What were the dimensions of the original rectangle?

Question

Alexandra started with a rectangle of paper. With one straight cut she divided it into a rectangle and a square. She took the rectangle and with one straight cut she divided it into a rectangle and a square, which was smaller than the previous one. She kept repeating this process until eventually the final rectangle was a square with sides 1 centimetre and she was left with a pile of squares of paper. The average area of the squares was a two digit number of square centimetres. What were the dimensions of the original rectangle? 

 

Show ALL working please to show how you hot to the answer. Thanks! 

 

Answer 1

1 by 1 cm^2 is the last square

2nd to last before the final cut was 2 1x1's

3rd from last was 2 by 2 square

4th last was 2 by 4

5th was 4 by 4

6th last was 8 by 4

7th 8 by 8

8th 16 by 8

9th 16 by 16

10th 32 by 16

11th 32 by 32

areas are

32x32 = 1024

16 x 16 = 256

8x8 = 64

4x4 = 16

2x2 = 4

1x1 = 1

average areas are

1, 5/2, 21/3, 85/4, 256/5, 1280/6

only 85/4 and 256/5 are 2 digit averages

21.25 or 51.2

16 by 16 or 8 by 8?