Six years ago Rahul's age was two times the square of Gaurav's age. Four years hence, Gaurav's age will be one fourths of Rahul's age. Find their present ages.

Let's denote Rahul's present age as R and Gaurav's present age as G.

According to the problem, we have two equations:

1. R - 6 = 2*(G - 6)^2
2. G + 4 = 1/4 * (R + 4)

We can solve these two equations to find the values of R and G.

Let's start with the first equation:

R - 6 = 2*(G - 6)^2 R = 2*(G - 6)^2 + 6

Now, let's substitute R in the second equation:

G + 4 = 1/4 * (2*(G - 6)^2 + 6 + 4) 4G + 16 = 2*(G - 6)^2 + 10 4G + 6 = 2*(G - 6)^2 2G + 3 = (G - 6)^2 2G + 3 = G^2 - 12G + 36 0 = G^2 - 14G + 33 0 = (G - 11)(G - 3)

So, G = 11 or G = 3.

Substitute G = 11 into the first equation:

R - 6 = 2*(11 - 6)^2 R - 6 = 2*25 R - 6 = 50 R = 56

Substitute G = 3 into the first equation:

R - 6 = 2*(3 - 6)^2 R - 6 = 2*9 R - 6 = 18 R = 24

So, the possible solutions are (R, G) = (56, 11) or (24, 3).

However, if we substitute these values into the second equation, we find that only (R, G) = (56, 11) is a valid solution.

So, Rahul is 56 years old and Gaurav is 11 years old.