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?
 A: $\frac{1}{6}$ 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.
A: Yes, the answer given in the book is correct, you just have to add up all the fractions in the following way;
$$
\frac{1}{6}+\frac{5}{6}\cdot \frac{1}{5}+\frac{5}{6}\cdot \frac{4}{5}\cdot\frac{1}{2}+\frac{1}{3}=1
$$
A: The easiest thing is to keep track of how much of the cake is left after each operation, not how much is consumed. To find the remaining part, one subtracts the eaten part from $1$ and multiplies out, so you get
$$
\left(1 - \frac{1}{6}\right) \times \left(1 - \frac{1}{5}\right)
                             \times \left(1 - \frac{1}{2}\right)
= \frac{5}{6} \times \frac{4}{5} \times \frac{1}{2}
= \frac{1}{3}
$$
A: Hint. I put the words into this equation
$$\frac{1}{6}+\frac{1}{5}\left(1-\frac{1}{6}\right)+\frac{1}{2}\left(1-\frac{1}{6}-\frac{1}{5}\left(1-\frac{1}{6}\right)\right)+x=1.$$
What is the value of $x$?
