### 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 evaluating a fraction that has ceiling functions in both the numerator and the denominator.
The ceiling function rounds up a number to the next integer, so for example $\lceil 3.2\rceil=4$ and $\lceil -1.8\rceil=-1$.
A possible strategy is to simplify the expressions inside the ceiling functions first, and then apply the ceiling function to the result.
To simplify $\cfrac{17}{7}-\left\lceil\cfrac{27}{17}\right\rceil$, I need to find the value of $\left\lceil\cfrac{27}{17}\right\rceil$.
Since $\cfrac{27}{17}$ is between 1 and 2, the ceiling function will round it up to 2.
So $\cfrac{17}{7}-\left\lceil\cfrac{27}{17}\right\rceil=\cfrac{17}{7}-2$.
To simplify this further, I need to find a common denominator for the two fractions.
The lowest common multiple of 7 and 1 is 7, so I can multiply the second fraction by $\cfrac{7}{7}$ without changing its value.
This gives me $\cfrac{17}{7}-\cfrac{14}{7}=\cfrac{3}{7}$.
Now I can apply the ceiling function to this result.
Since $\cfrac{3}{7}$ is between 0 and 1, the ceiling function will round it up to 1.
So $\left\lceil\cfrac{17}{7}-\left\lceil\cfrac{27}{17}\right\rceil\right\rceil=1$.
Next, I need to simplify $\cfrac{27}{7}+\left\lceil\cfrac{7\cdot17}{27}\right\rceil$.
To do this, I need to find the value of $\left\lceil\cfrac{7\cdot17}{27}\right\rceil$.
Since $\cfrac{7\cdot17}{27}$ is between 4 and 5, the ceiling function will round it up to 5.
So $\cfrac{27}{7}+\left\lceil\cfrac{7\cdot17}{27}\right\rceil=\cfrac{27}{7}+5$.
To simplify this further, I need to find a common denominator for the two fractions.
The lowest common multiple of 7 and 1 is 7, so I can multiply the second fraction by $\cfrac{7}{7}$ without changing its value.
This gives me $\cfrac{27}{7}+\cfrac{35}{7}=\cfrac{62}{7}$.
Now I can apply the ceiling function to this result.
Since $\cfrac{62}{7}$ is between 8 and 9, the ceiling function will round it up to 9.
So $\left\lceil\cfrac{27}{7}+\left\lceil\cfrac{7\cdot17}{27}\right\rceil\right\rceil=9$.
Finally, I can evaluate the fraction $\cfrac{\left\lceil\cfrac{17}{7}-\left\lceil\cfrac{27}{17}\right\rceil\right\rceil}{\left\lceil\cfrac{27}{7}+\left\lceil\cfrac{7\cdot17}{27}\right\rceil\right\rceil}$ by plugging in the values I found for the numerator and the denominator.
This gives me $\cfrac{1}{9}$.