Question

If am and gm of two quantities is 12:13 there ratio of number is

Answer 1

Harmonic mean: 2 / ( 1/a + 1/b) = 2 / ( a + b) / (ab) = 2*ab/ (a + b) 

Geometric mean = sqrt(a*b) 

HM/GM = 2*a*b) / (a + b) // sqrt(ab) = 2*a*b)//[(a + b) * sqrt(ab) ] = 12 / 13 

Notice that the denominator has sqrt(ab) in it. That can cancel with the ab on top. Also the 2 will go into 12. 

sqrt(ab) / (a + b) = 6/13 
Now use the reciprocal 

(a + b) / sqrt(ab) = 13/6 Now here's the tricky part. Divide the sqrt(ab) into (a + b) 

a/sqrt(ab) = sqrt(a/b) 
b/sqrt(ab) = sqrt(b/a) 

(a+ b)/sqrt(ab) = (sqrt(a/b) + sqrt(b/a) = 13/6 

Let the sqrt(a/b) = x 
Therefore the sqrt(b/a) = 1/x 

x + 1/x = 13/6 Multiply through by 6x 

6x^2 + 6 = 13x 
6x^2 - 13x + 6 = 0 
(3x - 2)(2x - 3) = 0 
x = 2/3 or x = 3/2 

Therefore 
sqrt(a/b) = 3/2 square both sides. 
a/b = 9/4 or 

sqrt(a/b) = 2/3 
a/b = 4/9