# Mixtures and Alligation — The Cross Method Made Easy

Source: https://skills101.in/learn/aptitude/mixtures-and-alligation — Skills101 free tutorial (Quantitative Aptitude). Updated 2026-11-09.

When two things with different prices (or strengths) are mixed, the mixture's price lies **between** them. **Alligation** tells you the ratio in which to mix them to get a target price.

## 1. The alligation cross

![Cross diagram: cheaper ₹40 and dearer ₹60 point to the mixture ₹45; the differences 60−45 = 15 and 45−40 = 5 give the ratio 3 : 1](https://skills101.in/learn-img/aptitude/alligation-cross.svg)

> **Cheaper : Dearer = (Dearer − Mean) : (Mean − Cheaper)**

Write the two prices at the top, the mixture (mean) price in the middle, and subtract **diagonally**.

### Example 1
*In what ratio should rice at ₹40/kg be mixed with rice at ₹60/kg to get a mixture worth ₹45/kg?*

Cheaper : Dearer = (60 − 45) : (45 − 40) = 15 : 5 = **3 : 1**.

Check: 3 kg at 40 + 1 kg at 60 = 180 for 4 kg = ₹45/kg ✓

## 2. Alligation also works for percentages

### Example 2
*How much water must be added to 30 litres of 80% milk to make it 60% milk?*

Treat water as 0% milk. Mix 80% (milk solution) with 0% (water) to get 60%.

Milk solution : Water = (60 − 0) : (80 − 60) = 60 : 20 = 3 : 1.

30 litres is 3 parts → 1 part = 10 → add **10 litres** of water.

### Example 3
*A shop sells part of its stock at 10% profit and the rest at 20% loss, making an overall loss of 5%. In what ratio were the parts?*

Mix +10 and −20 to get −5. Part at 10% : part at −20% = (−5 − (−20)) : (10 − (−5)) = 15 : 15 = **1 : 1**.

## 3. Milk and water ratio questions

### Example 4
*Two vessels contain milk and water in the ratios 3 : 1 and 5 : 3. Equal amounts are mixed. Find the new ratio.*

Make the totals equal: 3 : 1 has 4 parts, 5 : 3 has 8 parts. Double the first → 6 : 2 (8 parts).

Milk = 6 + 5 = 11, water = 2 + 3 = 5 → **11 : 5**.

### Example 5
*A 60-litre mixture has milk and water in the ratio 2 : 1. How much water must be added to make the ratio 1 : 1?*

Milk = 40 L, water = 20 L. For 1 : 1 we need 40 L of water → add **20 litres**.

## 4. Repeated replacement (a favourite)

A container has x litres of pure liquid. Each time, y litres are taken out and replaced with water. After n operations:

> **Pure liquid left = x × (1 − y/x)ⁿ**

### Example 6
*A 40-litre can is full of milk. 4 litres are taken out and replaced by water. This is done twice more (3 times in total). How much milk is left?*

40 × (1 − 4/40)³ = 40 × 0.9³ = 40 × 0.729 = **29.16 litres**.

## Common mistakes

- Subtracting straight down instead of **diagonally** in the cross.
- Forgetting that water is **0%** milk (or 0% acid, 0% alcohol).
- In two-vessel questions, adding ratios before making the totals equal.

## Quick recap

- Cheaper : Dearer = (Dearer − Mean) : (Mean − Cheaper).
- Works for price, %, speed, marks — any "average of two groups".
- Repeated replacement: x(1 − y/x)ⁿ.


Practice quiz and free certificate: https://skills101.in/learn/aptitude/mixtures-and-alligation