In what ratio must water be mixed with milk so that, by selling the mixture at cost price, a shopkeeper gains 60%?

3 : 5. Let the cost price of milk = Rs 1 per litre, and treat water as free (Rs 0 per litre). The mixture is sold at Rs 1 per litre, the same as milk's cost price, giving a gain of 60%. So the mixture's own cost price = 100/(100+60) per litre. By alligation, ratio of water to milk = (milk CP - mixture CP) : (mixture CP - water CP) = (1 - 100/(100+60)) : (100/(100+60) - 0) = 60 : 100. Dividing by the HCF 20 gives the simplest ratio 3 : 5. Check: mixing 3 parts water with 5 parts milk (Rs 1 per litre each part of milk) costs Rs 5 for 8 parts of mixture. Selling this mixture at Rs 1 per litre gives a gain of 60% on that cost, as required.

How many kg of rice costing Rs 10 per kg must be mixed with 45 kg of almonds costing Rs 19 per kg, so that a gain of 20% is made by selling the mixture at Rs 18 per kg?

36 kg. Let CP1 = Rs 10 per kg for rice, and CP2 = Rs 19 per kg for almonds. The mixture is sold at Rs 18 per kg with a gain of 20%. So the mixture's cost price = 100 x 18 / (100+20) = Rs 15 per kg. By alligation, ratio of rice to almonds = (CP2 - mixture CP) : (mixture CP - CP1) = (19-15) : (15-10) = 4 : 5. Dividing by the HCF 1 gives the simplest ratio 4 : 5. With 45 kg of almonds, let the rice quantity be x kg. Then x : 45 = 4 : 5. So x = (4/5) x 45 = 36 kg. Check: 36 kg of rice plus 45 kg of almonds costs Rs 1215 for 81 kg, which is Rs 15 per kg, matching the mixture's cost price and the 20% gain.

A container has 125 litres of pure milk. 25 litres are drawn off and replaced with water. This is done 2 times in total. How much pure milk is left in the container?

80 litres. Let the container start with x = 125 litres of pure milk. Each time, 25 litres of the mixture are drawn off and replaced with water. Fraction drawn off each time = 25/125 = 0.2. So the fraction of pure milk remaining after each operation = 1 - 0.2 = 0.8. This is repeated 2 times. After operation 1, pure milk = 125 x 0.8 = 100 litres. After operation 2, pure milk = 100 x 0.8 = 80 litres. By the repeated-dilution formula, pure milk left = 125 x (0.8)^2 = 80 litres.

A container has 120 litres of pure milk. 60 litres are drawn off and replaced with water. This is done 3 times in total. How much pure milk is left in the container?

15 litres. Let the container start with x = 120 litres of pure milk. Each time, 60 litres of the mixture are drawn off and replaced with water. Fraction drawn off each time = 60/120 = 0.5. So the fraction of pure milk remaining after each operation = 1 - 0.5 = 0.5. This is repeated 3 times. After operation 1, pure milk = 120 x 0.5 = 60 litres. After operation 2, pure milk = 60 x 0.5 = 30 litres. After operation 3, pure milk = 30 x 0.5 = 15 litres. By the repeated-dilution formula, pure milk left = 120 x (0.5)^3 = 15 litres.

Vessel A contains milk and water in the ratio 1 : 4, and vessel B contains milk and water in the ratio 2 : 3. In what ratio must liquids from A and B be mixed to get a new mixture with milk and water in the ratio 1 : 2?

1 : 2. Milk fraction in vessel A = 1/(1+4) = 0.2. Milk fraction in vessel B = 2/(2+3) = 0.4. Target milk fraction = 1/(1+2) = 0.33. By alligation, ratio of A to B = (B's fraction - target) : (target - A's fraction) = (0.4-0.33) : (0.33-0.2) = 0.07 : 0.13. Writing this over a common denominator and reducing gives the whole-number ratio 1 : 2. Check: mixing 1 part of A with 2 parts of B gives milk fraction (1x0.2 + 2x0.4) / (1+2) = 0.33, matching the target 0.33.

By the rule of alligation, if a cheaper ingredient costs c and a dearer one costs d, and the mean price of the mixture is m, what is the ratio of cheaper to dearer quantity?

(d - m) : (m - c). Let the cheaper ingredient cost c per unit, and the dearer one cost d per unit. Let the mean price of the mixture be m per unit. The rule of alligation gives cheaper quantity : dearer quantity = (d - m) : (m - c). Here (d - m) measures how far the dearer price sits above the mean. And (m - c) measures how far the mean sits above the cheaper price. A bigger gap on one side means less of that ingredient is required.

A container has x units of liquid. If y units are drawn off and replaced with water, and this is repeated n times, what is the correct formula for the pure liquid remaining?

x(1 - y/x)^n. Start with x units of pure liquid in the container. Each time, y units of the current mixture are drawn off and replaced with water. The fraction of liquid drawn off each time is y/x. So the fraction that stays pure after each operation is (1 - y/x). After one operation, pure liquid = x(1 - y/x). After n such operations, pure liquid = x(1 - y/x)^n.

Alligation or Mixture: Formulas, Shortcuts and Solved Examples

Alligation is a shortcut for mixing two things that differ in price or strength. It tells you the ratio of the mix without any long equations.

This lesson gives each formula and the reason it works. It then solves three examples and lists the traps that cost marks.

The idea in brief

Rule: cheaper : dearer = (d − m) : (m − c).
Mean price: always between c and d.
Gain: mean cost = SP × 100 / (100 + gain%).
Water: price zero.
Replacement: x × (1 − y/x)^n.

What Alligation Means

What is alligation?

Alligation tells you in what ratio to mix two things to hit a target price or strength. Take cheap rice at Rs 30 and costly rice at Rs 45. You want a blend worth Rs 36. Alligation gives the ratio in one step.

Which terms must you know?

Cheaper price (c): the lower price per unit. Dearer price (d): the higher price per unit. Mean price (m): the cost of one unit of the final mixture.

The mean price always lies between c and d. It can never be outside that range.

Check 1

The mean price can be higher than the dearer ingredient.

Core Formulas and Why They Work

What is the rule of alligation?

Cheaper quantity : Dearer quantity = (d − m) : (m − c).

Why it works: each cheap unit is (m − c) below the mean. Each dear unit is (d − m) above it. For the blend to average m, the total shortfall must equal the total excess.

So the ingredient closer to the mean is needed in larger amount.

Sources: NCERT Class 8 Mathematics, Direct and Inverse Proportions.

The cross method. Subtract along each diagonal. The answer sits on the same row as the ingredient it counts.

Item	Price per kg	Distance from mean
Cheaper	c = 30	45 − 36 = 9
Dearer	d = 45	36 − 30 = 6
Ratio		9 : 6 = 3 : 2

What are the steps to solve any alligation question?

Step 1: write the cheaper value c and the dearer value d on the left.

Step 2: write the mean m in the middle.

Step 3: subtract along the diagonals. The bigger value minus m goes on the cheaper row. m minus the smaller value goes on the dearer row.

Step 4: cancel any common factor to get the ratio.

Step 5: scale the ratio to the quantity given in the question.

The 3 : 2 mix drawn as one bar. More of the cheaper rice pulls the average closer to Rs 30.

How do you handle a gain or loss on the mixture?

Mean cost = Selling price × 100 / (100 + gain%). For a loss, use (100 − loss%).

Why it works: selling price is the cost plus the gain. So you undo the gain first. Then you apply the rule.

How do you add water to milk?

Treat water as an ingredient with price Rs 0.

Why it works: water adds volume but no cost. It fits the same rule with c = 0.

How do you handle repeated replacement?

Pure liquid left = x × (1 − y/x)^n. Here x is the start amount, y is drawn off each time, and n is the number of times.

Why it works: each round keeps the fraction (1 − y/x) of what was pure. After n rounds you multiply n times.

How do you mix two solutions of different strength?

Use the fraction of milk in each vessel as the “price”, then apply the same rule. The target fraction is the mean.

Why it works: the amount of milk is the quantity times its fraction, just like cost is quantity times price.

Check 2

Water can be treated as an ingredient with price zero.

Three Solved Examples

Easy: rice at Rs 30 and Rs 45 per kg is mixed to cost Rs 36. In what ratio?

Here c = 30, d = 45 and m = 36.

Cheaper : dearer = (45 − 36) : (36 − 30) = 9 : 6.

So the ratio is 3 : 2. Check: 3 kg at Rs 30 plus 2 kg at Rs 45 costs Rs 180 for 5 kg. That is Rs 36 per kg.

Medium: 12 kg of sugar at Rs 35 per kg is mixed with sugar at Rs 20 per kg. The blend sells at Rs 30 with a 20% gain. How much cheaper sugar is needed?

Mean cost = 30 × 100 / 120 = Rs 25.

Cheaper : dearer = (35 − 25) : (25 − 20) = 10 : 5 = 2 : 1.

The dearer sugar is 12 kg. So the cheaper is 2 × 12 = 24 kg.

Check: 24 × 20 + 12 × 35 = 900. And 900 ÷ 36 = 25.

Exam level: 80 litres of pure milk. 16 litres are drawn off and replaced with water. This is done twice. How much milk is left?

The fraction drawn each time is 16/80 = 0.2. So 0.8 stays pure each round.

After round one: 80 × 0.8 = 64 litres.

After round two: 64 × 0.8 = 51.2 litres.

The water is 80 − 51.2 = 28.8 litres. So milk : water = 16 : 9.

Step by step: 30 litres of milk costs Rs 40 per litre. How much water makes the mix cost Rs 32?

Step 1: milk has c = 0 for water and d = 40 for milk. The mean is m = 32.

Step 2: water : milk = (40 − 32) : (32 − 0) = 8 : 32 = 1 : 4.

Step 3: milk is 30 litres, which is 4 parts. One part is 7.5 litres.

So the water to add is 7.5 litres.

Check: total is 37.5 litres. Cost is 30 × 40 = 1200. And 1200 ÷ 37.5 = 32.

Check 3

Drawing off and replacing 20% twice leaves 60% of the milk.

Common Traps and Shortcuts

Which traps cost marks?

Reversing the ratio. The cheaper quantity goes with (d − m), not (m − c).

Using the selling price as the mean price. Remove the gain first.

Treating a replacement as simple subtraction. Two rounds of 20% do not remove 40%.

Which shortcuts save time?

Draw the cross: put c and d on top, m in the middle, and subtract diagonally.

The answer ratio is always in the order cheaper : dearer.

For equal parts, the mean is the plain average of c and d.

How do you know the ratio is the right way round?

Look at the mean. It sits closer to one ingredient. That ingredient is the one you need more of.

In the rice example, 36 is closer to 30 than to 45. So the Rs 30 rice is the bigger part: 3, not 2.

If your answer gives more of the ingredient farther from the mean, you have swapped the two numbers.

Check 4

The ingredient closer to the mean price is needed in larger quantity.

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