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A pencil, eraser and notebook cost a total of S1.00. If a pencil costs more than 2 erasers, 8 erasers cost more than 3 notebooks, and 5 notebooks cost more than 6 pencils, find the cost of a pencil.

Guest Feb 3, 2018
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 #1
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Let the cost ot a pencil be x

Let the cost of a notebook be y

Let the cost of an eraser be z 

 

So

 

x + y + z  =  1.00   ⇒  z  = 100 - x - y

x > 2z   ⇒  x > 2 (100 -x - y) ⇒ x > 200 - 2x - 2y ⇒ 2y > 200 - 3x  (1)

8z > 3y ⇒ 8(100 - x - y)  > 3y ⇒  800 - 8x - 8y  > 3y  ⇒ 800 - 8x > 11y   (2)    

5y > 6x   (3)

 

Here is the graph of (1), (2)  and (3) :

 

https://www.desmos.com/calculator/dfd3tgokb7

 

The integer solutions occur inside the triangle formed by the three inequalites (feasible region)

 

One is when  x  = 37, y = 45   ( and z = 18 )    

 

Proof

 

37 > 2(18)  ⇒  37 > 36

8(18) > 3(45)  ⇒ 144 > 135

5(45) > 6(37) ⇒  225 > 222

 

So..with this solution......the cost of a pencil  =  37 cents

 

 

 

cool cool cool

CPhill  Feb 4, 2018

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