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.

sqrt(a*b)*2/(a+b) = 3/5

10*sqrt(a*b) = 3*(a+b)

Divide both sides by b and let u = a/b 

10*sqrt(u) = 3(u + 1)

100*u = 9*(u^2 + 2u + 1)

9u^2 - 82u + 9 = 0

This factorises as: (9u - 1)(u - 9) = 0.  
 

u = 1/9 

Hence ratio of smallest to largest is a:b = 1:9