Question

Find two numbers whose mean proportional is 16 and the third proportional is 128.

Answer 1

Let x and y be two numbers
Their mean proportion = 16
and third proportion = 128
∴ `sqrt(xy)` = 16
⇒ xy = 256
⇒ x = `(256)/y`      ....(i)
and `y^2/x` = 128
⇒ x = `y^2/(128)`     ....(ii)
From (i) and (ii)
`(256)/y = y^2/(128)`
⇒ y³ = 256 x 128
= 32768
⇒ y³ = (32)³
⇒ y = 32
∴ x = `(256)/y`
= `(256)/(32)`
= 8
∴ Numbers are 8, 32.