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# Math

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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 has 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?

Apr 19, 2021

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Call the  number of marbles  that   Kaden  gave  away  =  G

Call the  number of marbles  that  John gave away   = (3/5)G

The  number of  marbles  that  John ends up with =  (3/5)G - 4

The  number of marbles  that  Kaden ends up with =  (2/5) [ (3/5)G - 4]

So

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

(3/5)G  - (6/25)G -  4  + 8/5  = 21

(15/25)G  - (6/25)G  - 4  + 8/5  = 21

(9/25)G  =  21   + 4  -8/5

(9/25)G  =  117/5

G =  (117/5) ( 25/9)  = 65

John  must  have  started  with     (3/5)G  + (3/5)G   - 4  =  (6/5)(65)  - 4  =  74

Kaden    must  have  started  with    G  + (2/5)[ (3/5)G - 4] =   65+  (2/5) [ (3/5)(65) - 4 ] =

65  + (2/5) [ 35] =

65  +  14   =

79

Proof

John started  with  74   and  gave  away (3/5)(65)  =  39

Kaden  started  with  79  and  gave  away  65

And

35  -14     =  21   Apr 19, 2021