Question

Find two numbers such that their mean proportional is 12 and third proportional in 96.

Answer 1

Let x and y be two given numbers.

Mean proportional of x and y is 12.

`sqrt(xy)` = 12

xy = 144   ...(1)

If 96 is the third proportional,

Then,

x : y = y : 96

⇒ `x/y = y/96`

⇒ x × 96 = y²

⇒ y² = 96x   ...(2)

Substitute equation (1) into (2)

xy = 144

x = `144/y`

Substitute in equation (2):

y² = 96`(144/y)`

y³ = 96 × 144

y³ = 13824

y = `root3 13824`

y = 24

Now find x:

x = `144/y`

= `144/24`

= 6

The two numbers are 6 and 24.