2017 AMC 12A Problem 4

Question

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\%

Answer: A

Concepts:Pythagorean Theorem percentage

Difficulty rating: 1200

Small hint:

Let the side be

x

; Jerry walks

2x

and Silvia walks the diagonal

Big hint:

The diagonal is

x\sqrt2,

so the fraction saved is

\dfrac{2x-x\sqrt2}{2x}=1-\dfrac{\sqrt2}{2}

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

\begin{aligned} &\dfrac{2x-x\sqrt2}{2x}=1-\dfrac{\sqrt2}{2} \\ &\approx 1-0.707=0.293. \end{aligned}

This is closest to

30\%.

Thus, the correct answer is A.