How many tiles was bought for the patio?

Question

A man bought square patio tiles in equal numbers of black and white for $600, each tile costing some dollars and no cents. At home, he lays a rectangle of black tiles and lays a border of uniform width of white tiles around it, using up all the tiles. How many tiles did he buy?

Were bought.

The man laid UNCUT square tiles, please! He doesn't want have to rent a tile-cutting saw.

Answer 1

If we wish to resolve it without guessing, we must follow Math Guy's approach and solve the Diophantine equation:
2(L - 1) + 2(W - 1) = (L - 2)(W - 2), or
LW = 4(L + W) - 8, solving for L we obtain
L = (4W - 8)/(W - 4) = (4W - 16 + 8)/(W - 4) =
= 4 + 8/(W - 4), so W - 4 must be a natural divisor of 8, or
W - 4 = 1, 2, 4, 8, or W = 5, 6, 8, 12,
then correspondingly L = 12, 8, 6, 5.
The sizes of the patio are either 12 x 5, or 8 x 6 so the tiles must be either 60 or 48, but 48 does not divide $600, so the only solution is 60 tiles.

Now I must hurry to buy some more gifts (I'm not going to buy tiles!) for $600, I almost have no time!
Merry Christmas to everybody!

Answer 2

L = length
W = width

2L + 2W - 2 = # of white tiles (Remember that in the corners, one corner is counted as both length and width.)

Since there are an equal number of black and white tiles,
L * W = 2*(# of white tiles)

Additionally, the area of the black tiles is
(L-2)(W-2)

So (L*W)/2 = 2L + 2W - 2 = (L-2)(W-2)


However, I am finding it easier to simply guess and check.
It is possible that he bought 30 tiles: 15 of each color. The patio could be 5 tiles by 12 tiles, so the black tiles cover an area of 3 tiles by 10 tiles.

This would mean that each tile cost exactly $20 (rather expensive, but it fits the criteria of being a whole dollar amount.)

I hope this helps!

Answer 3

The size of the tile is not significant because it is a square.

The man bought 60 tiles, 30 black and 30 white.

He made a 3 x 10 black rectangles and surrounded it with 30 white tiles.

The cost of each tile is $10.

Answer 4

ok if the tiles are 1 foot by 1 foot at a dollar a peice he bought 600 tiles i dont think you are given enough information though

like how many square foot is he covering with the tile or how big are the tile
the tile could be one inch by one inch or one foot by one foot
the rectangle could be 2 square tiles side by side

Answer 5

purchased 24 black and 24 white arranging black in inner rectangle of 6x4 and then single white tile border using all 24 outside. He bought 24+24= 48 tiles in total.

Answer 6

Yes, there is enough information to solve. It gets a little hairy.