+59 A rectangular room is completely tiled by 1-foot square tiles. All the adjacent to a door or wall are purple, and the rest of the tiles are white. If exactly 2/7 of the tiles are purple, then what is the smallest possible area of the room, in square feet?
+36 Suppose the total number of 1 ft2 squares is A = LW
We don't want to count the 4 corner squares twice, so since the length is L, Then there are 2(L-2) non-corner squares along the 2 lengths and 2(W-2) non-corner squares along the 2 widths. So there are 2(L-2)+2(W-2) + 4 corner squares. That's 2L-4+2W-4+4 = 2L+2W-4 squares.
2/7 * LW = 2L * 2W - 4
2LW = 14L + 14W - 28
LW = 7L + 7W - 14
LW = 7L - 7W = -14
Complete the rectangle. We want to find what we must add to both sides so that LW-7-7W will factor in the form
(L+p)(W+q)
Multiply that out: LW + qL + pW + pq
We see q=-7, p=-7, pq=49
Now we have to add 49 to both sides
LW - 7L - 7W+49=- 14+49
(L - 7) * (W - 7) = -14 + 49
(L - 7) * (W - 7) = 35
The factors of 35 are
35x1 = 7x5 Two possibilities if L > W
L-7=35, W-7=1 or L=42, W=8, LW = 336
L-7=7, W-7=5 or L=14, W=12, LW = 168
The smallest possible area is 168 ft2.
