The mathematician solves 112 problems that day.
We can rewrite the equation as (3p+7)(t-4) = 2tp, then we can solve for t with 4*(3-(14/p+7)). The expression (14/p+7) can only be 1, which results in a value of 7 for p. We can then solve the problem easily: (3*7 + 7)(8-4) = 28*4 = 112.