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