Why does this question use multiplication instead of subtraction? Ok so I'm reading a book right now that tackles pre-algebra and I stumbled on this example problem that goes like this. 
Jerry bought a pie and ate 1⁄5 of it. Then his wife Doreen ate 1⁄6 of what was
left. How much of the total pie was left?
The authors solution is to solve it like this.
First, subtract 1/5 from 1/1 (pie) which equals 4/5, and then multiply 1/6 to 4/5 to get how much Doreen ate from the whole pie, 4/30.
My way obviously is to solve it like this.
First subtract 1/1 by 1/5 (the same with the author), but then I subtract 4/5 with 1/6, which gives me a totally different answer, 19/30.
So my questions are these,
1.) Why does subtracting one fraction to another not work in this scenario. (Because I have a different result)
2.) Obviously (for me) I should subtract based on the question, but why does the author use multiplication instead? 
3.) And why does multiplying instead of subtraction work?
 A: Here, I recommend we use variables.
Let $p$ represent the pie that is being eaten (somewhat timidly).
Jerry eats $1/5$ of it, so $1-1/5 = 4/5$ of the original pie is left: $4p/5$.
The his wife comes and eats $1/6$ of what is left of the pie.
Take the remaining pie ($4p/5$), and take away $1/6$ of this new pie stuffs, not the original pie.
$(4p/5)(1-1/6) = 4p/5 \cdot 5/6 = 2p/3$. So two-thirds of the pie remains.


*

*You are subtracting two fractions of the entire pie. The question asks you first about the enitre pie, then the remainder.

*'Of' implies multiplication: $10\%\ \text{of}\ 500 = .1 \cdot 500 = 50$.

*Multiplication is the only way it will work, because subtraction is the incorrect way to think about the wording of the problem.
Cheers!
A: If you eat a fraction $q$ of something then what is left is $1-q$ of that something, that is to say you are left with the earlier amount of that something multiplied by $(1-q)$ 
In effect you subtract from $1$ and then multiply  
So 


*

*They start off with $1$ pie and Jerry eats $\dfrac15$ of it, leaving $1 \times \left(1-\dfrac15\right) =\dfrac45$ of a pie

*They then have  $\dfrac45$ of a pie and Doreen eats $\dfrac16$ of that, leaving $\dfrac45 \times \left(1-\dfrac16\right)=\dfrac23$ of a pie
