Problem 18 · 1995 AJHSME
Hard
Geometry & Measurement
area-fraction
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Answer: C — 50.
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Hint 1 of 2
The four L-shapes plus the center square fill the whole big square. So instead of measuring the center directly, find what FRACTION of the square the four L's eat up — the center is whatever fraction is left.
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Hint 2 of 2
Work in area first; only turn area into side length at the very end with a square root.
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Approach: leftover area (whole − the four L's), then square-root to a side
- Don't try to read the center square's side off the picture. Use the whole: the four L-regions and the center together make the full square (fraction 1). Each L is 316, so four of them cover 4 × 316 = 1216 = 34.
- That leaves the center square as 1 − 34 = 14 of the area.
- The big square is 100 × 100 = 10000 sq in, so the center is 14 × 10000 = 2500 sq in. Its side is √2500 = 50 inches.
- Why this transfers: for 'what's left in the middle' figures, fractions of area beat measuring lengths — and a center square that's exactly 14 the area has side exactly 12 the big side (since √(1/4) = 1/2). Sure enough, 50 is half of 100.
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