Math · Aptitude · Chapter 10

🥤 Mixture & Alligation

Mixing two prices/qualities — the alligation rule.

💡 Mixing Two Things at Different Prices

When you mix two ingredients (or items) at different prices/qualities, the average of the mixture lies between them.

Alligation is a fast way to find the ratio in which to mix.

Cheaper : Costlier = (CP of costlier − Mean) : (Mean − CP of cheaper)

Example: Tea A costs ₹20/kg. Tea B costs ₹30/kg. We want a blend at ₹24/kg. In what ratio should we mix?
Ratio A:B = (30 − 24) : (24 − 20) = 6 : 4 = 3 : 2.

Replacement / dilution problems — a classic SSC type:

After n replacements: pure portion = (1 − x/Total)ⁿ × initial

x = quantity replaced each time, n = number of times. The pure substance keeps reducing exponentially.

⚡ Cross visualization

Place cheaper price (top-left) and costlier price (top-right). Put mean below. Cross-subtract:

20 ___ 30
    \ /
   24
   / \
(30−24)=6    (24−20)=4

Ratio of cheaper : costlier = 6 : 4 = 3 : 2.

⚡ Milk-water dilution

A jar has 40L pure milk. 4L is replaced by water each time, repeated 3 times. How much milk left?

Use (1 − x/T)ⁿ: (1 − 4/40)³ × 40 = (0.9)³ × 40 = 0.729 × 40 = 29.16 L milk remaining.

⚡ Two mixtures combined

Mix X is 30% sugar. Mix Y is 60% sugar. Combine to get 50% sugar. Find ratio?
X : Y = (60 − 50) : (50 − 30) = 10 : 20 = 1 : 2.

🎬

The Alligation Cross

Animation

The differences "cross over" — that's why it's called the alligation cross.

🧮

Mixture Calculators

Calculator

⚗️ Alligation

Cheaper price ₹ · Costlier ₹ · Mean ₹

Cheaper : Costlier = 3 : 2

🥛 Milk-water dilution

Container has L milk. Each time L is replaced by water, times.

Milk remaining: 29.16 L · Water now: 10.84 L

Practice (SSC): In what ratio must rice at ₹30/kg be mixed with rice at ₹45/kg so that mixture is worth ₹35/kg?

By alligation: cheaper : costlier = (45 − 35) : (35 − 30) = 10 : 5 = 2 : 1.

Practice (Banking): A 60L mixture of milk and water has 80% milk. How much water needs to be added to make it 60% milk?

Pure milk = 80% of 60 = 48L. Water = 12L. We don't add milk — only water. So 48L milk should be 60% of new total. New total = 48/0.6 = 80L. Water needed = 80 − 60 = 20L.

Practice (CAT-level): A jar contains 60L of milk. 6L of milk is removed and replaced with water. This is done twice more. What's the % of milk left?

Use (1 − x/T)ⁿ × 100%. (1 − 6/60)³ × 100 = (0.9)³ × 100 = 0.729 × 100 = 72.9% milk left.

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Ages & Averages