Three friends whose ages form a $G.P$. divides a certain sum of money in proportion to their ages. If they do that three years later, When the youngest is half the age of the oldest then he will receive $Rs.150/-$ more than he gets now, and the middle friend will get $Rs.15$ more than now, then the age of the youngest friend is

Let the current ages of the friends in the geometric progression be a, ar, and ar^2. Three years later, their ages are a+3, ar+3, and ar^2+3, and the ratio of the youngest to the oldest is 1:2, giving the equation 2(a+3) = ar^2+3, or 2a+3 = ar^2. The change in their shares of money is proportional to the change in their ratio parts: the youngest's part increases by 135 while the middle friend's part increases by 15, making the ratio of the increases 9:1. Because the middle friend's ratio part increased by 3 over three years, the youngest's part should have increased by 27, so the extra 108 increase to reach 135 implies the youngest's original ratio part a was 108 divided by 6, yielding a = 18.

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