Both credits start from adjusted qualified education expenses: tuition and required enrollment fees actually paid for the academic period, minus tax-free assistance (scholarships, employer aid, veterans benefits) and refunds allocable to the period. The American opportunity base adds required course materials whether or not bought from the school; the lifetime base generally does not. Each student gets a separate Part III worksheet, and no student's AOTC line carries more than $4,000.
The AOTC then pays \(100\%\) of the first $2,000 plus \(25\%\) of the next $2,000, capped at $2,500 per student, with 40% of the credit refundable and the rest nonrefundable. The LLC pays \(20\%\) of the return-wide base up to $10,000, capped at $2,000 per return, entirely nonrefundable and phased out across its MAGI band. Both credits phase out over the same MAGI band: single, head-of-household, and qualifying-surviving-spouse filers phase out between $80,000 and $90,000, and joint filers between $160,000 and $180,000, with no credit at $90,000 or more ($180,000 or more joint). Refunds received after year-end but before filing shrink the prior year's base; refunds arriving after filing can force recapture of a credit already claimed.
Worked example
Hal files jointly with MAGI of $100,000 and claims the AOTC for his freshman daughter, who paid $4,500 in tuition and required books with no assistance or refunds. Adjusted expenses are $4,500, capped at $4,000 on her line. Her credit is \(100\% \times 2{,}000 + 25\% \times 2{,}000 = 2{,}500\), $2,500, of which up to 40% ($1,000) may be refundable. Had the family instead claimed the LLC for her alone, the same $4,500 base would yield \(20\% \times 4{,}500 = 900\), $900, nonrefundable — the comparison that shows why the choice is made per student.