+467 One day, the mathematician drinks some coffee and discovers that he can now solve \(3p+7\) problems per hour. In fact, he only works for \(t-4\) 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?
+129849 On a normal day, he solves pt problems
So.....on a day he drinks coffee he solves 2pt problems
So....we have that
(3p + 7) (t - 4) = 2pt simplify
3pt + 7t - 12p - 28 = 2pt
pt + 7t - 12p - 28 = 0
pt = 28 + 12p - 7t
pt = 7 ( 4 - t) + 12p
pt - 12p = 7 (4 - t)
p ( t - 12) = 7(4 - t)
p = 7 ( 4 - t)
_______
t - 12
4 - t
p/7 = _____
t - 12
p must be a multiple of 7...so....testing some values
p t
1 5
7 8
14 no integer
21 10
35 no integer
49 11
So ....we have these possibilities
2 (1)(5) = 10 problems
2pt = 2 (7)(8) = 112 problems
2pt = 2 ( 21) (10) = 420 problems
2pt = 2 (49)(11) = 1078 problems
