Mixtures and Alligation for CAT: 4 Question Shapes

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Two aspirants get the same mixture question in a mock. One writes a pair of equations, solves for two unknowns, and finishes in three minutes with the right answer. The other draws a small cross on the rough sheet, writes two numbers under it, and has the ratio in twenty seconds. Same marks, same paper, one of them still has two and a half minutes left for a DILR set.

That gap is what alligation buys you, and it is why mixtures keeps appearing in CAT papers. The topic looks like a formula chapter and behaves like a decision chapter. Most aspirants working through CAT ratio and proportion questions memorise four separate mixture formulas and still freeze, because nobody told them the formulas answer four different question shapes.

Why Alligation Feels Like a Trick Until It Is Not

Alligation gets taught as a diagram. Draw a cross, put the cheaper value bottom left, the dearer value bottom right, the mean in the middle, subtract diagonally, read the ratio. It works, so people use it, and almost nobody is shown where it comes from. That is a problem, because a rule you cannot rebuild is a rule you will misapply the moment a question changes shape.

Here is the derivation in one line. If quantity a has value x and quantity b has value y, the mean value m satisfies ax plus by equals m(a plus b). Rearranging gives a over b equals (y minus m) over (m minus x). That fraction is exactly what the cross computes. The diagram is bookkeeping for an equation you already know.

Once you see that, two things follow immediately. The method works for anything that averages, not just price. And the answer it produces is always a ratio of quantities, never the quantities themselves. The list below is the range of things CAT has treated as a mixable value.

• Price per unit, in the classic tea, rice and alloy questions.
• Concentration of a solution, where the mean is the final strength.
• Average speed across two stretches covered at different rates.
• Marks or scores across two groups of unequal size.
• Profit percentage on two batches sold together as one lot.

The Formula Bank for CAT Mixtures and Alligation

Everything the chapter asks sits in the table below. Copy it once by hand, because writing it out is what makes the third row stick, and the third row is the one that decides several marks a paper.

Situation	Relationship
Two components mixed	Quantity ratio equals (dearer minus mean) to (mean minus cheaper)
Mean value from known quantities	Mean equals (a times x plus b times y) divided by (a plus b)
Repeated replacement, n operations	Final pure quantity equals initial times (1 minus r/V) raised to the power n
Concentration after adding pure solvent	Solute stays constant, so new concentration equals old solute over new total
Three or more components	Apply the two component rule in pairs, then combine the resulting ratios

Notice that only one of those is a formula in the memorise-it sense. The other four are restatements of the weighted average, which is why a reader who understands row two can reconstruct rows one, four and five in about fifteen seconds on the rough sheet.

Use the bank as a lookup during practice and as a reconstruction test during revision. The distinction matters, because a bank you can only read is a bank that will fail you in an exam hall where the sheet is not in front of you.

• Row one gives a ratio of quantities. It never gives litres, kilograms or rupees on its own.
• Row two is the parent. If you can write it, you can derive rows one, four and five without help.
• Row three is the only genuine memorisation in the chapter, and it is worth drilling to reflex.
• Row four covers dilution and evaporation. Ask which substance is unchanged, then write it as a constant.
• Row five is a pairing instruction, not a formula. The pairing choice decides whether the arithmetic stays positive.

The Mixture Map: 4 Question Shapes

The reason the chapter feels bigger than it is comes down to shape recognition. CAT asks one of four things, and each one tells you which row of the bank to reach for before you write anything down.

Name the shape before you write anything. Shapes one and two both run on alligation and differ only in which term is unknown. Shape three needs no diagram at all, and shape four needs the one memorised formula. That single naming step is what stops aspirants reaching for simultaneous equations on a question the cross would have closed in twenty seconds.

Shapes 1 and 2: The Two Directions of Alligation

Shape one is the textbook case. Tea at 180 rupees per kilogram is mixed with tea at 240 to sell at 200. The ratio is (240 minus 200) to (200 minus 180), which is 40 to 20, so 2:1. Cheaper tea dominates, which matches intuition since 200 sits nearer 180.

Shape two reverses it. You are told the ratio is 2:1 and the mean is 200, and asked for the dearer price. Write 2 over 1 equals (y minus 200) over (200 minus 180), so y minus 200 equals 40 and y is 240. Same equation, different unknown, and the diagram is optional once you write the fraction.

• If the mean sits closer to one value, that component has the larger share. Use this as a sanity check before you commit.
• A ratio of 1:1 means the mean is the exact midpoint. If your answer says otherwise, you have flipped a subtraction.
• Percentages, concentrations and rates all substitute directly for price. The units never change the method.
• If three components appear, pair two of them first and treat the pair as a single component with the pair's mean value.

Repeated Replacement: The One Formula to Memorise Exactly

A vessel holds V litres of pure milk. You remove r litres and top it up with water, n times. The milk remaining is V times (1 minus r/V) to the power n. Everything else in the chapter can be rebuilt from the weighted average, and this cannot, so it earns its place in memory.

The reason the formula looks strange is that each operation removes a proportion, not a quantity. After the first top up the vessel is no longer pure, so the second removal takes some water with it. Multiplying the same fraction n times is what accounts for that, and it is why the answer is never simply V minus n times r.

Four Variations Worth Recognising

• Final quantity asked: substitute directly and compute the power.
• Final ratio asked: convert the ratio to a fraction of the total first, then take the nth root.
• Number of operations asked: express both sides as powers of the same fraction and compare exponents.
• Replacement volume asked: solve for r after taking the nth root, which usually lands on a friendly fraction.

Three Previous Year Style Questions, Solved

Read the Wrapper Before the Numbers

Recent CAT papers rarely label a mixture question as one. They wrap it in profit wording or in a concentration story. These three follow the shapes above and are worth doing on paper before you read the solutions.

The Three Wrappers CAT Uses Most

1. Profit wrapper. A trader mixes rice costing 40 and 60 rupees per kilogram, then sells the mixture at 66 for a 20 percent profit. Selling at 66 with 20 percent profit means cost is 55. Alligation on 40 and 60 with a mean of 55 gives 5:15, so 1:3.
2. Concentration wrapper. A 50 litre solution is 20 percent acid. How much water must be added to make it 10 percent? Acid is fixed at 10 litres. Ten litres must be 10 percent of the new total, so the new total is 100 and you add 50 litres.
3. Replacement wrapper. From 80 litres of milk, 16 litres are removed and replaced with water twice. Milk remaining is 80 times (4/5) squared, which is 51.2 litres.

All three took under a minute once the shape was named, and all three would take three or four minutes solved as simultaneous equations. That difference is the entire argument for learning the map before learning more questions.

Question two uses no formula at all. Identifying the quantity that stays constant, the acid, turns it into a one line percentage question, and that habit transfers straight into percentage questions for CAT preparation.

Where Mixtures Questions Go Wrong

Five failure points account for most of the marks lost here, and each is a reading error rather than a method error. Check for them before you start writing on the rough sheet.

Five Failure Points to Check First

• Ratio read as quantity. The cross gives parts. Multiply by the total before answering.
• Cost price and selling price confused. Alligation runs on cost. Convert the selling price first if a profit percentage is stated.
• Wrong constant identified. In dilution the solute is constant, in evaporation the solute is constant, in topping up the total volume is constant.
• Replacement counted wrongly. Removing and replacing once is n equal to 1. A question saying the process was repeated once often means n equal to 2.
• Three components paired badly. Pair the two whose values sit on the same side of the mean, or the second ratio will come out negative.

Build the Mixture Map Into Your CAT Preparation Week

Mixtures rewards a short, repeated block far more than a long one. The chapter is small enough to cover in two sittings and slippery enough to fade in a fortnight, so schedule it as maintenance rather than as a project.

1. Day one, shape sorting. Take fifteen questions and label each with a shape number. Solve none.
2. Day two, solve by shape. Work the same fifteen from your labels, and note any you mislabelled.
3. Weekly, five questions. Five mixed questions every week keeps the replacement formula from decaying.

Once the shapes are automatic, move to real papers. Mixture questions in topic-wise CAT exam previous year questions are phrased far less helpfully than textbook sets, and that wording gap is the last thing standing between recognition and marks.

Keep one page of notes, not five. The free CAT formula and cheatsheet library already holds the bank above in a revisable form, so your own page only needs the mistakes you personally keep making.