Let Amit's and his father's present ages be 2x and 5x, respectively.
In 4 years, their ages will be \[2x + 4\] and \[5x + 4\].
Given the ratio after 4 years is \[\frac{2x + 4}{5x + 4} = \frac{3}{7}\].
Cross-multiplying yields \[7(2x + 4) = 3(5x + 4)\], which simplifies to \[14x + 28 = 15x + 12\].
Rearranging terms, we get \[15x - 14x = 28 - 12\], so \[x = 16\].
Therefore, their present ages are Amit = \[2 \times 16 = 32\] and Father = \[5 \times 16 = 80\].
In 6 years, their ages will be \[32 + 6 = 38\] and \[80 + 6 = 86\].
The ratio of their ages after 6 years will be \[\frac{38}{86} = \frac{19}{43}\].