2015 AMC 10A Problems/Problem 2

Problem

A box contains a collection of triangular and square tiles. There are tiles in the box, containing edges total. How many square tiles are there in the box?

$\textbf{(A)}\ 3\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 7\qquad\textbf{(D)}}\ 9\qquad\textbf{(E)}\ 11$ (Error compiling LaTeX. Unknown error_msg)

Solution

Let be the amount of triangular tiles and be the amount of square tiles.

Triangles have edges and squares have edges, so we have a system of equations.

We have tiles total, so .

We have edges total, so .

Solving gives, and , so the answer is .

Alternate Solution

If all of the tiles were triangles, there would be edges. This is not enough, so there need to be some squares. Trading a triangle for a square results in one additional edge each time, so we must trade out triangles for squares. Answer:

See Also

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.