Mixture and allegation is one of the most scoring shortcut topics in AFCAT Numerical Ability. A single tool — the rule of allegation — replaces messy equations and answers questions on milk-and-water, price-mixing and alloys in under a minute. In this Cavalier guide you will learn the cross rule, the repeated-replacement formula, and the exact wording traps that cost candidates easy marks.
Why Mixture and Allegation Matters in AFCAT
The AFCAT Numerical Ability section is short but time-pressured, so paper-setters love topics that reward a clever shortcut over slow algebra. Mixture and allegation is exactly that. One or two questions appear almost every shift, and they overlap heavily with averages, ratio and proportion, percentage and profit and loss. Learning the cross rule here quietly speeds up several other chapters.
The good news is that nearly every mixture question reduces to the same picture: two ingredients of different value blended to a middle value. Once you can draw the allegation cross in ten seconds, you skip the equation-building stage entirely and read the answer straight off the diagram.
Students at The Cavalier are trained to recognise the trigger words — “mixed”, “blended”, “diluted”, “alloy”, “average price” and “in what ratio”. The moment one appears, the cross rule should be your first instinct.
AFCAT carries negative marking (typically −1 for a wrong answer against +3 for a correct one). The allegation method is fast, but always double-check which value is cheaper and which is dearer before placing them on the cross.
The Two Words You Must Define
The whole topic rests on two simple ideas, so fix them firmly.
- Mixture — the result of combining two or more ingredients (liquids, grains, metals or even money values) into a single batch with one overall value or strength.
- Mean value — the average value (price, or percentage of a component) of the final mixture, lying somewhere between the values of the two ingredients.
The mean value of a mixture always lies between the cheaper and the dearer value. If a calculated mean falls outside that range, you have made an arithmetic slip — recheck before answering.
Allegation is simply a rule that tells you, given the two ingredient values and the desired mean, in what ratio the two ingredients must be mixed. That ratio is the answer to most questions in this chapter.
The Rule of Allegation (The Cross Method)
This is the heart of the chapter. Suppose a cheaper ingredient has value c and a dearer ingredient has value d, and they are mixed so the mixture has mean value m (with c < m < d). Then:
Quantity of cheaper : Quantity of dearer = (d − m) : (m − c)
In words: the ratio of the two quantities is the difference on the opposite side. Each ingredient is matched with how far the mean sits from the other ingredient.
Draw it as a cross. Put the cheaper value (c) at top-left and the dearer value (d) at top-right. Put the mean (m) in the middle. Subtract diagonally, taking the smaller from the larger each time:
- Bottom-left (under the cheaper) = d − m → this is the share of the cheaper.
- Bottom-right (under the dearer) = m − c → this is the share of the dearer.
The difference computed on the left does not belong to the left ingredient — it gives the share of the ingredient on that same bottom corner. Many candidates flip the ratio. Always remember: the bottom-left number is the quantity of the cheaper, the bottom-right is the quantity of the dearer.
Allegation in Action: A Worked Example
Let us nail the cross method with a price-mixing sum.
In what ratio must rice at ₹30/kg be mixed with rice at ₹42/kg so that the mixture is worth ₹36/kg?
So the two varieties must be mixed in the ratio 1 : 1. That makes sense — ₹36 is exactly midway between ₹30 and ₹42.
Now an off-centre case so you see the rule really working:
In what ratio must tea at ₹200/kg be mixed with tea at ₹260/kg to get a mixture worth ₹215/kg?
Mix the ₹200 and ₹260 teas in the ratio 3 : 1. Because the mean is close to the cheaper value, we naturally need more of the cheaper tea — a quick sanity check that the answer feels right.
Milk and Water Problems
The classic AFCAT mixture sum involves milk diluted with water. The trick is to treat water as the “free” ingredient with value zero (it costs nothing, or contributes 0% milk).
If a shopkeeper buys milk at ₹x per litre and wants to sell the diluted mixture at ₹x per litre yet still make a certain profit, you can apply allegation with the water priced at 0.
In what ratio should water be mixed with milk costing ₹20/litre so that by selling the mixture at ₹20/litre the seller gains 25%?
Add 1 part water to 4 parts milk. The seller charges the milk price on water that cost nothing, which is exactly where the 25% gain comes from.
When water is added free of cost and the mixture is sold at the milk price, profit % = (water ÷ milk) × 100 using the quantity ratio. Here 1/4 × 100 = 25%, which matches.
The Repeated Replacement Formula
A favourite AFCAT twist: a vessel is full of pure liquid; you draw out some, replace it with water, and repeat. You must find how much pure liquid is left.
If a vessel holds V litres of pure liquid and x litres is withdrawn and replaced with water n times, then:
Pure liquid left = V × (1 − x/V)n
Equivalently, Final pure : Total = (V − x)n : Vn.
A 40-litre vessel is full of milk. 8 litres is drawn out and replaced with water. This is done twice. How much milk remains?
So 25.6 litres of milk remains and 14.4 litres has been replaced by water.
Do not subtract 8 litres each time linearly (40 − 8 − 8 = 24). After the first replacement the vessel already contains water, so the second draw removes a milk-water mix, not pure milk. Only the power formula handles this correctly.
Mixing Three or More Ingredients
When a question gives three ingredients, allegation still works — you just pair them around the target. The usual approach is to group ingredients into two camps: those cheaper than the mean and those dearer, or pair them two at a time so that each pair straddles the mean.
- If two ingredients are both below the mean and one is above, combine the two cheaper ones first (their internal ratio is often given), then allegate that combined value against the dearer one.
- If the question fixes the final ratio of all three, set the price equation directly: total cost ÷ total quantity = mean.
For a straightforward three-part check, use the weighted-average identity: Mean = (q₁v₁ + q₂v₂ + q₃v₃) ÷ (q₁ + q₂ + q₃), where q is quantity and v is value. Allegation is just the two-ingredient shortcut for this same identity.
Because allegation and weighted average are two faces of one rule, you can always fall back on the average formula if the cross diagram feels ambiguous. They will never disagree.
Allegation as a Reverse Average
It helps to see why the cross rule works. A mixture's mean value is simply the weighted average of its ingredients. Allegation runs that average backwards: instead of finding the mean from known quantities, it finds the quantities from a known mean.
Start from Mean = (cheaper×Q₋ + dearer×Q₌) ÷ (Q₋ + Q₌). Rearranging gives Q₋(m − c) = Q₌(d − m), so Q₋ : Q₌ = (d − m) : (m − c). That is exactly the cross rule — no magic, just the average formula solved for the ratio.
The ingredient values can be prices, percentages of a component, or even concentrations. As long as you compare like with like (all prices, or all milk-percentages), the same cross rule applies.
This is why allegation also cracks questions on alloy composition, acid concentration and even average speed for two stretches of a journey — any “blend two things to a middle value” situation is fair game.
Speed Shortcuts for the Exam Hall
To finish each sum in 30–45 seconds, drill these reflexes.
- Symmetry check: if the mean is exactly midway between the two values, the answer is instantly 1 : 1 — no subtraction needed.
- Closer-is-bigger rule: the ingredient whose value is nearer the mean is needed in the larger quantity. Use this to eliminate flipped-ratio options at a glance.
- Water = 0, profit-by-dilution: profit % when selling water at the liquid price equals (water ÷ pure liquid) × 100 by quantity.
- Replacement: remember the fraction (1 − x/V)n gives the share of the original liquid, so 1 minus that gives the water fraction.
- Scale freely: the cross gives a ratio; multiply both parts by any constant to match a given total quantity.
Always reduce the final ratio to lowest terms before matching options — AFCAT lists answers in simplest form, and an un-reduced 6 : 6 can trick you into missing the 1 : 1 choice.
Traps That Catch Most Aspirants
Mixture sums are easy marks only if you sidestep a few classic traps.
Ratio vs quantity: allegation gives a ratio, not an actual amount. If the question asks “how many litres”, you must use the given total to convert the ratio into a real quantity.
- Wrong corner: putting the cheaper value where the dearer should be inverts the answer. Always write the smaller value on the left.
- Unit mismatch: mixing ₹/kg with ₹/litre, or a percentage with a price, gives nonsense. Compare like with like.
- Linear replacement: as seen earlier, never just keep subtracting the drawn amount — use the power formula.
- Profit/cost confusion: in dilution sums, the “mean” for allegation is the cost price of the mixture, found from the selling price and profit %, not the selling price itself.
- Allegation ratio (cheaper : dearer) = (d − m) : (m − c) — opposite-side differences.
- The mean always lies between the two ingredient values.
- The ingredient closer to the mean is needed in greater quantity.
- For dilution, water has value 0; mixture cost = SP ÷ (1 + profit).
- Repeated replacement: pure left = V(1 − x/V)n.
- Allegation gives a ratio — scale it with the total to get actual amounts.
Practise a Previous-Year Style Question
Time yourself on this AFCAT-style item using the methods above.
Q A vessel contains 60 litres of pure milk. 12 litres is removed and replaced with water; the same operation is repeated once more. The quantity of milk now left in the vessel is:
Answer: Using milk left = V(1 − x/V)n with V = 60, x = 12, n = 2: 60 × (1 − 12/60)2 = 60 × (4/5)2 = 60 × 16/25 = 38.4 litres.
If the same question instead asked for the ratio of milk to water at the end, it would be 38.4 : 21.6 = 16 : 9 — which is exactly (V − x)2 : [V2 − (V − x)2] = 16 : 9. Spotting this saves a full computation.
Frequently asked questions
What is the rule of allegation in simple words?
It is a shortcut that tells you in what ratio to mix two ingredients of different value to get a desired average value. The ratio of cheaper to dearer equals (dearer value − mean) : (mean − cheaper value) — the differences taken on opposite sides of a cross diagram.
How many questions on mixture and allegation come in AFCAT?
Typically one to two questions per shift in the Numerical Ability section. They are high-value because the allegation shortcut lets you answer each in well under a minute, and the same idea also speeds up averages, ratio and profit-and-loss sums.
Why can't I just subtract the drawn amount in replacement problems?
After the first replacement the vessel already holds a milk-water mixture, so the next draw removes both milk and water, not pure milk. You must use the formula pure liquid left = V(1 − x/V)^n, where V is the volume, x the amount replaced each time and n the number of operations.
How is allegation related to averages?
The mean value of a mixture is the weighted average of its ingredients. Allegation simply runs that average backwards: given the mean and the two ingredient values, it returns the quantity ratio. That is why the cross rule and the weighted-average formula always agree.
Does allegation work for percentages and concentrations too?
Yes. As long as you compare like with like — all prices, all milk-percentages, or all acid-concentrations — the same cross rule applies. This is why it cracks alloy composition, acid-dilution and even average-speed questions, not just price mixing.
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