Not Quite.
As you say, if you want 4,000 for a credit, and the tuition expense totaled 4,947,
the student would only be left with 947 that they paid out of the scholarship.
Since they received 10,249 and paid 947, they walked away with 9,302.
The student would need to claim 9,302 as income for you to get 4,000 to use for a credit.
If the amount in Box 1 is 4,947 as you say, and the amount in Box 5 is 10,249, that would mean that the student received 10,249. You are saying that you paid 4,000 so that you can take the full American Opportunity Tax Credit, and that leave only 947 that the student can say they paid with the scholarship.
Of course there might also be expenses for books and supplies.
The student in this example claims 9,302. If there is any tax due on the student's return would depend on the student's other income. If this is the only income for the student to claim, there would be no tax liability and the student needn't file.
In the second example you show, the tuition expense (Box 1) is 2,859. The scholarship was 10,802. If the student took the 10,802 and paid the tuition amount of 2,859, they would walk away with 7,943.
If someone want to use the tuition for a credit, since that tuition was only 2,859, that is the most a person could use for a credit, 2,859, so that person claims the 2,859 towards a credit.
If all the tuition (2,859) goes towards a credit, the student did not pay any of it, so the student claims the entire 10,802 as income.
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