2020 AMC 8 Problem 25

Below is the video solution and professionally curated solution for Problem 25 of the 2020 AMC 8, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2020 AMC 8 solutions, or check the answer key.

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Concepts:system of equationsrectangle

Difficulty rating: 1370

25.

Rectangles R1R_1 and R2,R_2, and squares S1,S2,S_1,\,S_2,\, and S3,S_3, shown below, combine to form a rectangle that is 3322 units wide and 2020 units high. What is the side length of S2S_2 in units?

651651

655655

656656

662662

666666

Video solution:
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Written solution:

We represent the lengths of each square as s1,s2,s_1,s_2, and s3s_3 respectively. The length of the rectangle is s1+s2+s3s_1+s_2+s_3 as these 33 squares span the entirety of a side of the large rectangle. Therefore, s1+s2+s3=3322.s_1+s_2+s_3=3322.

Also, the height of the large rectangle is the sum of the height of R2R_2 and S3.S_3. Now, note that the sum of height of R2R_2 and s2s_2 is s1,s_1, so height of R2R_2 is equal to s1s2.s_1-s_2. Therefore, the height of the large rectangle is s1s2+s3,s_1-s_2+s_3, which means s1s2+s3=2020.s_1-s_2+s_3=2020. Subtracting both of our results yields 2s2=33222020=1302.2s_2=3322-2020=1302. This would mean s2=651.s_2=651.

Thus, the correct answer is A.

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