In the figure below, a rectangle has been cut into eleven squares of various sizes. The smallest square measures 9 cm x 9 cm. Determine the dimensions of the large rectangle.
Solution to Problem:
The answer is 176 cm x 177 cm.
Recall that the square shaded in grey is 9 cm x 9 cm.
Let x represent the small length on the right just above
the 9 x 9 cm square (it is marked in red in the diagram below).
Then the square shaded in green has sides of length (9 + x) cm.
Keep working from square to square to represent the length of each side:
the next square would be 18 + x, then 27 + x, then 27 + 2x, then
36 + x, then 36 + 3x, then 3x, then 36 + 6x, then 63 + 2x, and
finally 36 + 9x (the large square in the upper left).
Since the large figure is a rectangle, equate the top length and the
bottom length and solve for x:
(36+9x) + (36+6x) = (63+2x) + (27+x) + (18+x) + (27+2x)
so, 72 + 15x = 135 + 6x
therefore, x = 7.
Now substitute 7 for x in the two sides of the rectangle to get
176 cm x 177 cm.
Correctly solved by:
| 1. Richard Johnson | La Jolla, California |
| 2. Rick Jones | Kennett Square, Pennsylvania |
| 3. Keith Mealy | Cincinnati, Ohio |
| 4. Sue Mealy | Cincinnati, Ohio |
| 5. Jeff Gaither | Winchester, Virginia |
| 6. George Gaither | Winchester, Virginia |