I have looked to get my head round this myself. You do this problem your way and i'll do it my way:

The pot is $800, we check, our opponent bets $100 and we call expecting to win 10% of the time (that's 9:1 against).

We expect to break even:

i.e. EV = 0.1($800 + $100 + $100) - $100 = $0.

I think the confusion here comes from working in terms of ratios a lot.

When our opponent bets $800, the pot is $900. We have 10% equity so that makes us 9:1 against. We're getting 9:1 pot odds so we call and break even.