Problem 15 · 2025 Math Kangaroo
Medium
Geometry & Measurement
ratioarea-fraction
Anna is looking at a picture on her smart phone. The format is 16:9 and fills the entire screen. If she turns the smart phone, the picture becomes smaller. What proportion of the screen is needed for the smaller picture?
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Answer: E — \(\frac{81}{256}\)
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Hint 1 of 2
Turning the phone rotates the picture to a 9:16 shape, which must still fit inside the same 16:9 screen.
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Hint 2 of 2
The limiting dimension is the screen's short side (height 9); scale the rotated picture so its tall side just fits, then compare areas.
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Approach: fit the rotated frame and compare areas
- Take the screen as \(16\times 9\). Rotated, the picture has shape \(9:16\); to fit, its long side must equal the screen's height \(9\), so it scales to \(9\times k\) by \(16\times k\) with \(16k=9\), giving \(k=\tfrac{9}{16}\).
- Picture area \(=9k\times 16k=144k^2=144\cdot\tfrac{81}{256}\); screen area \(=144\).
- The proportion is \(\tfrac{144\cdot 81/256}{144}=\tfrac{81}{256}\), answer E.
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