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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.

What integer should replace Z so that the values below are in ascending order: 57%, Z/10, 0.68?

Learn how to determine the integer Z to place values 57%, Z/10, and 0.68 in ascending order. Explore percentage-to-decimal conversion and inequality solving techniques suitable for middle school students in Grades 6-8.

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