Mixture & Alligation: The Cross Method That Saves You Minutes
Master mixtures and the rule of alligation for SSC-CGL with a simple cross-method shortcut, ratio formulas, and the classic repeated-replacement trick.
By the PadhoDost Team · 📖 7 min read · Updated 4 August 2026
Part of SSC CGL & Banking prep🧠 The chai stall test
Imagine a chaiwala mixing cheap tea leaves at ₹200/kg with premium leaves at ₹300/kg to sell a blend at ₹260/kg. He is not doing hard maths in his head. He simply feels out 'how far each price is from the target' and pours accordingly. That instinct is exactly what Alligation formalises. Once you see mixtures as a tug-of-war between two prices pulling toward a middle value, the whole chapter becomes one clean shortcut.
A 'mixture' problem gives you two (or more) ingredients of different values (price, purity, concentration, or speed) combined into one blend. Your job is usually to find the ratio in which they were mixed, or the resulting average value. Alligation is the fast tool that connects the two.
The Rule of Alligation
When two ingredients at prices (or purities) 'Cheaper' and 'Dearer' are mixed to give a 'Mean' value, the quantities are inversely proportional to their distances from the mean. Cheaper stuff sits farther, so you use less of the dearer and more of the cheaper depending on where the mean lands.
The cross method (do this in 10 seconds)
- 1Write Cheaper value on the top-left, Dearer value on the top-right.
- 2Write the Mean (target) value in the middle.
- 3Bottom-left = Dearer - Mean. Bottom-right = Mean - Cheaper.
- 4Read the two bottom numbers: that is the ratio Cheaper : Dearer.
- 5Always take positive (absolute) differences so the ratio makes sense.
📝 Worked example: the chai blend
Cheaper tea = ₹200/kg, Dearer tea = ₹300/kg, Mean = ₹260/kg.
Dearer - Mean = 300 - 260 = 40.
Mean - Cheaper = 260 - 200 = 60.
Ratio Cheaper : Dearer = 40 : 60 = 2 : 3.
So for every 5 kg of blend, use 2 kg cheap and 3 kg premium.
Check: (2 x 200 + 3 x 300) / 5 = (400 + 900) / 5 = 1300 / 5 = ₹260. Correct.
The repeated-replacement (removal) formula
A favourite SSC trap: a vessel has pure milk; you remove some, replace it with water, and repeat this a few times. After each cycle the milk gets diluted. There is a neat formula for the milk left.
📝 Removal example
A 40-litre vessel is full of milk. 8 litres is removed and replaced by water, done twice.
Fraction retained each time = 1 - 8/40 = 1 - 1/5 = 4/5.
Milk left = 40 x (4/5)^2 = 40 x 16/25 = 25.6 litres.
So water present = 40 - 25.6 = 14.4 litres.
Remember these
- ✓Alligation ratio = distances from the mean, taken inversely (Dearer-Mean : Mean-Cheaper).
- ✓The bottom-left of the cross belongs to the Cheaper quantity, bottom-right to the Dearer.
- ✓Alligation also works for purity %, concentration %, and even average speed - anything that averages.
- ✓Removal-and-replace uses (1 - r/V)^n; the fraction, not the litres, is raised to the power.
- ✓Always sanity-check by computing the weighted average back to the mean.
⚡ Quick check
In what ratio must rice at ₹30/kg be mixed with rice at ₹40/kg so the mixture is worth ₹36/kg?
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