A sheet of paper of thickness of 0.5 mm repeatedly folded to fill up a gap of size 1.28 cm. The required number of folds is: (A) 12 (C) 8 (B) 10 (D) 7

We are tasked with determining the number of times a sheet of paper needs to be folded so that its thickness fills a gap of 1.28 cm, given that the initial thickness of the paper is 0.5 mm.

### Steps:

1. **Convert units**: The initial thickness is given in millimeters, and the gap size is in centimeters. Let's convert everything to the same unit (millimeters in this case):
   \[
   \text{Initial thickness of paper} = 0.5 \, \text{mm}
   \]
   \[
   \text{Gap size} = 1.28 \, \text{cm} = 12.8 \, \text{mm}
   \]

2. **Thickness after \(n\) folds**: Every time the paper is folded, its thickness doubles. After \(n\) folds, the thickness of the paper will be:
   \[
   \text{Thickness after } n \text{ folds} = 0.5 \times 2^n \, \text{mm}
   \]
   We need to find the smallest \(n\) such that the thickness equals or exceeds 12.8 mm:
   \[
   0.5 \times 2^n \geq 12.8
   \]
   
3. **Solve for \(n\)**: Divide both sides by 0.5 to isolate \(2^n\):
   \[
   2^n \geq \frac{12.8}{0.5} = 25.6
   \]

4. **Estimate \(n\)**: Now, find the smallest \(n\) such that \(2^n \geq 25.6\).

   \[
   2^4 = 16, \quad 2^5 = 32
   \]
   Therefore, \(n = 5\), as \(2^5 = 32\), which is greater than 25.6.

Thus, after 5 folds, the thickness of the paper will exceed 12.8 mm.

### Conclusion:
The required number of folds is **5**. However, since the provided options are different, there seems to be an issue with the given answers or assumptions.

Do you want to review more details or have questions on this? 

Here are some relative questions:
1. What is the role of exponential growth in folding problems?
2. How do changes in units affect calculations in such problems?
3. What would happen if the initial thickness of the paper were doubled?
4. How can this problem be solved graphically?
5. What would be the number of folds required to reach 10 cm?

**Tip**: Always ensure unit conversions are consistent when solving word problems involving measurements.

Learn how to calculate the number of folds needed for a sheet of paper with a thickness of 0.5 mm to fill a gap of 1.28 cm. This solution covers concepts of exponential growth, unit conversion, and inequalities. Suitable for students in Grades 8-10.

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