Math

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