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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
 #1
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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

 

cool cool cool

 Dec 22, 2020

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