The Alligation Cross

Mixtures and Alligations · 20 min

Aptitude · Mixtures and Alligations

Two prices, one target, and a ratio in ten seconds

A mixture question asks what ratio of cheap and expensive gets you a chosen average. Solved with algebra it takes two minutes. Solved with the alligation cross it takes one subtraction each way — provided you respect the one rule about units.

Set two values and a target, and watch the cross fill in →
A mixture is the physical situation. Alligation is the shortcut for solving it. They are not the same word and not the same thing.

01 The idea

The average tells you the ratio, backwards

Mix cheap rice at ₹40 a kilo with premium at ₹60 a kilo and the blend costs something between the two. Which value between them depends entirely on how much of each you used. Alligation runs that sentence in reverse: given the blend price you want, it tells you the ratio that produces it.

The mechanism is easier to feel than to state. If your target of ₹45 sits close to the cheap rice, the blend must be mostly cheap rice. If it sat at ₹58 it would be mostly premium. The distance from the target to each price is what fixes the proportions, and it fixes them in the opposite order — the far side gets the small share.

So the cross does exactly that. Dearer minus mean gives the parts of the cheaper item; mean minus cheaper gives the parts of the dearer. Both differences are taken larger-minus-smaller so no negative ever appears, and the answers land on the opposite diagonal from the value they came from.

One condition governs the whole chapter, and it is the source of most wrong answers: all three numbers must be the same kind of quantity. Three cost prices, or three percentages, or three speeds. Put a selling price in the middle of two cost prices and the cross returns a confident, meaningless ratio.

Dearer − mean gives the parts of the cheaper. Mean − cheaper gives the parts of the dearer. The differences cross over — that is the whole rule.
MixtureThe physical act in the question: milk with water, cheap rice with premium, a 20% solution with a 50% one. It is the situation, not the technique.
AlligationThe shortcut used to solve it — a cross of two subtractions that returns the required ratio directly, without setting up or solving any equation.
Mean valueThe average the blend must come to. It must lie strictly between the two ingredient values; a target outside that range is impossible, not merely difficult.

02 Worked example

₹40 and ₹60 rice, blended to ₹45

This blend runs through the whole lesson and the next one. In what ratio must rice at ₹40/kg be mixed with rice at ₹60/kg so the mixture is worth ₹45/kg?

1
Set the three values in placeCheaper left, dearer right, target in the middle. All three are prices per kilo, so the units condition is satisfied.cheaper = 40    mean = 45    dearer = 60
2
Dearer minus meanSixty minus forty-five. This difference belongs to the cheaper item, on the opposite diagonal.60 − 45 = 15  ← parts of the ₹40 rice
3
Mean minus cheaperForty-five minus forty, which belongs to the dearer item.45 − 40 = 5  ← parts of the ₹60 rice
4
SimplifyFifteen to five, cheaper to dearer.15 : 5 = 3 : 1
5
Check it against common senseThree parts cheap to one part premium. Verify by averaging: three kilos at ₹40 and one at ₹60 costs ₹180 over 4 kilos.(3 × 40 + 1 × 60) / 4 = 180 / 4 = ₹45 ✓

The target ₹45 sits only ₹5 above the cheap rice but ₹15 below the premium, so it must be mostly cheap rice — and the ratio 3 : 1 says exactly that. Use this as your instinct check: the ingredient whose price is nearer the target is the one you need more of. If your answer says otherwise, you have written the differences on the wrong sides.

03 The method

The rule, the formula, and the unit condition

The cross is a picture of the formula below. Either is fine; the cross is faster and the formula is what to fall back on when the picture confuses you.

Quantity of cheaper / Quantity of dearer = (dearer − mean) / (mean − cheaper). Both differences are taken larger minus smaller, so neither is ever negative.
The unit condition, which is where the marks go: all three values must be the same kind of quantity. Three cost prices, three percentages, three purities, three speeds. If the question hands you a selling price and two cost prices, convert the selling price to a cost price before you draw anything.
Question is aboutCheaper / dearer areMean is
Blending goods by priceThe two cost prices per unitThe blend's cost per unit
Acid or alcohol strengthThe two percentagesThe target percentage
Average weight or marksThe two group averagesThe combined average
Gold purityThe two carat valuesThe target carat
Coins of two valuesThe two coin valuesTotal value / number of coins
Selling price givenConvert to cost price firstNever mix SP with CP

05 Cheat sheet

Alligation on one page

Two rows of rule and four rows of the disguises the same rule appears in.

CaseCross withGives
Ratio wanted(d−m) : (m−c)cheaper : dearer
Quantities wantedshare the total by that ratioactual amounts
Percentage solutionsthe three percentagesratio of the two solutions
Combined averagesthe two group averagesratio of the group sizes
Coins of two valuesmean = total value / countratio of the counts
Purity or caratthe three puritiesratio of the two alloys
Mean outside the rangeimpossibleRe-read the question
The differences cross overDearer minus mean is the quantity of the cheaper item. Writing each difference under the value it came from inverts every answer you produce.
Same kind of quantity, all threeAll cost prices or all percentages. A selling price in the middle of two cost prices gives a confident wrong answer, which is worse than an obvious one.
The nearer side needs moreThe mean is always closer to whichever ingredient dominates the blend. One glance at the distances tells you which of two options can be right.

06 Where & why

Where the cross shows up

Alligation is a weighted-average tool, so it appears any time two groups combine into one — which is far beyond the chapter it is taught in.

TCS NQT · Accenture
Straight blending questions

Two prices and a target, ratio wanted. Twenty seconds with the cross against two minutes with simultaneous equations.

Bank PO · SSC CGL
Percentage solutions and coins

Acid strengths and two-denomination coin problems are the same cross with a hidden mean — for coins, total value divided by the number of coins.

Any averages question
Combined average, group sizes wanted

Given two group averages and the overall average, the cross returns the ratio of the group sizes immediately. Most students set up algebra for this.

Profit and loss overlap
Blending then selling

Papers combine the two chapters deliberately, which is exactly where the selling-price trap gets set. Convert to cost price first, every time.

If a question gives you two of something and an average of the combination, reach for the cross before you reach for x and y — even when the chapter heading says averages rather than mixtures.

07 Interview questions

What gets asked

Ten, starting from the distinction in the title and ending at the condition that breaks it.

What is the difference between a mixture and alligation?
A mixture is the physical situation in the question — combining milk and water, or two grades of rice. Alligation is the mathematical shortcut used to solve it. One is the problem, the other is the method, and the chapter title names both.
State the alligation rule.
The quantity of the cheaper item to the quantity of the dearer equals (dearer − mean) to (mean − cheaper). Both differences are taken larger minus smaller so neither is negative, and each lands on the diagonal opposite the value it came from.
Why do the differences cross over rather than stay on their own side?
Because the closer the target is to one ingredient's value, the more of that ingredient the blend must contain. The small difference belongs to the far ingredient. So the difference computed from the dearer value is the quantity of the cheaper one — it crosses.
Mix rice at ₹40 and ₹60 to get ₹45. What ratio?
3 : 1, cheap to premium. The differences are 60 − 45 = 15 and 45 − 40 = 5, giving 15 : 5. Check by averaging: three kilos at 40 plus one at 60 is ₹180 over 4 kilos, which is ₹45.
What is the one condition on the three values?
They must all be the same kind of quantity — three cost prices, or three percentages, or three purities. The most punished violation is putting a selling price in the middle of two cost prices; convert it to a cost price first.
What happens if the mean is outside the two values?
The question is impossible. A blend's average always lies between its ingredients, so a target above both or below both cannot be produced. Mechanically the cross returns a negative quantity, which is the signal to re-read the question rather than to drop the minus sign.
A merchant has 100 kg of tea, part at ₹110 and the rest at ₹130, averaging ₹118. How much is the cheaper tea?
Sixty kilos. The cross gives (130 − 118) : (118 − 110) = 12 : 8 = 3 : 2. That is 5 parts covering 100 kg, so one part is 20 kg, and the ₹110 tea is 3 parts — 60 kg.
₹12.40 is made of 80 coins, either 10 paise or 20 paise. Find the ratio of the counts.
9 : 11. The mean is hidden: 1240 paise over 80 coins is 15.5 paise per coin. Then the cross on 10, 15.5 and 20 gives 4.5 : 5.5, which is 45 : 55 and simplifies to 9 : 11. Spotting that the mean must be computed is the whole difficulty.
How would you use alligation on an averages question?
Identically. If girls average 45 kg, boys average 60 kg and the class averages 50 kg, the cross gives (60 − 50) : (50 − 45) = 10 : 5 = 2 : 1 for girls to boys. Alligation is a weighted-average tool, so any two-group averaging problem is in scope.
When is alligation the wrong tool?
When more than two ingredients are combined and you are not given enough structure to pair them up, and when the quantity being averaged is not additive — average speed over equal distances, for instance, is not a plain weighted average of the two speeds, so the cross does not apply directly. In both cases go back to the definition of the average.

08 Practice problems

Six crosses

All six are one cross each. The work is deciding what the three values are, not doing the subtraction.

Straight blend

Easy
In what ratio must a grocer mix sugar at ₹18/kg and ₹24/kg so the mixture is worth ₹20/kg?
Follow-up
Check your answer against the distances before you write it: ₹20 is much nearer ₹18, so the cheaper sugar must dominate.
Show the hint
The two differences are 24 − 20 and 20 − 18.

Percentages instead of prices

Easy
Bottle A holds a 20% acid solution and bottle B a 50% one. In what ratio must they be mixed to get a 30% solution?
Follow-up
Nothing changes because the quantities are percentages rather than rupees — which is the point. The cross does not care what is being averaged.
Show the hint
Use 20, 30 and 50 in exactly the positions you used prices.

Clear the decimals

Medium
In what ratio must pulses costing ₹64.50/kg and ₹84.00/kg be mixed so the mixture is worth ₹72.50/kg?
Follow-up
The cross gives a ratio with a decimal in it, which is correct but not an answer. Multiplying both sides by the same number is always allowed and is how you finish.
Show the hint
You will get 11.5 : 8 — double both sides.

Find the hidden mean

Medium
A sum of ₹12.40 is made up of 80 coins, each either 10 paise or 20 paise. Find the ratio of the number of 10-paise coins to 20-paise coins.
Follow-up
The mean is not in the question. You have to build it, and you must work in a single unit throughout — mixing rupees and paise here is fatal.
Show the hint
Convert ₹12.40 to paise and divide by the number of coins to get the value of an average coin.

Averages, not mixtures

Medium
The average weight of the girls in a class is 45 kg and of the boys 60 kg. The whole class averages 50 kg. Find the ratio of girls to boys, and then the number of each in a class of 45.
Follow-up
There is no mixture anywhere in this question, and the cross still solves it in one line. The second half is the unitary step that turns a ratio into counts.
Show the hint
Treat the two group averages exactly as you treated two prices.

Two crosses and a trap

Hard
A shopkeeper blends tea costing ₹80/kg with tea costing ₹110/kg. (a) Find the ratio that makes the blend cost ₹90/kg. (b) He then sells this blend at ₹99/kg. Find his profit percentage. (c) A student attempts part (a) by putting ₹99 in the middle of the cross instead of ₹90, reasoning that ₹99 is the price the mixture is 'worth'. Explain precisely what is wrong with that, and what ratio their method would have returned.
Follow-up
Part (c) is the lesson's unit condition, set as a trap rather than stated as a rule. ₹99 is a selling price and the other two numbers are cost prices, so the cross is being asked to average incompatible quantities — and it will answer anyway, which is what makes the error dangerous.
Show the hint
For (c), work out what the student's cross actually returns, then check whether that blend really costs ₹90 to make.