Gus and Ike have a 5:3 ratio of cards. if Gus gives Ike 15 cards they now have the same amount. how many cards to they have and how many did they start with?

I will call x to be the "unit" of the ratio: by this I mean that if the ratio of what Gus and Ike have is 5:3, then Gus has 5x, and Ike 3x cards, and the total amount of cards is 5x+3x=8x.

the second sentence gives us this knowledge:

5x-15=3x+15

and now we count:

5x-3x=15+15

2x=30

x=15.

so in the beginning Gus had 5x=5*15=75 and Ike 3x=3*15=45, and now they both have 60 cards each!