Problem 13 · 2011 AMC 8
Medium
Geometry & Measurement
overlap-regionarea-ratio
Two congruent squares, ABCD and PQRS, have side length 15. They overlap to form the 15 by 25 rectangle AQRD shown. What percent of the area of rectangle AQRD is shaded?
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Answer: C — 20%.
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Hint 1 of 2
The shaded piece is where the two squares overlap. You don't need its dimensions — just notice that adding both squares' areas covers the overlap twice, while the rectangle covers everything exactly once.
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Hint 2 of 2
Overlap = (area of both squares added) − (area of the rectangle they fill). This is inclusion-exclusion: the "extra" from double-counting is exactly the shared region.
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Approach: inclusion-exclusion — the overlap is the double-counted area
- Each square is 15 × 15 = 225, so the two together account for 225 + 225 = 450 of area — but the overlap got counted in both, i.e. twice.
- The rectangle they actually fill is 25 × 15 = 375 (each point counted once). The difference is the part that was double-counted: overlap = 450 − 375 = 75.
- Shaded fraction of the rectangle: 75 / 375 = 1/5 = 20%.
- Worth keeping: for any two overlapping regions, (sum of the two areas) − (area of their union) = area of the overlap — no need to measure the overlap directly.
Another way — find the overlap's dimensions directly:
- The union is 25 wide; the two 15-wide squares stick out 25 − 15 = 10 on each side, so they share a strip of width 15 − 10 = 5.
- Overlap = 5 × 15 = 75, giving 75/375 = 20%.
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