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Set Theory
Every CAT Set Theory formula on one page — Venn diagrams, De Morgan's laws, cardinality formulas, and set identities — each with a worked example.
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Mixtures & Alligations
CAT'26 QUANT CHEATSHEET
Every mixtures & alligations formula you need for CAT 2026 — on one page.
Mixtures and Alligations on CAT is really one idea wearing several costumes: milk and water, two grades of alloy, cheap and expensive stock, or a solution being diluted over repeated replacements. Every one of these collapses to the same alligation ratio once you identify the higher value, the lower value, and the target mean. This sheet lays out every formula you need for the CAT quant section: weighted averages, the alligation rule itself, concentration and pure-quantity tracking, the repeated-replacement formula, and the profit-through-mixing pattern examiners keep reusing, each with a worked example in real numbers. Keep it open while you practise, and after a mock check where you stand on the CAT score predictor to see which idea is costing you marks.
Dev note: the canonical route
/cheatsheets/optima-learn-mixtures-alligations-cheatsheet is not live yet
(no /cheatsheets hub in the current sitemap). Built production-ready as a drop-in once it ships.Mixtures & Alligations: every formula you need
1Core Idea
A mixture combines two or more quantities into one.
Questions cover concentration, replacement, average value, profit mixing and ratio determination
Example: milk and water combined, or two grades of an alloy blended, are both mixture setups.
CAT Hack: Most mixture questions solve faster with alligation than with algebra.
2Weighted Average Concept
The average of a mixture weighted by each part's quantity.
Average = (Total Value) ÷ (Total Quantity)
Example: 2L milk at ₹40/L and 3L at ₹60/L → (2·40+3·60)/5 = ₹52/L.
3Alligation Rule
Turns a target average into a clean mixing ratio.
Ratio = (Higher − Mean) : (Mean − Lower)
Example: ₹80/L milk mixed with ₹50/L milk to give ₹60/L → (80−60):(60−50) = 20:10 → Costlier:Cheaper = 1:2.
CAT Favourite: The single most important formula in this chapter — almost every mixture question reduces to it.
4Concentration Formula
What fraction of the mixture is the substance of interest.
Concentration % = (Solute ÷ Mixture) × 100
Example: 20L solution with 5L alcohol → 5/20 × 100 = 25%.
5Mixing Two Solutions
Add up the pure substance from each part separately.
Pure Quantity = Quantity × Concentration
Example: 10L of 20% + 20L of 40% → 10·0.20 + 20·0.40 = 2+8 = 10L pure alcohol.
6Replacement Formula
Repeatedly removing and replacing part of a mixture.
Final Quantity = Initial × (1 − x/V)n (V=total volume, n=operations)
Example: 40L milk, 10L removed and replaced twice → 40·(3/4)2 = 22.5L milk left.
CAT Insight: x/V is the fraction removed each time, not the amount — always divide by the total volume V.
7Profit Through Mixing
Blend a cheap and an expensive variety, then sell at the average price.
Profit % = (SP − CP) ÷ CP × 100
Example: mix expensive and cheap varieties, sell at the average price → profit comes from ratio manipulation, not the individual prices.
CAT Hack: The classic CAT setup: mix, sell at the blended average, and the profit hides in the mixing ratio.
8Alcohol-Water Shortcut
The two components of a solution always add to 100%.
Alcohol % + Water % = 100%
Example: if alcohol is 35% of a solution, water must be 65% — no extra calculation needed.
9Alligation with Percentages
The same ratio rule, applied directly to percentage solutions.
Ratio = (Higher% − Required%) : (Required% − Lower%)
Example: 20% and 50% solutions mixed to give 30% → (50−30):(30−20) = 20:10 = 2:1.
10Reverse Compounding
The replacement formula viewed from the other side.
Fraction Left = (1−x/V)n | Fraction Removed = 1 − (1−x/V)n
Example: after 2 replacements with x/V = ¼, fraction left = (3/4)2 = 9/16.
CAT Favourite: Frequently tested in CAT — don't confuse fraction left with fraction removed, they are complements.
CAT exam shortcuts, traps & revision
11
CAT Exam Shortcuts
- Use alligation whenever a mixing ratio is asked
- Think in weighted averages, not simple averages
- Memorise the replacement formula: Initial × (1−x/V)^n
- Always track the pure quantity, not just the total volume
- Convert percentages into fractions before combining them
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Most Common CAT Traps
- Reversing the alligation ratio (costlier and cheaper swapped).
- Averaging the two values directly instead of using alligation.
- Forgetting to compute the pure quantity separately for each part.
- Using volume where concentration was actually required.
- Confusing fraction left with fraction removed in replacement problems.
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30-Second Revision Box
- Average = Total Value ÷ Total Quantity
- Alligation ratio = (Higher−Mean) : (Mean−Lower)
- Concentration = Solute÷Mixture ×100
- Pure Quantity = Quantity × Concentration
- Replacement = Initial × (1−x/V)^n
- Use alligation for speed; always track the pure substance
This topic rewards recognising the alligation pattern over setting up equations from scratch — once you spot the higher value, lower value and mean, the ratio falls out in one line. Drill this sheet until the alligation rule and the replacement formula are reflex, then test them on full sets and track progress with the CAT score predictor. For more topic guides, browse the Optima Learn blog or explore every study guide, and work through the full CAT exam hub for section-wise strategy. When you want structured, mentor-led prep, the team at Optima Learn can map out your plan — book a free CAT 2026 call and line up your next eight weeks.
