Mixture and Alligation Cheat Sheet

Mixture and Alligation Cheat Sheet

Quick Revision

Alligation is a fast rule for problems that mix two things of different values (prices, strengths or speeds) to reach a given average, called the mean price.

1. The Rule of Alligation

When two ingredients are mixed, the ratio of their quantities is:

Cheaper quantity     Dearer price − Mean price
----------------  =  -------------------------
Dearer quantity      Mean price − Cheaper price

In words: each quantity is proportional to the difference on the opposite side of the mean.

2. The Cross Diagram

Cheaper (C)           Dearer (D)
        \             /
         \           /
          Mean (M)
         /           \
        /             \
 (D − M)               (M − C)

So Cheaper : Dearer = (D − M) : (M − C).

Example

Rice at ₹20/kg is mixed with rice at ₹30/kg to sell at ₹24/kg.

Cheaper : Dearer = (30 − 24) : (24 − 20) = 6 : 4 = 3 : 2

3. Mean Price

The mean price is the cost of one unit of the final mixture:

Mean price = Total cost of ingredients / Total quantity

4. Repeated Replacement (dilution)

If a container holds V units of liquid and x units are drawn out and replaced with water, done n times, the pure liquid left is:

Pure liquid left = V × (1 − x/V)ⁿ

Example

16 litres of milk; 4 litres are replaced with water twice:

= 16 × (1 − 4/16)² = 16 × (3/4)² = 16 × 9/16 = 9 litres of milk

5. Exam Quick Tips

• Alligation gives a ratio, not a quantity — scale it to the total afterwards.
• The differences go on the opposite sides (cross), a common slip.
• Alligation also works for average speed, percentages and strengths, not just price.
• For dilution, use V(1 − x/V)ⁿ for the pure liquid remaining.

Keep revising with the rest of the Maths section:

• All Maths topics
• Ratio and Proportion Cheat Sheet
• Average Cheat Sheet

How this was made: Drafted with AI assistance; spot-checked by the editor on 31 October 2026.