Fraction problem I have been reading Basic Math and Pre Algebra for Dummies, the question below has me stumped!

David bought a cake for himself and his friends. He cut a piece for himself that was $1/6$ of the total cake. The Sharon cut a piece that was $1/5$ of what was left. Then Armand cut a piece that was $1/2$ of what was left. How much of the cake was left after all three friends had taken their pieces?

The answer in the book is $1/3$, is this a mistake ? 
When he ate $1/6$ there was $5/6$ left, so  $5/6 - 1/5  = 19/30$ 
$19/30  - 1/2$?   am I on the wrong path here?

$\frac{1}{6}$ is taken off the bat, so we're left with $\frac{5}{6}$ of a cake. Then $\frac{1}{5}$ of that is taken, so $\frac{1}{5}\times\frac{5}{6}=\frac{1}{6}$ of the original cake is taken. Now we have $1-2\times\frac{1}{6}=\frac{2}{3}$ of the original cake left. Finally, half of the remaining cake is taken, so $\frac{1}{2}\times\frac{2}{3}=\frac{1}{3}$ of the original cake is taken. Thus, we are left with $\frac{2}{3}-\frac{1}{3}=\frac{1}{3}$ of the original cake left.
