The expression given is:

\[
\lfloor n \cdot p \rfloor - 10\lfloor n \cdot p \rfloor
\]

### Breaking it down:
1. **\(\lfloor n \cdot p \rfloor\)** represents the **floor function** of \(n \cdot p\), which means it is the greatest integer less than or equal to \(n \cdot p\).
   
2. The second term is **10 times** the same floor value, \(10\lfloor n \cdot p \rfloor\).

Thus, the whole expression simplifies to:

\[
\lfloor n \cdot p \rfloor - 10\lfloor n \cdot p \rfloor = -9\lfloor n \cdot p \rfloor
\]

### Conclusion:
The expression simplifies to:

\[
-9\lfloor n \cdot p \rfloor
\]

Would you like more details on how the floor function works or on a specific part of this solution?

Here are 5 related questions to explore:
1. What is the floor function and how does it differ from the ceiling function?
2. How does multiplying within a floor function affect the output?
3. Can you generalize this expression for other integer multipliers?
4. How does this expression behave for non-integer values of \(n\) and \(p\)?
5. What happens if we apply a different mathematical function instead of the floor function?

**

Understand how to simplify the expression ⌊n*p⌋ - 10⌊n*p⌋ by using the floor function. This problem highlights important concepts in integer functions and multiplication. Suitable for students in Grades 9-12.

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