CAT Quant · Study & Practice

Mixtures & Alligations

AreaArithmetic DifficultyModerate CAT weightage1–3 questions (directly + folded into Ratio, Averages, Profit/Loss, SI/CI)

Mixtures & Alligations is the chapter where ratios, averages and percentages all meet in one place — which is exactly why CAT, XAT, SNAP and NMAT keep reusing it. At its heart sits one idea: whenever two things of different "strengths" (price, concentration, purity, speed, or any per-unit value) are blended, the strength of the blend is the weighted average of the two, with the quantities acting as the weights. The alligation rule is simply that weighted average read backwards — instead of finding the average from the quantities, you find the ratio of quantities from a known average. That single reversal turns a messy two-variable problem into a ten-second cross-calculation, and it is the most exam-efficient tool in all of Arithmetic. This chapter builds the idea in four steps: mixing two substances directly, the alligation cross-method, concentration and dilution (when you add or remove pure substance), and the high-yield repeated-replacement model where a fixed amount is drawn off and refilled again and again. You will also see how the same alligation cross solves average-age, average-price, average-speed and even simple/compound-interest blending problems — so the payoff extends well beyond this one chapter.

Topics

1 Mixing Two Substances 4 worked · 5 practice · 8-Q test Start → 2 Alligation Rule 4 worked · 5 practice · 8-Q test Start → 3 Concentration & Dilution 4 worked · 5 practice · 8-Q test Start → 4 Repeated Replacement 4 worked · 5 practice · 8-Q test Start →

⚡ CAT shortcuts & speed methods

The fastest ways to crack this chapter under time pressure — the techniques that separate a 95+ percentiler from the rest.

  • Alligation cross: quantity of each ingredient ∝ the distance of the OTHER ingredient from the mean — ratio = (c₂ − M) : (M − c₁).
  • When water is added free, treat its value/concentration as 0 and run the same alligation cross.
  • Repeated replacement: final = initial × (1 − x/V)ⁿ. Use the amount drawn EACH time, with V the constant total volume.
  • In dilution, the PURE substance is conserved — write every equation on the fixed pure amount, not the changing total.
  • Alligation gives only a RATIO; you need one absolute datum (a total, a difference, or one quantity) to get actual amounts.
  • Profit-on-mixture: first back out the true cost of the blend (SP ÷ profit-multiplier), then alligate around that mean.

⚠️ Common mistakes & traps

CAT is designed so that careless errors here cost you marks. Internalise each trap before the exam.

  • Using total amount drawn instead of the per-operation amount x in final = initial × (1 − x/V)ⁿ.
  • Reading the alligation ratio the wrong way round — the cheaper quantity pairs with (dearer − mean), not (mean − cheaper).
  • Adding water but forgetting the pure substance stays fixed, so basing the new % on a wrong numerator.
  • Treating the alligation ratio as actual litres/kg without using a given total to scale it.
  • In a profit/loss mixture, alligating around the selling price instead of the cost price of the blend.

📈 CAT exam insight & PYQ analysis

In recent CAT papers mixtures appear less as standalone arithmetic and more woven into ratio, averages and profit-loss sets, but the repeated-replacement model (draw x, refill with water, n times) remains a near-certain TITA or MCQ favourite across CAT, XAT, SNAP and NMAT. Examiners love two twists: giving the final milk : water ratio and asking for n or for the draw size, and chaining a mixture into a profit problem where you must first recover the blend’s cost. Difficulty is Moderate; the marks go to students who reach for the alligation cross and the (1 − x/V)ⁿ formula instantly instead of setting up linear equations.

🎴 Flashcards — instant recall

Tap a card to reveal the answer. Drill these until they are automatic.

Alligation ratio q₁ : q₂ = ?Tap to reveal
(c₂ − M) : (M − c₁)
Repeated-replacement final pure quantity?Tap to reveal
final = initial × (1 − x/V)ⁿ
Fraction of pure substance left after n replacements?Tap to reveal
(1 − x/V)ⁿ
Weighted average (mean) of a two-substance blend?Tap to reveal
(q₁c₁ + q₂c₂)/(q₁ + q₂)
Where must the mean M always lie?Tap to reveal
Strictly between c₁ and c₂
New concentration after adding w litres of water to V at c%?Tap to reveal
(V·c/100) / (V + w) × 100
Value to use for water in a price alligation?Tap to reveal
0 (it is free)
What conserved quantity drives all dilution equations?Tap to reveal
The amount of pure substance
80 L milk, draw 8 L & refill water twice — milk left?Tap to reveal
80 × 0.9² = 64.8 L
Alligation gives you a ratio or an amount?Tap to reveal
A ratio — you need one absolute datum for amounts
Mix 30 and 45 to mean 36 — cheaper : dearer ratio?Tap to reveal
(45−36):(36−30) = 9:6 = 3:2
In a mixture sold for profit, alligate around which price?Tap to reveal
The cost price of the blend (= SP ÷ profit multiplier)

📌 Quick revision

A mixture’s strength is the weighted average of its parts: M = (q₁c₁ + q₂c₂)/(q₁ + q₂), and M always lies between c₁ and c₂. Run that average backwards with the alligation cross to get the quantity ratio q₁ : q₂ = (c₂ − M) : (M − c₁) — the fastest tool in Arithmetic, but it yields only a ratio. In dilution the pure substance is conserved, so build every equation on that fixed amount. The headline result is repeated replacement: drawing x and refilling with water n times leaves final = initial × (1 − x/V)ⁿ. Watch the traps — read the cross the right way round, use the per-operation draw, and for profit-mixtures alligate around the COST of the blend.

Chapter test

📝 Mixtures & Alligations — Chapter Test
25 fresh questions · timed · full solutions
Start chapter test →
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🏆 Vidaara CAT success checklist

You have truly mastered Mixtures & Alligations when you can tick every box below.

  • Recall every formula in this chapter without looking them up
  • Solve each topic’s practice set with at least 80% accuracy
  • Use the chapter shortcuts to cut your solving time in half
  • Spot and avoid every common trap listed above
  • Score 80%+ on the timed chapter test

📋 Chapter mastery scorecard

Track where you stand. Aim for the target before moving to the next chapter.

Skill checkpointTarget
Concept theory & formulas understood100%
Topic practice sets attempted (4 topics)4/4
Best topic-test score— → 80%+
Chapter test score— → 80%+
Flashcards drilled to instant recall12 cards