If 6 buttons were removed from Box A, the ratio A : B would be 2 : 5.

If 9 buttons were removed from Box B instead, the ratio A : B would be 6 : 11.

Based off the first case, box A had (2 units + 6) buttons whle box B had 5 units buttons originally.

Based off the second case, box A had (6 parts buttons while box B had (11 parts + 9) buttons originally.

Note that both cases correspond to the exact same original case, so we can say that

- Box A had (2 units + 6) buttons or 6 parts buttons.

- Box B had 5 units buttons or (11 parts + 9) buttons.

So, we have

(1) 2u + 6 = 6p
(2) 5u = 11p + 9

If 2 units + 6 is equivalent to 6 parts of buttons, then we can say that 1 unit + 3 is equivalent to 3 parts. Or, with 3 fewer buttons, 1 unit is equivalent to 3 parts - 3. In short, we have

(1, modified) u = 3p - 3

Now, the u is the same throughout this question, while the p is the same throughout this question (though u may or may not be equal to p).

If u = 3p - 3, then we have

(2) 5u = 11p + 9
u + u + u + u + u = 11p + 9
3p - 3 + 3p - 3 + 3p - 3 + 3p - 3 + 3p - 3 = 11p + 9
15p - 15 = 11p + 9
15p = 11p + 9 + 15
15p = 11p + 24
15p - 11p = 24
4p = 24
p = 6

So each "part" of buttons means 6 buttons.

Then,

u = 3p - 3
= 3 * 6 - 3
= 18 - 3
= 15

So each "unit" of buttons means 15 units.

Box B had 5 units buttons originally.

Number of buttons in Box B (at first)
= 5 * 15
= 75

Thank you so much