Ryan had some blue and green buttons. 3/7 of the buttons are blue and the rest were green. He gave away 2/5 of his buttons. 80 of the buttons given away were blue. 2/3 of the buttons left were green. How many buttons did Ryan have altogether at first?
Let the total buttons = T
If (2/3) of the buttons left were green, then (1/3) left were blue
So
Blue Buttons left = (1/3) (of what remains)
(3/7)T - 80 = (1/3) [ T - (2/5)T] simplify
(3/7)T - 80 = (1/3 ) (3/5)T
(3/7)T - 80 = (1/5)T
(3/7)T - (1/5)T = 80
[ 15 - 7 ] T / 35 = 80
8/35 T = 80
T = 80 (35/8) = 350 buttons at first
Proof
(2/5)(350) = 140 were given away
210 were left
140 were green
70 were blue
Total blue = 80 + 70 = (3/7)(350) =150
Total green = (4/7) (350) = 200
150 + 200 = 350