There are three types of rectangular tiles : 3' x 3', 3' x 7' and 3' x 11'. An area of rectangular shape of dimensions 3' x 100' is to be covered using these tiles without breaking them. If x and y are the maximum and minimum numbers of tiles of various sizes, respectively, that can be used to cover the area exactly, then x - y is

Correct Answer: (A) 20

Because the area width is 3' and every tile has width 3', the problem reduces to partitioning the length of 100' using tiles of lengths 3', 7', and 11': $3a + 7b + 11c = 100$. To maximize tiles ($x$), use the smallest 3' tiles as much as possible: with $b = 1, c = 0$, $3(31) + 7(1) = 100$, giving $x = 31 + 1 = 32$ tiles. To minimize tiles ($y$), use the largest 11' tiles as much as possible: with $c = 8, a = 4, b = 0$, $11(8) + 3(4) = 88 + 12 = 100$, giving $y = 8 + 4 = 12$ tiles (fewer than 12 tiles cannot total 100). Thus, $x - y = 32 - 12 = 20$.