Billy has a collection of baseball cards that he has separated into two different value categories. His common cards are worth $0.50 each, while his premium cards are worth $1.00 each. Together, Billy's entire baseball card collection of 124 cards is worth $81.50.

Let Billy have x common cards, and y premium cards:

(i) Billi has 124 cards:

x+y=124

(ii) the total value of the cards is 81.50$:

0.5x $ + 1.00*y $ = 81.50 $, so

0.5x+y=81.5

we can write y=81.5-0.5x and substitute in (i):

x+y=124

x + (81.5-0.5x) = 124

x+81.5-0.5x=124

x-0.5x=124-81.5

0.5x=42.5

x=42.5/0.5=85

y=124-85=39

Answer: Billy has 85 common cards and 39 premium cards