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2023 Math Kangaroo

Problem 28

Problem 28 · 2023 Math Kangaroo Stretch
Geometry & Measurement Fractions, Decimals & Percents area-fraction

A regular hexagon is split into four quadrilaterals and a smaller regular hexagon. The ratio area of the dark sectionsarea of the small hexagon = 43. How big is the ratio area of the small hexagonarea of the big hexagon?

Figure for Math Kangaroo 2023 Problem 28
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Answer: A — 311
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Hint 1 of 2
Call the small hexagon's area S; then the dark sections total (4/3)S.
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Hint 2 of 2
The big hexagon = small hexagon + four quadrilaterals; express the quadrilaterals using the dark/light split.
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Approach: write all areas in terms of the small hexagon's area
  1. Let the small hexagon have area S. By symmetry the four quadrilaterals are equal; two of them are dark, two light.
  2. Dark = 2 quadrilaterals = (4/3)S, so one quadrilateral = (2/3)S and all four total (8/3)S.
  3. Big hexagon = small hexagon + four quadrilaterals = S + (8/3)S = (11/3)S.
  4. So small : big = S : (11/3)S = 3/11, option A.
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