Concept 1
A mixture combines quantities with different values
The “value” may be price per kilogram, percentage concentration, purity, marks, speed or any comparable rate. The mean value of a mixture must lie between the component values when all quantities are positive.
Balance rule: total value = sum of (quantity × component value). Mean value = total value ÷ total quantity.
Concept 2
Use a weighted average when quantities are known
Mix 6 kg rice at ₹40/kg with 4 kg at ₹55/kg. Mean price = (6×40 + 4×55)/(6+4) = ₹46/kg.
A simple average works only for equal quantities. Unequal quantities must carry unequal weights.
Concept 3
Alligation finds the required mixing ratio
To mix a cheaper value C and dearer value D to obtain mean M, the quantity ratio is cheaper:dearer = (D−M):(M−C).
| Component | Value | Opposite difference |
|---|---|---|
| Cheaper | ₹30 | 50−42 = 8 |
| Dearer | ₹50 | 42−30 = 12 |
Thus the ratio of ₹30 material to ₹50 material for a ₹42 mean is 8:12 = 2:3.
Concept 4
Track pure substance, not just total liquid
A 20-litre solution containing 30% acid has 6 litres of acid and 14 litres of water. Adding 5 litres of water leaves acid at 6 litres but makes total volume 25 litres, so the new concentration is 24%.
When two solutions are mixed, treat their percentage concentrations as component values and their volumes as weights.
Concept 5
One replacement removes both ingredients proportionally
From a well-mixed vessel of volume V, removing x litres leaves the fraction (1−x/V) of every original ingredient. Replacing the removed amount with a new liquid restores total volume but not the removed ingredient.
For a 40-litre milk vessel, remove 10 litres and replace with water. Original milk remaining = 40×(1−10/40) = 30 litres.
Concept 6
Repeated equal replacements use a retention factor
If x is removed and replaced n times from a vessel of fixed volume V, original substance remaining = initial amount × (1−x/V)ⁿ.
From 64 litres of pure milk, replace 16 litres with water twice. Milk remaining = 64×(3/4)² = 36 litres; water = 28 litres.
Worked methods
Examples with reasoning
Known quantities and prices
8 kg at ₹35 and 12 kg at ₹45 cost ₹820 total. Mean = 820/20 = ₹41/kg.
Find a mixing ratio
Mix 20% and 50% solutions to obtain 32%. Ratio low:high = (50−32):(32−20) = 18:12 = 3:2.
Find an unknown component value
5 kg at ₹24 and 3 kg at ₹x average ₹30. Then 120+3x=240, so x=₹40/kg.
Dilution with water
15 litres at 40% contains 6 litres pure substance. To make it 30%, total volume must be 6/0.30=20 litres, so add 5 litres water.
Repeated replacement
From 50 litres, replace 10 litres twice. Original liquid remains 50×(4/5)²=32 litres.
Exam traps
Errors that upset the balance
Taking a simple mean
Average component values directly only when their quantities are equal.
Writing the alligation ratio backward
Use opposite differences and keep the component order consistent.
Mean outside the range
A positive mixture mean cannot be below both values or above both values.
Removing only one ingredient
A sample removed from a uniform mixture contains ingredients in the same ratio.
Adding percentages
Convert percentages into actual pure quantities before changing volume.
Using n instead of an exponent
Repeated replacement multiplies the retention factor, producing a power.
Original practice
Check understanding after learning
Original syllabus-aligned questions—not claimed previous-year questions.
Rapid revision
The one-minute sheet
- Weighted mean = Σ(quantity × value) ÷ Σquantity.
- Cheaper:dearer = (dearer−mean):(mean−cheaper).
- The mean must lie between the component values.
- Pure substance = total mixture × concentration fraction.
- After one replacement, original fraction remaining = 1−x/V.
- After n equal replacements, original amount = initial × (1−x/V)ⁿ.
Source ledger
Official foundations
- SSC CGL official syllabus: Mixture and Alligation under Mathematical Abilities
- NCERT official mathematics exemplar: ratio and proportion foundation
- NCERT official textbook portal
Explanations, worked examples and practice are original. Official references are credited and not reproduced. Drafted 25 September 2026.
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