Dollars and cents swap after spending half

A shopper walks into a store with some dollars and cents (not zero) and spends exactly half of it. On leaving, the number of cents he holds equals the number of dollars he started with, and the number of dollars he holds is half the number of cents he started with. How much did he have on entering?

$99.98. Start with $x$ dollars and $y$ cents and leave with $\tfrac y2$ dollars and $x$ cents. In cents, $100x+y=2(50y+x)$, so $98x=99y$. With $x,y\le99$ the only non-zero solution is $x=99$, $y=98$, and indeed half of $99.98 is $49.99.