Infinite Squares

Algebra Level 3

What proportion of the (largest) square is shaded?

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

7 8 \frac{ 7}{8} 3 4 \frac{3}{4} 6 7 \frac{ 6}{7} 4 5 \frac{4}{5} 5 6 \frac{5}{6}

Chung Kevin by Chung Kevin , 45, Australia

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2 solutions

Ruslan Abdulgani
Mar 24, 2015

Let the area of the biggest square is a2, then the area of the biggest shaded part is 3(1/2 a)^2 = (3/4) a^2, The area ratio of the two successive shaded part is 1/16. So the total shaded area is (3/4) a^2/(1 – 1/16) = (4/5) a^2,

3 Helpful 0 Interesting 0 Brilliant 0 Confused
Lorenzo Moulin
Apr 17, 2015

1 Helpful 0 Interesting 0 Brilliant 0 Confused

Handwriting programs tend to be hard to read. Maybe if you wrote it out and then took a picture of your solution instead, that will be easier to understand?

Chung Kevin - 6 years, 1 month ago

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