Mixture and Alligation is one of those topics where SBI PO loves to hide a five-step word problem inside what should be a fifteen-second calculation. The moment you spot it for what it is, the problem collapses.
Here is the core idea: you are blending two (or more) things — milk and water, two varieties of rice, two types of chemical solutions — and you want to know either (a) the resulting concentration or cost, or (b) in what proportion you must take the two things to hit a target average.
The word "alligation" itself comes from the Latin "alligare" (to bind together). Think of it as the rule that binds two averages into one.
The everyday analogy: Imagine pouring ₹10 tea and ₹16 tea into one kettle to sell at ₹13 a cup. You do not pour equal amounts — you pour more of the cheaper tea. The alligation rule tells you exactly how much more. It is just a structured shortcut for the weighted average formula, expressed visually as a cross.
The weighted average formula underneath it all is:
Alligation rearranges this into:
That ratio on the right — (higher value minus mean) : (mean minus lower value) — is the alligation cross, and it is the only thing you actually need to memorize.
The "mixture" part of the topic covers a separate but related situation: a container has a mixture, you remove some, add something pure (or another mixture), and you need to track concentrations through multiple steps. The two sub-types look different but both reduce to the same ratio logic.
Draw a cross. Put the mean (target) value in the centre. Put the two ingredient values at the top-left and top-right. Subtract diagonally — always subtract the centre from the corners (and keep the result positive). The two results at the bottom are the quantities in the required ratio.
A₁ A₂
\ /
\ /
A_mean
/ \
/ \
(A₂ - A_mean) (A_mean - A₁)
The ratio of ingredient 1 to ingredient 2 is (A₂ - A_mean) : (A_mean - A₁).
Example: Two varieties of rice priced at ₹40/kg and ₹55/kg are mixed to get a mixture priced at ₹46/kg.
Done. No equation, no algebra.
Type 1 — Find the resulting average/concentration
Two known quantities of known averages are mixed. Find the new average. Use the weighted average formula directly:
This is rarely asked in isolation at SBI PO level — it is usually embedded inside a harder problem.
Type 2 — Find the mixing ratio given the target
This is the classic alligation cross problem. You know both ingredient values and the target average. Apply the cross, read off the ratio.
Key point: the "value" can be a price per kg, a concentration fraction, a percentage, even a speed — as long as the same quantity is being mixed.
Type 3 — Repeated dilution / removal problems
A container holds a mixture. You remove a fraction, replace it with pure water (or another liquid), and repeat. The formula for the final concentration of the original liquid after n such operations:
Where r is the volume removed each time and V is the total volume of the container (constant, because you top it up).
Look — this formula is worth memorizing exactly as written. In SBI PO, removal problems are sometimes disguised: they say "25% mixture is removed" instead of giving you a fraction explicitly. "25% removed" means r/V = 1/4, so the multiplier is 3/4.
When the problem involves concentration ratios (like water:alcohol = 4:3), convert to a single fraction first before applying the alligation cross.
Water fraction in a 4:3 water-alcohol mixture = 4/(4+3) = 4/7.
Then apply alligation on the fractions. This is the most common source of errors — students try to apply the cross to the ratio numbers (4 and 3) directly, which is wrong.
When three ingredients are involved, SBI PO typically gives you a constraint that reduces it to a two-ingredient alligation. Read carefully: one ingredient's amount is usually fixed, and you solve for the other two. Alternatively, pair any two of the three and solve sequentially.
A very common SBI PO pattern: the selling price and profit percentage are given, so the cost price of the mixture is implied. Compute that CP first, then apply alligation on the two ingredient CPs. This is exactly what the 2019 gram question does (see Solved PYQs below). Always extract the mean CP before touching the alligation cross.
When applying the alligation cross, always subtract diagonally toward the center, never away from it. To avoid sign errors: write (Higher − Mean) and (Mean − Lower) and label them before assigning to ratios. This eliminates the single most common mistake in alligation — swapping which ratio belongs to which ingredient. Standard method with algebra: ~50 seconds. Alligation cross with this labeling habit: ~12 seconds. The ratio you get directly maps to the ingredient at the opposite diagonal corner.
Whenever a problem gives a concentration as a ratio (e.g., milk:water = 3:2), convert to a fraction (milk fraction = 3/5) before applying the cross. Write the fraction in a common denominator form if you are comparing two mixtures — the subtraction in the cross becomes trivial and you avoid mis-subtraction of ratio numbers. Example from the 2019 PYQ: water fractions 4/7, 2/3, and 13/21 — once you express all three over 21, the cross subtraction is (14−13):( 13−12) = 1:1. No algebra needed. Standard approach tracking both components separately: ~90 seconds. This approach: ~25 seconds.
If a problem gives SP and profit %, compute CP of mixture first: CP = SP × 100 / (100 + profit%). Only then apply the alligation cross. Attempting to apply alligation on the SP directly — a trap students walk into — gives a wrong ratio every time. The step count: (1) CP = SP×100/(100+profit%), (2) write alligation cross with ingredient CPs and mixture CP, (3) read ratio. Three steps, no equation. Standard algebraic setup: 6-7 steps with two unknowns.
For repeated removal problems, chain the multiplier: if 25% is removed each time, the multiplier is 3/4. After n removals and replacements, multiply the original quantity by (3/4)^n. For a single removal (n=1) this is just direct multiplication — no formula needed. For n=2, square the fraction. Commit (3/4)² = 9/16 and (2/3)² = 4/9 to memory — these two appear most often. Translating a word problem into the formula: ~15 seconds. Setting up simultaneous equations from scratch: ~75 seconds.
Alligation applies any time the question asks "in what ratio" or "how much of each" when two sources combine to a known average. The disguise: the "average" might be a speed, an age, a score, a profit percentage — not just a price or concentration. Before attempting algebra on any "two groups, one combined result" problem, check if alligation applies. If it does, the cross gives the answer in two subtractions versus solving a two-variable system. The elimination is of unnecessary algebra, saving 30-60 seconds per question.
In the exam hall, run this decision tree the moment you see a mixture or blending question:
Step 1 — Identify what is being mixed. Is it two liquids? Two products with a price? Two solutions with concentrations? Name the "value" property (price per kg, fraction of one component, percentage strength).
Step 2 — Is a quantity being removed and replaced? If yes, use the removal formula: C_final = C_initial × (1 − r/V)^n. If this is a one-step removal with no repetition, just multiply by (1 − r/V) once and track quantities directly.
Step 3 — Is there a profit/loss or markup in the problem? If yes, back-calculate the cost price of the mixture before touching alligation.
Step 4 — Are you given concentrations as ratios? Convert each ratio to a single fraction (part of interest ÷ total parts) before applying the cross.
Step 5 — Apply the alligation cross. Write the two ingredient values, the target mean in the centre. Subtract diagonally. The two results are the mixing ratio (lower ingredient : higher ingredient = right diagonal : left diagonal).
Step 6 — Verify units. The ratio is of quantities (kg, litres), not of the value property. Make sure you assign the ratio to the right ingredient.
If the answer is not matching an option, check: did you convert ratio to fraction correctly? Did you use SP instead of CP? Did you swap the ratio sides?
Why this question: This tests the removal-and-replacement sub-type — not the classic alligation cross. SBI PO uses this to separate students who only memorize the cross from those who understand the underlying tracking logic.
Solving path: Initial water = 140 − 80 = 60 litres. Removing 25% of the entire mixture removes 25% of each component proportionally, so water removed = 60 × (1/4) = 15 litres. Remaining water = 60 − 15 = 45 litres. Adding 19 litres of pure water: final water = 45 + 19 = 64 litres. The trap here is removing 25% only from water — the question says 25% of the mixture, which contains both milk and water in proportion.
Why this question: This is the classic profit-percentage-disguised-alligation problem. The selling price and profit are given; you must derive the mixture's cost price before applying the cross. This pattern appears repeatedly in SBI PO and IBPS PO from 2018 onward.
Solving path: CP of mixture = 767 × 100/118 = ₹650/kg. Apply alligation cross: ingredient A at ₹614, ingredient B at ₹695, mean at ₹650. Diagonal from B: 695 − 650 = 45. Diagonal from A: 650 − 614 = 36. Ratio A:B = 45:36 = 5:4. Amount of A is 35 kg, which corresponds to 5 parts, so 1 part = 7 kg. Amount of B = 4 × 7 = 28 kg. The common error is computing 767 − 614 and 695 − 767 directly (using SP instead of CP), which gives a nonsensical or wrong ratio.
Why this question: This tests whether you can apply alligation when concentrations are given as ratios, not direct fractions. The conversion step is what SBI PO is actually testing.
Solving path: Water fraction in Pot 1 = 4/7. Water fraction in Pot 2 = 2/3. Target water fraction = 13/21. Express all over 21: Pot 1 → 12/21, Pot 2 → 14/21, target → 13/21. Apply alligation cross: diagonal from Pot 2: 14/21 − 13/21 = 1/21. Diagonal from Pot 1: 13/21 − 12/21 = 1/21. Ratio of Pot 1 : Pot 2 = (1/21) : (1/21) = 1:1. When both diagonal differences are equal, the answer is always 1:1 — a useful pattern to recognize instantly without completing the full subtraction.
Applying the cross to ratio numbers instead of fractions. If a mixture is milk:water = 3:2, using "3" and "2" directly in the cross gives wrong results. Always convert to the fraction of one component first (e.g., milk fraction = 3/5).
Swapping which ratio belongs to which ingredient. The number you get on the right side of the cross (from the right ingredient's diagonal subtraction) belongs to the left ingredient, and vice versa. This is not intuitive, which is why labeling the cross carefully before reading off the ratio is essential.
Using selling price instead of cost price when profit is given. If the mixture is sold at a profit, the SP is not the "value" you plug into the alligation cross for pricing problems. Back-calculate CP first, every time.
Forgetting that removal is proportional. When a fraction of a mixture is removed, both components reduce by the same fraction. Students often subtract the removed volume only from one component (usually water, since it feels like the "non-valuable" one).
Treating repeated removal as a single removal. If 20% is removed twice and replaced with water, the final milk is not 60% of original — it is 0.8 × 0.8 = 64% of original. Each step multiplies by the same factor; they do not simply add.
Ignoring the total volume constraint in replacement problems. In removal-and-replacement, the total volume stays constant (you top up what you removed). If the total volume changes, you are in a different type of problem and need to track absolute quantities instead of using the repeated removal formula.