A mathematician works for $t$ hours per day and solves $p$ problems per hour, where $t$ and $p$ are positive integers. One day, the mathematician drinks some coffee and discovers that he can now solve $4p+7$ problems per hour. In fact, he only works for $t-3$ hours that day, but he still solves twice as many problems as he would in a normal day. How many problems does he solve the day he drinks coffee?
The number of problems he solves on a normal day = p*t
On the day he drinks coffee he solves 2pt problems....so...
(4p + 7) ( t - 3) = 2pt
4pt +7t - 12p - 21 = 2pt
2pt + 7t - 12p - 21 = 0
2pt + 7t = 12p + 21
t = [ 12p + 21 ] / [ 2p + 7 ]
Note that this makes t an nteger when p= 7 and t = 5
So.....he solves 2(7)(5) = 70 problems on the day he drinks coffee