Let's analyze the problem.

Problem Details:

1. Alan's new rectangle:

   • Alan cuts his rectangle horizontally into two halves, each of size $n \times 5$.
   • He then places these halves side by side, creating a new rectangle of size $2n \times 5$.
   • Perimeter of Alan's new rectangle: $2(2n + 5) = 4n + 10$.

2. Jenny's new rectangle:

   • Jenny cuts her rectangle vertically into two halves, each of size $\frac{n}{2} \times 10$.
   • She places these halves one below the other, creating a new rectangle of size $\frac{n}{2} \times 20$.
   • Perimeter of Jenny's new rectangle: $2\left(\frac{n}{2} + 20\right) = n + 40$.

Goal:

Find the value of $n$ such that the two perimeters are equal: $4n + 10 = n + 40.$

Solving the Equation:

$4n + 10 = n + 40$ $4n - n = 40 - 10$ $3n = 30$ $n = 10.$

Final Answer:

The value of $n$ that makes the two perimeters equal is $\mathbf{10}$.

Let me know if you'd like further explanation or additional details!
Here are five related questions to expand on this concept:

1. How does changing the orientation of cutting affect the shape and perimeter of a rectangle?
2. What would happen if Alan cut his rectangle vertically instead of horizontally?
3. Can you derive a general formula for the perimeter of a new rectangle created this way?
4. What is the area of each rectangle before and after cutting?
5. How would the solution change if Jenny and Alan had rectangles of different dimensions?

Tip: Always verify formulas for geometric transformations like cutting and rearranging by sketching diagrams.