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# help needed

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Kaden and John had some marbles. John gave away 3/5 as many marbles as Kaden and was left with 4 fewer marbles than the marbles he gave away. Kaden had 2/5 as many marbles left as John. The difference between the number of marbles they had in the end was 21. How many marbles did Kaden have at first?

May 28, 2021

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Let   the  number  that  John started  with  = J

And  let  the  number  that  Kaden  started  with  =  K

Let  the  number  that Kaden  gave  away   =  G

Let  the  number  that John gave  away  =   (3/5)G

So

J  -  (3/5)G  =   (3/5)G  - 4        (1)

And

K  -  G    =   (2/5) [ (3/5)G  - 4  ]       (2)

And

[(3/5)G  - 4]  -  (2/5) [ (3/5) G - 4 ]   =  21      (3)

Simplifying  the  3rd  equation, we  have  that

(3/5)G  - 4  - (6/25)G  + 8/5  =  21                  multiply  through  by 25

15G   -  100  - 6G  +  40  =  525

9G   -  60  =  525

9G   =  585

G =  585 /  9   =  65

Sub  this  into  the  second  equation  for   G

K  -  65    = (2/5)  [  (3/5) 65  - 4 ]

K  -  65  =  (2/5)  [ 39 - 4 ]

K - 65  =  (2/5) ( 35)

K  -  65   =  14

K   =  14  + 65   =    79   =  what Kayden started  with

Proof

Kayden    is left with  (2/5)(35)   =  14

John    is  left  with (3/5)G - 4  = (3/5)(65) - 4  =  39 - 4 =    35 =  21  more  than Kayden   May 28, 2021