"8 to 1" is a not very exact, but it means in general that you win 1 times while you lose 8 times. Thus, you have around 1 times from 9 tries. It is like throwing a 9-sided dice:
In the context of this task, it means that you win 9 times of your bet, if you win. It is a different thing. The win is always smaller as your chance.
The problem doesn't say, what is the chance, but it is not needed here, because it says, how many times he won.
Because he won 10 times, he gets \$90 from the wins. He also lost 100 in the bets. His total result is a \$10 loss.
P.s. Don't gamble. Never ever. You are a very beginner in probability theory - if you wouldn't, you would understand why you shouldn't ever gamble. No, there is no some "tricky formula" with what you could win. And it is not a complex proof, it is a very-very basic thing in the probability theory. You could ask a new question from the details.