How can I calculate how much of the funds from scholarships and grants went to my room and board?

It does not matter which funds were actually used for which charges. At tax time, you are free to allocate your expenses for the best tax outcome.  There is one exception. If any part of the scholarship was "restricted" to being used for tuition, that much, of the scholarship,  must be allocated to tuition.

What you want to do is allocate enough qualified educational expenses (QEE) (tuition, fees. books and a computer) to the  AOC tuition credit, to get the maximum credit (it takes $4000 of QEE to get the max $2500 AOC).  Then allocate to the scholarship to keep it tax free.  Room and board are not QEE. 

You can put all the expenses (tuition, fee on Form 1098-T, books and supplies not on the 1098-T and room and board expenses) and scholarships/Grants that you received on paper. 

Allocate any way you choose.

Only a distribution from an education savings account reported on Form 1099-Q can be offset by room and board. 

If you only have education expenses and scholarships/Grants  and NO 1099-Q, room and board is a bit irrelevant unless you PURPOSELY want to make some of your scholarship taxable in order to free-up expenses for a credit. 

For example, if you are eligible for and wish to claim the maximum American Opportunity Tax Credit, you paid 10,000 tuition and got 10,000 scholarships, you could allocate 4,000 scholarship to room and board (which makes that 4,000 taxable) and use the 4,000 "freed-up" expenses for a credit. Usually the credit is worth more than the tax. 

You do not need to figure out what dollars went where directly, you can allocate as you wish.

@rucbresee24