Averages, Mixtures, Alligation for CAT 2026: 15 PYQs
Averages, mixtures, and alligation together produce 3 to 4 CAT questions every paper, and yet dedicated cheatsheet content for this combined cluster barely exists online. Most aspirants treat these three sub-topics as separate, miss the underlying weighted-average logic that unifies them, and leave 3 to 4 reliable QA marks on the table every cycle. The averages CAT 2026 question type rewards recognition more than algebra, and the see-saw method visual cuts solve time on every weighted-average problem by 60 to 70 percent. This 18-formula cheatsheet maps the averages-mixtures-alligation cluster, walks through the see-saw method, and closes with 15 CAT-level PYQs across all three sub-topics.
The mixtures alligation formulas CAT 2026 question pattern leans on three recognition cues: a weighted-average problem stated directly, a mixture composition asked in reverse (alligation), and a successive replacement scenario (compound dilution). The 18 formulas below are organised so each maps to one of these cues, not to a textbook chapter order. The CAT previous year papers 2017-2025 reference shows the frequency of each pattern across 9 cycles.
Why Averages-Mixtures-Alligation is a Top-3 ROI CAT QA Cluster
Three reasons make this cluster a top-3 ROI target in CAT 2026 QA preparation. The cluster never goes below 3 questions in any CAT paper. The 18 formulas can be drilled in 3 to 4 study days for engineers and 6 to 8 days for non-engineers. And the see-saw weighted-average shortcut reduces solve time below 60 seconds for the most common question type.
Compared to harder QA topics, this cluster has the lowest cognitive load per mark. CAT QA syllabus data shows it sits in the top-5 most-tested arithmetic clusters along with TSD, Percentages, Ratio-Proportion, and Profit-Loss. Aspirants who skip it as "too easy" forget that easy-to-medium questions in CAT are precisely where the sectional percentile gets built.
Averages CAT 2026: The 18 Formulas Across Three Blocks
Block 1 — Averages (6 formulas)
Recognition cue: any problem asking for arithmetic mean, weighted mean, or aggregate of a group.
| # | Formula | Use case |
|---|---|---|
| 1 | Average = Sum of terms / Number of terms | Basic mean definition. |
| 2 | Sum = Average × Count | Reverse formula when total needed. |
| 3 | Weighted Avg = (q1·v1 + q2·v2) / (q1 + q2) | Two-group weighted mean. |
| 4 | Avg of first n natural numbers = (n + 1) / 2 | Speed shortcut for sequence averages. |
| 5 | Avg of n consecutive integers from a = a + (n − 1) / 2 | Consecutive integer averages. |
| 6 | New avg after addition/removal = (Old sum ± added/removed value) / (n ± 1) | Group addition or removal problems. |
Block 2 — Mixtures (5 formulas)
Recognition cue: two or more components combined; find composition or concentration.
| # | Formula | Use case |
|---|---|---|
| 7 | Mixture concentration = (Sum of pure component) / (Total mixture) | Concentration from quantities. |
| 8 | Quantity of pure A in mixture = Concentration × Total quantity | Reverse from concentration. |
| 9 | If x litres pure A added to y litres mixture with concentration c, new concentration = (x + cy) / (x + y) | Adding pure component. |
| 10 | 3-component mixture concentration = (q1·c1 + q2·c2 + q3·c3) / (q1 + q2 + q3) | Three or more components. |
| 11 | Successive dilution: Final pure = Initial × (1 − removed/total)n | n-time replace-and-refill scenarios. |
Block 3 — Alligation (7 formulas + 3 types)
Recognition cue: given final mixture value, find component ratio.
| # | Formula | Use case |
|---|---|---|
| 12 | Alligation rule: q1 / q2 = (v2 − vmean) / (vmean − v1) | Standard alligation ratio. |
| 13 | vmean is always between v1 and v2; if not, no solution | Validity check before solving. |
| 14 | Price alligation: q1 / q2 = (price2 − pricemean) / (pricemean − price1) | Cost-based alligation (Type 1). |
| 15 | Concentration alligation: q1 / q2 = (c2 − cmean) / (cmean − c1) | Quality-based alligation (Type 2). |
| 16 | Replacement alligation: same rule applied to before-and-after concentrations | Successive dilution as alligation (Type 3). |
| 17 | 3-component alligation: apply alligation rule twice, between adjacent components | Three or more components. |
| 18 | Profit alligation: cost ratio mirrors profit-margin difference ratio | Hidden alligation in profit-loss problems. |
The See-Saw Method: Visual Shortcut for Weighted Average
The see-saw method visualises weighted average as a balance beam. The mean value sits at the fulcrum; the two component values sit at the ends. The distances from the fulcrum represent (vmean − v1) and (v2 − vmean). The quantities act as weights at each end. The balance condition states quantity1 × distance1 equals quantity2 × distance2.
distance from mean: (vmean − v1) (v2 − vmean)
quantities: q1 q2
Example: A trader mixes rice at Rs 30/kg with rice at Rs 50/kg to sell at Rs 38/kg. Find the quantity ratio. Using see-saw: distance left = 38 − 30 = 8; distance right = 50 − 38 = 12. Ratio q1:q2 = 12:8 = 3:2. The trader uses 3 parts of Rs 30 rice for every 2 parts of Rs 50 rice. Solve time: 25 seconds. No algebra needed.
3-Type Alligation Recognition for CAT 2026
CAT 2026 tests three alligation types. Recognition before solving cuts time per question by 30 to 40 percent.
| Type | Recognition cue | Variable interpretation | Typical CAT frequency |
|---|---|---|---|
| Type 1: Price | Cost or selling price per unit; trader mixes goods | v = price per unit; q = quantity | 1-2 per paper |
| Type 2: Concentration | Percentage purity, alcohol-water, milk-water mixtures | v = concentration %; q = volume | 1 per paper |
| Type 3: Replacement | "Removes x litres, replaces with water, repeats n times" | v = before/after concentration; q = mixture, water | 1 in 2-3 papers |
Type 3 (replacement) is the highest-difficulty alligation type because it combines the alligation rule with the successive dilution formula. Formula 11 and Formula 16 work together for these questions: first apply the successive dilution formula to find the post-replacement concentration, then apply the alligation rule if the question asks for the additive ratio. CAT 2018 and CAT 2024 both included one Type 3 question in QA, each scoring 1 mark with a 90 to 120 second solve target under the see-saw plus dilution combined approach.
15 CAT-Level Questions With Solutions
Avg of 5 numbers is 18. If one number is removed, avg becomes 16. Find the removed number.
Old sum = 5 × 18 = 90. New sum = 4 × 16 = 64. Removed = 90 − 64 = 26. Answer: 26
A class of 30 has avg height 156 cm. A new student of height 168 cm joins. Find new avg.
Old sum = 30 × 156 = 4680. New sum = 4680 + 168 = 4848. New avg = 4848 / 31 ≈ 156.4 cm. Answer: ~156.4 cm
A class has 20 boys with avg 70 marks and 30 girls with avg 75 marks. Find the combined avg.
Weighted avg = (20·70 + 30·75) / 50 = (1400 + 2250) / 50 = 3650/50 = 73. Answer: 73
Find the average of the first 50 natural numbers.
Formula 4: (n + 1) / 2 = 51 / 2 = 25.5. Answer: 25.5
A 40-litre milk-water solution is 60 percent milk. Find the litres of milk.
Milk = 0.60 × 40 = 24 litres. Answer: 24 litres
20 litres of 60 percent alcohol solution. 5 litres of pure alcohol added. Find new concentration.
Old alcohol = 12 L. New alcohol = 17 L. Total = 25 L. New concentration = 17/25 = 68%. Answer: 68%
A vessel has 80 litres milk. 10 litres removed and replaced with water, repeated 3 times. Find final milk quantity.
Formula 11: Final = 80 × (1 − 10/80)3 = 80 × (7/8)3 = 80 × 343/512 ≈ 53.6 L. Answer: ~53.6 L
Rice at Rs 30/kg mixed with rice at Rs 50/kg to sell at Rs 42/kg. Find the ratio.
See-saw: distances 12 and 8. Ratio q1:q2 = 8:12 = 2:3. Answer: 2:3
25% alcohol solution mixed with 75% alcohol solution to get 45% alcohol. Find the ratio.
See-saw: distances 20 and 30. Ratio q1:q2 = 30:20 = 3:2. Answer: 3:2
A shopkeeper sells two grades of tea at Rs 18 and Rs 24 per kg. Average sale price Rs 21. Cost is Rs 15 and Rs 20 respectively. Profit margins are 20% and 20%. Find quantity ratio.
Standard price alligation: distances 3 and 3. Ratio 1:1. Answer: 1:1
Avg score of a cricketer in 10 innings is 50. Avg rises to 51 after the 11th innings. Find the score in the 11th innings.
Formula 6: 11th score = 11 × 51 − 10 × 50 = 561 − 500 = 61. Answer: 61
Three solutions of acid concentration 20%, 30%, 50% mixed in ratio 2:3:5. Find mixture concentration.
Formula 10: (2·20 + 3·30 + 5·50) / 10 = (40 + 90 + 250) / 10 = 380/10 = 38%. Answer: 38%
A trader wants to mix rice at Rs 20/kg and Rs 30/kg to sell at Rs 35/kg. Find the ratio.
Formula 13 validity check: vmean = 35 is greater than v2 = 30. No solution exists. Answer: Impossible (vmean outside [v1, v2])
A vessel has 60 L of 80% alcohol. 15 L removed and replaced with water. Find new concentration.
Formula 11: Alcohol fraction = 0.80 × (1 − 15/60) = 0.80 × 0.75 = 0.60. New concentration = 60%. Answer: 60%
Avg weight of 15 students is 40 kg. A new student joins, raising avg to 41 kg. Find the new student's weight.
New sum = 16 × 41 = 656. Old sum = 15 × 40 = 600. New student weight = 656 − 600 = 56 kg. Answer: 56 kg
Three Mistakes Aspirants Make
- Algebra-first thinking. Setting up two-variable equations for every alligation problem instead of applying the see-saw rule. The fix is to name the cue first and reach for the rule before the variables.
- Skipping validity checks. Attempting alligation when vmean is outside the [v1, v2] interval. Formula 13 takes 3 seconds to check and eliminates impossible scenarios before solving.
- Forgetting the successive dilution formula structure. Solving Type 3 replacement problems iteratively (one replacement at a time) instead of applying Formula 11 in one step. The compound formula saves 60 to 90 seconds per question.
Where Averages-Mixtures-Alligation Fits in the CAT 2026 Plan
This cluster sits in the early Arithmetic block of the QA preparation sequence, typically weeks 2 to 4 of a 7-month plan. The 18 formulas can be drilled in 3 to 4 study days for engineers and 6 to 8 days for non-engineers. Pair with the Ratio and Proportion cheatsheet since alligation reduces to ratio calculation. The Optima Learn CAT preparation hub sequences the arithmetic cluster, and the CAT 2026 personalised planner adjusts the per-topic days based on diagnostic results.
Lock the Averages-Mixtures-Alligation Reflex Into Your CAT 2026 Stack
Stop solving alligation problems with two-variable algebra. Move to see-saw recognition in under 30 seconds per question and unlock 3 to 4 reliable QA marks every CAT paper.
Wire My Alligation ReflexCommon Doubts About Averages, Mixtures, and Alligation
Are CAT alligation questions getting harder?
The base difficulty has been stable across CAT 2017-2025, but the integration with other QA topics has deepened. Recent papers overlay alligation on profit-loss (Q10 style), on percentages (concentration to percentage gain), and on time-speed-distance (mixture of two speeds over different durations). The fix is recognising the alligation layer hidden inside other problem types and reaching for the see-saw rule whenever a weighted-mean structure surfaces, regardless of how the question is wrapped.
Should I memorise the see-saw method?
Memorise the visual, not the equation. Once the see-saw mental model is anchored, the algebra emerges naturally for any 2-component problem. For 3-component problems, apply the see-saw rule twice between adjacent components, using the first two components to find an intermediate weighted average and then combining that with the third component. The visual approach also makes validity-check Formula 13 instinctive: if vmean is outside the interval, the see-saw cannot balance.
How many PYQs should I solve in this cluster?
Aim for 25 to 30 PYQs across CAT 2017-2025, roughly 10 averages, 8 mixtures, and 12 alligation problems. The Optima Learn questions library tags each PYQ with the cue type and trap structure, which speeds up Phase 1 untimed drilling because the recognition cue is visible before the solve attempt begins.
Where does this cluster overlap with profit-loss?
Profit-loss problems often hide an alligation structure when the trader sells two grades at one combined price. Formula 18 covers this overlay. Recognising the hidden alligation cuts solve time by 40 to 50 percent on these hybrid problems. The CAT preparation blogs library has the dedicated profit-loss cheatsheet for the rest of the structure.
Final note. Averages CAT 2026 reduces to 18 formulas across three blocks, unified by the weighted-average logic. The see-saw method is the highest-leverage shortcut in the cluster. Drill formulas block-by-block, run 25 to 30 PYQs, and the reflex compounds across the May-to-November plan for 3 to 4 reliable marks every CAT paper.
