2017 AMC 12A Problem 4

Below is the professionally curated solution for Problem 4 of the 2017 AMC 12A, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2017 AMC 12A solutions, or check the answer key.

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Concepts:Pythagorean Theorem percentage

Difficulty rating: 1200

Jerry and Silvia wanted to go from the southwest corner of a square field to the northeast corner. Jerry walked due east and then due north to reach the goal, but Silvia headed northeast and reached the goal walking in a straight line. Which of the following is closest to how much shorter Silvia's trip was, compared to Jerry's trip?

$30\%$

$40\%$

$50\%$

$60\%$

$70\%$

Solution:

If the square has side $x,$ Jerry walks $x+x=2x,$ while Silvia walks the diagonal $\sqrt{x^2+x^2}=x\sqrt2.$

The fraction by which Silvia's trip is shorter is $\dfrac{2x-x\sqrt2}{2x}=1-\dfrac{\sqrt2}{2}\approx 1-0.707=0.293.$

This is closest to $30\%.$

Thus, the correct answer is A.

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