2017 AMC 10A Problem 7

P1-10~1 minPrint

Triangles: Area & Pythagorean Ratios, Percents & Averages Bound the quantity above/below or estimate to pin it down Set up the equation/formula and compute; no special trick needed

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?

Show solution without answering

Key insight

Pick an easy side length, 1. Jerry walks 2, Silvia walks the diagonal, about 1.41. She saves about 0.59 out of Jerry's 2, which is about 30%.

Solution

The answer is a percent, so the size of the field does not change it. Pick the easiest size: each side is $1$ .

1. Jerry's trip. He walks one full side east and one full side north: $1 + 1 = 2$ .
2. Silvia's trip. She walks straight across the square, corner to corner. That diagonal is the long side of a right triangle whose two short sides are both $1$ . By the Pythagorean theorem,
$\text{diagonal} = \sqrt{1^2 + 1^2} = \sqrt{2} \approx 1.414.$
3. How much shorter. $2 - 1.414 = 0.586$ .
4. As a percent. "Shorter compared to Jerry's trip" means divide by Jerry's distance:
$\frac{0.586}{2} \approx 0.293 = 29.3\%.$
5. The closest choice is $30\%$ .

The answer is $\boxed{\textbf{(A)}\ 30\%}$ .

Why this works

When every answer is a percent, choose simple numbers for the lengths; the percent comes out the same for any size. A percent decrease is always measured against the starting amount, here Jerry's longer trip. It also helps to remember $\sqrt{2} \approx 1.414$ , the diagonal of a square with side $1$ .

Concise solution

A shorter write-up, the way contest solutions are usually written. It may take a different route from the solution above.

The size of the field does not matter, so let each side be $1$ .

Jerry walks two full sides: $1 + 1 = 2$ .

Silvia walks the diagonal, the hypotenuse of a right isosceles triangle with legs $1$ , so her distance is $\sqrt{1^2 + 1^2} = \sqrt{2} \approx 1.414$ .

She saves $2 - 1.414 = 0.586$ , which as a fraction of Jerry's trip is
$\frac{0.586}{2} \approx 0.293 \approx 29\%.$

The closest choice is $\boxed{\textbf{(A)}\ 30\%}$ .

Review in the AoPS books

Introduction to Geometry, Chapter 4: Perimeter and Area (§4.2)
Introduction to Algebra, Chapter 6: Ratios and Percents
Introduction to Geometry, Chapter 6: Right Triangles (§6.1, §6.2, §6.3)
Introduction to Algebra, Chapter 7: Proportion (§7.1, §7.2, §7.3)