The given expression is:

\[
\left\lfloor \frac{n}{p} \right\rfloor - 10 \left\lfloor \frac{n}{10p} \right\rfloor
\]

where \( \left\lfloor x \right\rfloor \) represents the floor function, which returns the greatest integer less than or equal to \(x\).

### Steps to Simplify:
1. **Floor Function Definition**: 
   - \( \left\lfloor \frac{n}{p} \right\rfloor \) gives the integer part of the division of \(n\) by \(p\).
   - \( \left\lfloor \frac{n}{10p} \right\rfloor \) gives the integer part of the division of \(n\) by \(10p\).

2. **Interpretation**:
   - The first term, \( \left\lfloor \frac{n}{p} \right\rfloor \), calculates how many times \(p\) fits into \(n\).
   - The second term, \( \left\lfloor \frac{n}{10p} \right\rfloor \), calculates how many times \(10p\) fits into \(n\), and this result is multiplied by 10.

Thus, this expression provides a comparison between the number of times \(p\) divides into \(n\) and how much of that gets reduced when considering multiples of \(10p\). 

### To Proceed:
- If you have specific values for \(n\) and \(p\), I can calculate the exact result.
- Would you like a more detailed explanation or examples?

### Expansion:
Here are five questions that could expand this topic:

1. How does the floor function behave with non-integer division?
2. What happens to the expression if \(n\) is much larger than \(p\)?
3. How does this expression change if \(p\) is negative?
4. Can this expression be simplified further if \(p\) is a power of 10?
5. How would the result change if we used the ceiling function instead of the floor function?

**Tip**: The floor function is useful for rounding down values and can simplify algorithms involving discrete steps.

Explore the mathematical expression ⌊n/p⌋ - 10⌊n/10p⌋, which uses the floor function to round down divisions. This detailed solution breaks down how the floor function applies to both n/p and n/10p, helping students understand how these concepts work together. Suitable for students in Grades 9-12.

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