What fraction of a pie does each person receive? 
Two identical pies are cut into $16$ equal parts. If each part is then
  split equally among three people, what fraction of a pie does each
  person receive?

So, after cutting the pies, there are $8$ equal parts  of each pie. If each part is then split among $3$ people, each person receives ${1\over   3}$ of $1$ part of a pie. That is a person receives 8 parts of a 24 part of a pie. That is ${1\over  3 }$ of a pie. But, the answer is given as ${1\over  24}$, which is to me as "one part is divided among 3 people, not each". So, what is the difference here?
 A: Two pies are split into 16 equal parts, which means each individual pie is split into 8 equal parts, or every three people gets $\frac{1}{8}$ of the pie. Then we divide that eighth of a piece by three, or, because dividing is equal to multiplying by the reciprocal, we multiply by $\frac{1}{3}$. So, $\frac{1}{8} \times \frac{1}{3}$ is our problem. To multiply fractions, we multiply the tops and then multiply the bottoms. $1\times 1 = 1$ and $8\times 3=24$. So the answer is $\frac{1}{24}$ - each person gets only a twenty-fourth of the pie. 
A person does not receive $8\times \frac{1}{3} = \frac{8}{3}$ of the pie. To see why, let's think through this. We are trying to divide a piece into 3 parts. This piece just so happens to be an eighth of the overall thing we're cutting up. If we want a piece of a piece, shouldn't that piece be smaller than the original piece? Your solution is larger than $\frac{1}{8}$, which doesn't make sense logically. So instead of multiplying by the whole number $8$, we want to multiply by the fraction $\frac{1}{8}$. 
You are also somewhat misinterpreting the problem. You say

which is to me as "one part is divided among 3 people, not each"

Thing is, we do want that one part (in this case $\frac{1}{8}$) to be divided among 3 people. So sometimes, in cases like these, it might help to reread the problem.
Hope this helps!
