Infinite Rectangles

Algebra Level 2

What proportion of the (largest) rectangle is shaded?

Each smaller rectangle has half the horizontal length of the rectangle that it is in.
Assume that the pattern continues, even though the eye is unable to see it.

4 5 \frac{ 4}{5} 3 4 \frac{3}{4} 1 2 \frac{1}{2} 2 3 \frac{2}{3}

Chung Kevin by Chung Kevin , 45, Australia

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2 solutions

Lorenzo Moulin
Apr 17, 2015

Write a solution.

2 Helpful 0 Interesting 0 Brilliant 0 Confused
Ruslan Abdulgani
Mar 24, 2015

Let area of the biggest rectangle is ab, where a is the length.The length ratio is (1/8 a)/(1/2 a) = ¼ a, so the total shded area is (1/2 ab)/(1 – ¼) = (2/3) ab. The proportion is [(2/3) ab]/[ab] = 2/3.

2 Helpful 0 Interesting 0 Brilliant 0 Confused

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