2011/074) If the ratio of GM and HM of 2 numbers is 5: 4 find the ratio of numbers?

without loss of generality let the numbers be a and at

as GM^2 = AM * HM
GM/HM = AM/GM = (a+at)/(2a sqrt(t)) = 5/4

so (1+t)/( 2sqrt(t) = 5/4 and putting sqrt(t) = x

(1+x^2) / (2x)= 5/4

or 4(1+x^2) = 10 x
o (4x^2-10x + 4) = 0 or (2(2x-1)(x-2) = 0 or x = 1/2 or 2 or t = 1/4 or 4

so ratio = 1 : 4 (or 4:1)