To solve the problem of determining which integer should replace $Z$ to keep the numbers in ascending order, let's first express each value consistently, either in decimal or percentage form.

1. Convert each value:

   • $57\%$ is equivalent to $0.57$ as a decimal.
   • $0.68$ is already in decimal form.
   • $\frac{Z}{10}$ is a fraction. To compare it, we can express it as $0.Z$ (i.e., $Z/10$) in decimal form.

2. Set the inequality: For the values to be in ascending order, we need: $0.57 < \frac{Z}{10} < 0.68$ This gives two inequalities: $0.57 < \frac{Z}{10}$ and $\frac{Z}{10} < 0.68$

3. Solve for $Z$:

   • From $0.57 < \frac{Z}{10}$, multiplying both sides by 10 gives: $5.7 < Z$ This means $Z > 5.7$, so $Z \geq 6$.
   • From $\frac{Z}{10} < 0.68$, multiplying both sides by 10 gives: $Z < 6.8$ So $Z$ must be less than 6.8, meaning $Z \leq 6$.

4. Conclusion: The integer $Z$ must be $6$, as it satisfies both conditions.

Would you like further details or clarification on any part? Here are some related questions you could explore:

1. What if $Z$ were a decimal rather than an integer?
2. How would the order change if we had percentages instead of decimals for all values?
3. Can you find the value of $Z$ if the inequality changes to descending order?
4. How does the placement of $Z/10$ affect the range of possible values for $Z$?
5. What are other ways to approach inequalities involving fractions?

Tip: When comparing percentages, decimals, or fractions, always convert all values into a common form, like decimals, to make it easier to compare.