4A. Bridge to Ratio & Mixtures — Seeing the Same Idea Twice

Before moving to practice, it is worth explicitly seeing how a combined-average question and a mixture-and-alligation question are, mathematically, the identical problem wearing different clothing. This recognition will save you time in later chapters.

Solved Example (Average phrasing): A class has boys and girls in some ratio. The average marks of boys is 72 and of girls is 84. The combined average of the class is 78. Find the ratio of the number of boys to girls.

By alligation: ratio of boys : girls = (Girls’ average − Combined average) : (Combined average − Boys’ average) = (84−78):(78−72) = 6:6 = 1:1.

Solved Example (Mixture phrasing, same numbers): In what ratio should a solution costing ₹72/litre be mixed with a solution costing ₹84/litre to obtain a mixture costing ₹78/litre?

By alligation: ratio = (84−78):(78−72) = 6:6 = 1:1.

Both questions use the identical alligation cross-diagram and produce the identical ratio — because “average marks weighted by number of students” and “average cost weighted by quantity in litres” are the same weighted-average structure. Once you reach the Mixtures & Alligations chapter, you will find that mastering weighted average here means that chapter’s core formula is already second nature to you.