Proportion 3 : 6 :: 6 : a

In a continuous proportion $x:y::y:z$, the product of extremes equals the product of means: $x \times z = y^2$. Here, $3 \times a = 6 \times 6 = 36$.

Students sometimes mistakenly assume the ratio $3:6$ is the same as $6:a$ resulting in $a=12$, failing to account for the property of mean proportionals where $y^2 = xz$.

This concept is foundational for understanding Geometric Progression (GP) where $B^2 = AC$, often appearing in compound interest and sequence-based questions.