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# A mathematician works for \$t\$ hours per day and solves \$p\$ problems per hour, where \$t\$ and \$p\$ are positive integers and \$1

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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?

Dec 22, 2020

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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   Dec 22, 2020