what number, when added to both the numerator and to the denominator of 8/11, gives 2/3?

We have

[8 + N] / [11 + N] = 2/3 cross-multiply

2[11+N]= 3[8 + N] simplify

22 + 2N = 24 + 3N subtract 24, 2N from both sides

-2 = N

Check

[8 - 2]/ [11- 2] =

6/9 =

2/3