2025 AMC 10 A

Complete problem set with solutions and individual problem pages

Problem 13 Easy

The ratio of the side length of a square to the side length of the next inner square is k, where 0< k< 1. The area of the shaded portion of the figure is 64% of the area of the original square. What is k?

 

  • A.

    \frac { 3 } { 5 }

  • B.

    \frac { 1 6 } { 2 5 }

  • C.

    \frac { 2 } { 3 }

  • D.

    \frac { 3 } { 4 }

  • E.

    \frac { 4 } { 5 }

Answer:D

Solution 1:

Notice that, the shaded area and the non-shaded area are also similar, with the length ratio 1:k, and therefore an area ratio 1:k^2.

Thus the shaded area is \dfrac{1}{1+k^2}=64\%, giving k=\dfrac{3}{4}.

Solution 2:

The total area forms an infinite geometric series: 1 - k^2 + k^4 - k^6 + k^8 - k^{10} + \cdots = \frac{1}{1-(-k^2)} = \frac{1}{1+k^2}

Setting \frac{1}{1+k^2} = 0.64 yields k = \frac{3}{4}.

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