2017 AMC 10A Problem 7

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 estimation

Difficulty rating: 960

Small hint:

Let the square side length be

s

Big hint:

Compare

2s

with

s\sqrt2

as a percent of Jerry’s trip

Solution:

Let

s

be the side length of the field. Then Jerry traveled

2s

and Silvia traveled

s\sqrt{2}

from the Pythagorean theorem.

The desired value is

\begin{aligned} \dfrac{2s - s\sqrt{2}}{2s} &= \dfrac{2 - \sqrt{2}}{2} \\ &\approx \dfrac{2 - 1.4}{2} \\&= 0.3\\ &= 30\%. \end{aligned}

Thus, A is the correct answer.