Question 1208784
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<pre>Adam, Ben and Calvin had a total of 540 marbles. 
Adam gave 1/4 of his marbles to Ben.                             (1st exchange)
Then, Ben gave 1/3 of his total number of marbles to Calvin.     (2nd exchange)
In the end, all the boys had the same number of marbles.
(a) How many marbles did each boy have in the end? 
(b) How many marbles did Ben have at first?
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                     <U>Solution</U>


(a)  Part (a) is easy.

     Indeed, at the beginning, the total number of marbles was 540.

     In process of exchanges, the total number of marbles remains the same:
     it did not change.

     Therefore, since at the end all the boys had the same number of marbles, 
     this final number of marbles, which every boy had at the end was  540/3 = 180.


     It is the <U>ANSWER</U> to question (a).



(b)  At the last exchange, Ben gave 1/3 of his marbles to Calvin.

     So, we can write this equations, describing this exchange

         b - {{{(1/3)b}}} = 180,   (1)

         c + {{{(1/3)b)}}} = 180,   (2)


     where "b" and "c" are the numbers of marbles that Ben and Calvin had before 2nd exchange.


     From equation (1),  {{{(2/3)b}}} = 180,  hence  b = {{{180*(3/2)}}} = 270.

     Then equation (2) becomes 

         c + {{{(1/3)*270}}} = 180,

     or

         c + 90 = 180,




     giving  c = 180 - 90 = 90.


     So,  before 2nd exchange and after 1st exchange Ben had 270 marbles;  Calvin had 90 marbles.


     Now for/from the 1st exchange, we have this equation

         {{{x - (1/4)x}}} = 180,    (3)  

         {{{y + (1/4)x}}} = 270.    (4)


     where "x" and "y" are the numbers of marbles that Adam and Ben had before 1st exchange 
     (i.e. originally).


     From equation (3),  we have 

         {{{(3/4)x}}} = 180,  x = {{{180*(4/3)}}} = 60*4 = 240.


     From equation (4), we have

         {{{y + (1/4)*240}}} = 270,

         y + 60 = 270, 

         y = 270 - 60 = 210.


At this point, the solution is complete.


<U>ANSWER</U>.  Initially,  Adam had 240 marbles;  Ben had 210 marbles;  Calvin had 90 marbles.


<U>CHECK</U>.  1st exchange:  {240,210,90) --->  (180,270,90).

        2nd exchange:  (180,270,90)  --->  (180,180,180).   ! correct !
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Solved.