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Problem 522

Geometry Difficulty 2.5 Multiple choice CEMC Fermat · Canada · 2022

Seven identical rectangles are used to create two larger
rectangles, as shown in Figure A and Figure B.

Figure A

Figure B

The ratio of the perimeter of Figure A to the perimeter of Figure B
is

Pick one

Official solution

Suppose that each of the small rectangles has shorter sides of
length $x$ .

Then the height of the rectangle in Figure A is $2x$ , which means that the longer side of
each small rectangle has length $2x$ .

Therefore, the rectangle in Figure A has height $2x$ and width $2x + x = 3x$ , which means that its
perimeter is $2(3x+2x) = 10x$ . Also,
the rectangle in Figure B has height $2x$ and width $x + 2x + x = 4x$ , so its perimeter is
$2(4x + 2x) = 12x$ .

Therefore, the ratio of the perimeter of Figure A to the perimeter of
Figure B is $10x : 12x$ which is
equal to $5 : 6$ , since $x \neq 0$ .

Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.