Averages, Mixtures & Alligation: Connected Concepts Made Easy

Averages, Mixtures, and Alligation are deeply connected. Master the alligation (criss-cross) method and you can solve all three instantly.

Alligation (Criss-Cross) Method

For mixing two quantities at different values to get a mean value: Draw a cross. Put the two values on the left, mean in the center. Cross-subtract to get the ratio on the right. This ratio is the mixing ratio.

Application to Averages

If Group A has average 40 (n1 people) and Group B has average 60 (n2 people), combined average is 45: Alligation gives ratio n1:n2 = (60-45):(45-40) = 15:5 = 3:1.

Application to Mixtures

Mix 20% solution with 50% solution to get 30% solution: Ratio = (50-30):(30-20) = 20:10 = 2:1. So mix 2 parts of 20% with 1 part of 50%.

Repeated Dilution Formula

After n operations of removing x litres and replacing with water from a container of y litres: Remaining quantity = y × (1 - x/y)^n.