Let B represent the price of the bottle, C the price of the cork. We have that...

B + C = 1.10 ~~~~~~~~~~~~~~~~~~~~~~~~~~ from the first sentence, and
B = C + 1.00 ~~~~~~~~~~~~~~~~~~~~~~ from the second sentence, thus
2 B + C = C + 2.10 ~~~~ by the principle of 'equals added to equals,' and
2 B = 2.10 ~~~ by the principle of 'equals subtracted from equals,' thus
B = 1.05 ~~~~ by the principle of 'equals divided by equals,' so then
C = 0.05 ~~~~~~~~~~~~~~~~ that's a nickel, folks -- not a dime.

Here's a new challenge:

Try making up a problem which relates the prices of all these various liquids. (See Bottle Royal).