Let the fraction is x/y

Given, in a fraction, twice the numerator is 2 more than the denominator

=> 2x = y + 2 ...........1

Again, if 3 is added to numerator and denominator the new fraction is 2/3

=> (x + 3)/(y + 3) = 2/3

=> 3(x + 3) = 2(y + 3)

=> 3x + 9 = 2y + 6  ............2

From equation 1,

     2x = y + 2

=> y = 2x - 2   .........3

Put value of y in equation 2, we get

      3x + 9 = 2(2x - 2) + 6

=> 3x + 9 = 4x - 4 + 6

=> 3x + 9 = 4x + 2

=> 4x + 2 - 3x = 9

=> x + 2 = 9

=> x = 9 - 2

=> x = 7

From equation 3, we get

     y = 2*7 - 2

=> y = 14 - 2

=> y = 12

So, the fraction = 7/12