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📖 Mixture & Alligation

The Milkman’s Secret

Imagine a clever milkman who wants to earn a little extra. He has pure milk that costs ₹50 per litre. He mixes some water (cost ₹0 per litre) into it. If he sells the mixture at ₹50 per litre, customers won’t suspect anything, but his profit shoots up because part of that mixture was free. Now here’s a useful fact: you can calculate exactly how much water he added without a single x or y, in under 30 seconds, using a technique traders have used for centuries. That technique is Alligation. By the end of this chapter, you’ll be faster than the milkman himself.

🧠 Before you read on: If the milkman mixes 3 litres of water with 7 litres of milk and sells the whole 10 litres at ₹50 per litre, what is his profit compared to selling 10 litres of pure milk? (Keep that thought.)

The Struggle is Real — And it Ends Here

If your heart sinks the moment you see “In what ratio must rice at ₹40 per kg be mixed with rice at ₹60 per kg…”, you’re not alone. Most students treat Mixture & Alligation like a guessing game. Some try equations, get tangled in variables, and waste precious minutes. But here’s the truth: exam setters don’t want you to form equations. They want to test whether you can use a shortcut that works for weighted averages, mixtures, and even profit and loss. Once you see the pattern, these questions become predictable.

Take a deep breath. This chapter is going to replace confusion with a simple, visual tool. No more fear.

Meet the Alligation Cross — Your New Superpower

We have two ingredients: one cheaper (C), one dearer (D). We mix them to get a mean price (M). The alligation rule gives the ratio in which they must be mixed:

(Quantity of cheaper) : (Quantity of dearer) = (D – M) : (M – C)

Yes, just subtract crosswise. Draw a cross:

C D
\ /
M
/ \
(D-M) (M-C)

The ratio lands naturally in your hands. No equations, no stress.

🧠 Active Recall: Cover the screen. What goes on the left side of the ratio — D – M or M – C? And which ingredient does it belong to? Uncover and check.

The “Difference Dances Diagonally” Trick

Students often flip the two differences. Here is the fix. Look at the cross again: the difference written below the Dearer price (D – M) actually gives the quantity of the Cheaper ingredient, and the difference below the Cheaper price (M – C) gives the quantity of the Dearer ingredient. Each difference travels diagonally to the opposite side.

Remember the phrase: “Difference dances diagonally.”

A funny visual to lock it in: imagine a cow (Cheaper) and a deer (Dearer) standing at the top of the cross. The cow holds a placard saying “D – M” and the deer holds one saying “M – C”. Each animal reads the other’s difference. Silly, but you’ll never flip the ratio again.

Weighted Average — The Mother Concept

Alligation is nothing but an application of Weighted Average. When you mix 2 kg of ₹40/kg rice with 3 kg of ₹60/kg rice, the average price = (2×40 + 3×60) / (2+3) = 260/5 = ₹52/kg. The alligation cross simply reverses this: if the average is given, it finds the weights.

So if a question says, “In what ratio must a grocer mix two pulses costing ₹15 and ₹20 per kg to get a mixture worth ₹16.50 per kg?”, draw the cross and subtract:

Ratio of ₹15 pulses to ₹20 pulses = (20 – 16.50) : (16.50 – 15) = 3.50 : 1.50 = 7 : 3.

That’s it. Not a single equation was harmed.

🧠 Recall prompt: A mixture of two liquids costs ₹30/L. The cheaper liquid is ₹24/L, the dearer is ₹36/L. Without a pen, what ratio do you get? (Answer: (36–30) : (30–24) = 6 : 6 = 1 : 1.)

Replacement Problems — The “Take Out and Top Up” Dance

A classic trap: “A container has 40 litres of milk. 4 litres are withdrawn and replaced with water. This is repeated twice more. How much milk is left?” Students panic and start messy step-by-step calculations. But there’s a golden formula:

Amount of pure substance left after n operations = Initial quantity × [1 – (amount replaced each time / total quantity)]ⁿ

Here, milk left = 40 × (1 – 4/40)³ = 40 × (0.9)³ = 40 × 0.729 = 29.16 litres. Done. Notice that the same amount is replaced each time. If different volumes are removed, apply the process step by step — but competitive exams usually stick to equal replacements.

🧠 Recall: A jar has 80 L of alcohol. 8 L is drawn out and replaced with water, and the process is repeated 2 times. Find the alcohol left. (Answer: 80 × (1 – 8/80)² = 80 × 0.81 = 64.8 L.)

Alligation Beyond Liquids — With One Caution

Alligation isn’t just for liquids. It works for average marks, speeds, ages — anywhere a weighted average hides. But be careful with profit and loss questions.

⚠️ Important caution: You may apply the alligation cross directly on profit percentages only when the cost prices of both items are equal. If the cost prices differ, the overall profit percentage is NOT the weighted average of the individual profit percentages. In that case, first convert everything to profit per kg (or to selling prices) and then alligate those actual values.

Example (equal cost prices): A trader mixes two varieties of tea, both costing ₹100/kg. He sells one at 20% profit and the other at 12% profit. If the blend is sold at 15% profit overall, the ratio is (20 – 15) : (15 – 12) = 5 : 3 — the first variety to the second is 3 : 5 (remember, the difference dances diagonally).

Example (different cost prices): If the teas cost ₹130/kg and ₹180/kg, do NOT alligate 20% and 12% directly. Compute profit per kg first: 20% of ₹130 = ₹26, and 12% of ₹180 = ₹21.60. Then work with these actual profit amounts (or with the selling prices ₹156 and ₹201.60) against the mixture’s actual profit or selling price per kg.

Memory Trick for the Replacement Formula

To remember [1 – x/total]ⁿ, think of a mischievous monkey who steals a fixed fraction of your patience every study session and replaces it with daydreams. Patience left = Initial patience × (1 – fraction stolen)ⁿ. Comical, but it sticks.

When a Mixture of Mixtures Confuses You

Sometimes you mix two vessels that already contain components in different ratios. For instance, Vessel A has milk and water in the ratio 2:3, and Vessel B has them in the ratio 4:1. They are mixed to get a final mixture with ratio 1:1. In what ratio should the contents be taken?

Compute the concentration of one component (say milk) in each vessel:

• A: milk = 2/5
• B: milk = 4/5
• Target: milk = 1/2

Now alligate on these fractions:
(4/5 – 1/2) : (1/2 – 2/5) = (3/10) : (1/10) = 3 : 1.

So take 3 parts from Vessel A and 1 part from Vessel B. Neat.

🧠 Active Recall: What is the concentration of water in Vessel A if milk : water is 2:3? (Answer: 3/5.) And why did we alligate milk concentration and not water? (We could use either — the final ratio comes out the same.)

Summary Table

Two Days Later…

You’ve absorbed a mountain today. But the brain discards what it doesn’t revisit. In two days, come back and do just three things: 1) Redraw the alligation cross and recall the ratio rule. 2) Solve one replacement problem using the formula. 3) Explain to a friend why “difference dances diagonally”. That short review will lock this chapter into your long-term memory. See you then — you’re on your way to becoming an Alligation Artist.