This question was previously asked in

UPSSSC Lower PCS Prelims Official Paper 1 (Held On : 15 July 2018)

Option 2 : 4 and 64

**Given:**

Mean proportion = 16

Third proportion = 1024

**Formula Used:**

Mean proportion = √ ab

a and b are ratios

**Calculation:**

Let the two numbers be a and b

According to question

Mean proportion to a and b = 16

a : 16 :: 16 : b

⇒ ab = 256

⇒ a = 256/b ----(I)

Third proportion to a and b = 1024

a : b :: b : 1024

⇒ a/b = b/1024 ----(II)

substituting the value of I in II

⇒ 256/b^{2} = b/1024

⇒ b^{3} = 1024 × 256

⇒ b = 64

⇒ a = 256/64 = 4

**∴ The two numbers are 4 and 64**