Question

The ratio of the geometric mean and arithmetic mean of two numbers is 3:5, find the ratio of the smaller number to the larger number.

Answer 1

`(1)/(9)`

Explanation:

Let the numbers be x,y, where x>y

The geometric mean is

`√(xy)`

The Arithmetic mean is

`(x + y)/(2)`

The ratio of the geometric mean and arithmetic mean of two numbers is 3:5.

`( √(xy) )/( (x + y)/(2) ) = (3)/(5)`

We can write the equation;

`√(xy) = 3`

or

`xy = 9 - - - (2)`

l

and

`(x + y)/(2) = 5`

or

`x + y = 10 - - - (2)`

Make y the subject in equation 2

`y = 10 - x - - - (3)`

Put equation 3 in 1

`x(10 - x) = 9`

`10x - {x}^(2) = 9`

`{x}^(2) - 10x + 9 = 0`

`(x - 9)(x - 1) = 0`

`x =1 \: or \: 9`

When x=1, y=10-1=9

When x=9, y=10-9=1

Therefore x=9, and y=1

The ratio of the smaller number to the larger number is

`(1)/(9)`