What is the rule of alligation?

A short cut that gives the ratio in which two items must be mixed to get a mixture at a given mean price: cheaper : dearer = (dearer price − mean) : (mean − cheaper price).

What is the mean price of a mixture?

The cost of one unit of the mixture, found as a weighted average: total cost ÷ total quantity.

What is the formula for repeated replacement?

Liquid left = V(1 − x/V)ⁿ, where V is the volume, x the amount removed each time and n the number of times.

Alligation and Mixture: Rule, Mean Price, Solved Problems

When two ingredients with prices (or strengths) c and d are mixed, the mixture's mean price m lies between them. The rule of alligation says the quantities must be in the ratio (d − m) : (m − c), cheaper to dearer. It is a short cut for the weighted average m = (q₁c + q₂d) ÷ (q₁ + q₂). For repeated replacement, if x units are drawn from a vessel of V units and replaced with water n times, the original liquid left is V(1 − x/V)ⁿ.

Mean price of a mixture

The mean price is the cost of one unit of the mixture. It is a weighted average:

m = (q₁ × c + q₂ × d) ÷ (q₁ + q₂)

where q₁ units cost c each and q₂ units cost d each. The mean always lies between c and d, and it sits closer to the price of the larger quantity.

Example: 3 kg at ₹40 and 1 kg at ₹60 → m = (120 + 60) ÷ 4 = ₹45.

The rule of alligation

Alligation is a quick rule to find the ratio in which two ingredients must be mixed to get a mixture of a given mean price.

Quantity of cheaper : quantity of dearer = (d − m) : (m − c)

Set it out as a cross:

   c (cheap)       d (dear)
           \     /
             m
           /     \
   (d − m)         (m − c)

Why it works: at balance, the money 'lost' on the dear part equals the money 'gained' on the cheap part: q₁(m − c) = q₂(d − m). Divide both sides to get q₁ : q₂ = (d − m) : (m − c). It is the seesaw (moment) rule.

The rule works for anything that averages by weight: price per kg, % concentration, speed, temperature, marks.

Mixtures of liquids and concentrations

Treat water as a 'liquid of price 0' or a 'solution of 0% strength'. Example: milk costs ₹60/L. How much water must be mixed so the mixture costs ₹48/L? Water (0) and milk (60), mean 48: water : milk = (60 − 48) : (48 − 0) = 12 : 48 = 1 : 4.

For selling at a profit: if a seller wants a 25% profit by selling the mix at the cost price of milk, the mix's cost price is 60 ÷ 1.25 = ₹48, which is the same sum.

Repeated replacement (removal and refill)

A vessel holds V litres of pure liquid. Each time, x litres are taken out and replaced with water. After n such operations:

Pure liquid left = V × (1 − x/V)ⁿ

Each step keeps the fraction (1 − x/V) of what was there, so the amount shrinks like compound interest in reverse. Example: V = 40, x = 4, n = 2 → 40 × 0.9² = 32.4 L.

Try it: predict a temperature

Mix 3 cups of hot water at 60 °C with 1 cup of cold water at 20 °C. Alligation: hot : cold = (m − 20) : (60 − m) = 3 : 1, so m = 50 °C. Measure and compare. A little heat is lost to the cup, so your reading may be slightly lower.

Key formulas and definitions

Mean price m = (q₁c + q₂d) ÷ (q₁ + q₂)
Rule of alligation: cheaper : dearer = (d − m) : (m − c)
Balance: q₁(m − c) = q₂(d − m)
Water has price 0 (or 0% strength) in dilution problems
Repeated replacement: liquid left = V(1 − x/V)ⁿ
Profit case: cost price of mixture = selling price ÷ (1 + profit%)

Worked examples

1. Rice at ₹40/kg and ₹60/kg is mixed to get ₹45/kg. Find the ratio.

Cheap : dear = (60 − 45) : (45 − 40) = 15 : 5 = 3 : 1.

2. Find the mean price when 5 kg of sugar at ₹42/kg is mixed with 3 kg at ₹50/kg.

m = (5 × 42 + 3 × 50) ÷ 8 = (210 + 150) ÷ 8 = 360 ÷ 8 = ₹45/kg.

3. In what ratio must water be mixed with milk costing ₹60/L to get a mixture worth ₹50/L?

Water (0) and milk (60), mean 50. Water : milk = (60 − 50) : (50 − 0) = 10 : 50 = 1 : 5.

4. A 30% acid solution and a 70% acid solution are mixed to make 40 L of 45% solution. How many litres of each?

Ratio 30% : 70% = (70 − 45) : (45 − 30) = 25 : 15 = 5 : 3. 30% solution = 40 × 5/8 = 25 L; 70% solution = 15 L.

5. How many kg of tea at ₹300/kg must be mixed with 12 kg of tea at ₹420/kg so that the mixture is worth ₹350/kg?

Cheap : dear = (420 − 350) : (350 − 300) = 70 : 50 = 7 : 5. Cheap = 12 × 7/5 = 16.8 kg.

6. A milkman buys milk at ₹50/L, adds water and sells the mixture at ₹50/L, gaining 25%. Find the ratio of water to milk.

Cost price of mixture = 50 ÷ 1.25 = ₹40/L. Water : milk = (50 − 40) : (40 − 0) = 10 : 40 = 1 : 4.

7. A 40 L can of milk: 4 L is removed and replaced with water. This is done twice. How much milk is left?

Milk = 40 × (1 − 4/40)² = 40 × 0.81 = 32.4 L. Water = 7.6 L.

8. A 20 L vessel of pure juice: 5 L is drawn out and replaced with water three times. Find the ratio of juice to water at the end.

Juice = 20 × (1 − 5/20)³ = 20 × 0.421875 = 8.4375 L. Water = 11.5625 L. Ratio = 8.4375 : 11.5625 = 27 : 37.

Common mistakes

Writing the ratio the wrong way round: the CHEAP quantity goes with (dear − mean), not (mean − cheap).
Using the selling price as the mean price in a profit question; first convert to cost price.
Subtracting only once for repeated replacement; the factor (1 − x/V) must be raised to the power n.
Taking the simple average of prices when quantities are different; use the weighted average.

Practice quiz

Practice: answer these yourself

Type or choose your answer, then press Check. Use a hint if you are stuck; the full solution appears after you answer.

Alligation and Mixture

🎬 Step-by-step story

🤔 Common doubts, cleared

Why can't I just average the two prices?

Why does the cheap amount go with the dear gap?

Where does a mixture's mean price lie?

Why is the replacement formula a power and not a subtraction?

Can the mean equal one of the prices?