Find the fourth proportional to 4, 9, 12 and third proportional to 16 and 36.

Given:

Numbers: 4, 9, 12, 16, and 36

Formula Used:

Fourth Proportional:

If a, b, c, and d are four numbers, then d is the fourth proportional to a, b, and c if (a : b :: c : d).

Third Proportional:

If a, b, and c are three numbers, then c is the third proportional to a and b if (a : b :: b : c).

Calculations:

The fourth proportional to 4, 9, and 12:

Using the formula for fourth proportional: (4 : 9 :: 12 : d)

So, 4/9 = 12/d

⇒ d = 108/4

⇒ d = 27

Third proportional to 16 and 36:

Using the formula for third proportional: (16 : 36 :: 36 : c)

So, 16/36 = 36/c

⇒ c = (36 × 36)/16

⇒ c = 81

∴ The fourth proportional is 27 and the third proportional is 81.