What is the ratio of the mean proportional between 4.5 and 0.5 and the third proportional to 4.5 and 9.0?

What is the ratio of the mean proportional between 4.5 and 0.5 and the third proportional to 4.5 and 9.0? Correct Answer 1 : 12

Let mean proportion between 4.5 and 0.5 be x, then

4.5 : x :: x : 0.5

⇒ 4.5/x = x/0.5

x² = 4.5 × 0.5 = 2.25

x = √2.25

x = 1.5

Let third proportion between 4.5 and 9 be y, then

4.5 : 9 :: 9 : y

4.5/9 = 9/y

y = (9 × 9)/4.5

y = 18

Ratio of x : y = 1.5 : 18 = 1 : 12

Short trick:

Mean proportional between 4.5 and 0.5 = √(4.5 × 0.5) = 1.5

Third proportional between 4.5 and 9 = (9 × 9)/4.5 = 18