100)/5 = ₹44/kg. SP = 50. Profit % = (50−44)/44 × 100 ≈ 13.64%.
Five Trick Patterns IPMAT Loves
- Disguised alligation — students fail to recognise alcohol/water concentration problems as alligation
- Three-ingredient mixing — split into two pairs, apply alligation pairwise
- Wrong identification of cheaper/dearer — alligation requires a < m < b; if not, swap
- Replacement vs simple removal — formula only works when removed amount is replaced (volume constant)
- Profit-on-cost vs profit-on-SP — IPMAT mixes both within the same problem; read carefully
30 Practice Problems — IPMAT Mixtures and Alligation
[cg_quiz id=”cg-ipm-mixtures-2027″]
Frequently Asked Questions
When does the replacement formula apply?
When a fixed volume V of liquid X has x litres removed and immediately replaced by another liquid (typically water), with the process repeated n times. After n repetitions the remaining X equals V × (1 − x/V)ⁿ.
Can alligation be used for three ingredients?
Yes — split into two pairs. Combine any two ingredients first to get a known mean, then alligate that mean with the third ingredient to reach the final target.
What happens if mean price equals one of the ingredients?
The mixture must be entirely that ingredient. The other ingredient’s share is zero. Algebraically (m − a) or (b − m) becomes zero, giving ratio 1:0 or 0:1.
Can I use the rule for percentages instead of prices?
Yes. The rule applies to any concentration measure — alcohol percentage, salt percentage, average marks, average age, etc. The key is that all three quantities (a, b, m) are measured on the same scale.
Continue Your IPMAT 2027 Prep
Bottom line: Memorise the alligation rule, the replacement formula, and the five trick patterns. Practise 30 problems till the cross-diagram becomes automatic — IPMAT mixture questions then take under 30 seconds each.