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?
+128631 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
