Question

The sum of two numbers is 6 times their geometric mean, show that numbers are in the ratio .

Answer 1

Let, two numbers be aand b.

Now, the geometric mean of the numbers is,

G.M= ab

According to the given condition,

a+b=6 ab (1)

Squaring both of above equation, we get

( a+b ) 2 =36( ab )

We know that,

( a−b ) 2 = ( a+b ) 2 −4ab

Substitute the value of ( a+b ) 2 in above equation, we get

( a−b ) 2 =36ab−4ab ( a−b ) 2 =32ab a−b= 32 ab a−b=4 2 ab (2)

Adding equation (1) and (2), we get

2a=( 6+4 2 ) ab a=( 3+2 2 ) ab

Substitute the value of a in equation (1), we get

b=6 ab −( 3+2 2 ) ab b=( 3−2 2 ) ab

Now, the ratio of a and b is,

a b = ( 3+2 2 ) ab ( 3−2 2 ) ab a b = ( 3+2 2 ) ( 3−2 2 )

Thus, the required ratio is ( 3+2 2 ) ( 3−2 2 ) .