Let N be the number of Noah's stickers and J the number of Jackson's, then N/J=12/7. Later the ratio becomes (N+28)/(J-32)=(12J/7+28)/(J-32)=(12J+196)/(7J-224). Then the question breaks down, but we know we have to evaluate N-J=(12/7-1)J=5J/7.
J>35 and N>60 as the table below suggests:
| n |
N |
J |
N-J |
| 5 |
60 |
35 |
25 |
| 6 |
72 |
42 |
30 |
| 7 |
84 |
49 |
35 |
| 8 |
96 |
56 |
40 |
| 9 |
108 |
63 |
45 |
n is the common factor for the other 3 columns. The table continues indefinitely.
The ratio after the buying and giving away becomes (12n+28)/(7n-32).
Suppose the question continued: "five stickers were lost, and Noah and Jackson had 124 stickers left between them, how many more stickers did Noah have than Jackson in the beginning?", the answer would be 35, because before the 5 stickers were lost, there would have been 129 stickers=N+28+J-32=N+J-4=19J/7 - 4, so 19J/7=133 and J=49, so N=84 and N-J=35.