### Instruction: Evaluate $\cfrac{\left\lceil\cfrac{17}{7}-\left\lceil\cfrac{27}{17}\right\rceil\right\rceil}{\left\lceil\cfrac{27}{7}+\left\lceil\cfrac{7\cdot17}{27}\right\rceil\right\rceil}$.### Response: This problem involves the ceiling function, which rounds up any fraction to the next integer.
For example, $\left\lceil\cfrac{5}{2}\right\rceil=3$ and $\left\lceil\cfrac{11}{4}\right\rceil=3$.
To simplify the expression, I will start by evaluating the innermost ceilings first.
In the numerator, $\left\lceil\cfrac{27}{17}\right\rceil=2$, since $\cfrac{27}{17}$ is between 1 and 2.
Then, $\cfrac{17}{7}-\left\lceil\cfrac{27}{17}\right\rceil=\cfrac{17}{7}-2=\cfrac{3}{7}$, which is less than 1.
Therefore, $\left\lceil\cfrac{17}{7}-\left\lceil\cfrac{27}{17}\right\rceil\right\rceil=1$, since the ceiling of any fraction less than 1 is 1.
In the denominator, $\left\lceil\cfrac{7\cdot17}{27}\right\rceil=5$, since $\cfrac{7\cdot17}{27}$ is between 4 and 5.
Then, $\cfrac{27}{7}+\left\lceil\cfrac{7\cdot17}{27}\right\rceil=\cfrac{27}{7}+5=\cfrac{62}{7}$, which is between 8 and 9.
Therefore, $\left\lceil\cfrac{27}{7}+\left\lceil\cfrac{7\cdot17}{27}\right\rceil\right\rceil=9$, since the ceiling of any fraction between 8 and 9 is 9.
Now, the expression is reduced to $\cfrac{1}{9}$, which is the final answer.