Find two numbers such that their mean proportional is 6 and third proportional is 20.25.

4 and 9

Let, the two numbers be x and y respectively,

Mean proportional between x and y = 6

⇒ x : 6 = 6 : y

⇒ xy = 36

⇒ x = 36/y      ----(1)

Third proportional to x and y =  20 1/4 = 81/4

⇒ x : y = y : 81/4

From equation(1), 

⇒ y² = (81/4) × (36/y)

⇒ y³ = 81 × 9 

⇒ y3 = 9 × 9 × 9 = (9)³

⇒ y = 9

Again, From equation(1), we get,

x = 36/y = 36/9 = 4