Question

A rectangular floor is fully covered with square tiles of identical size. The tiles on the edges are white and the tiles in the interior are red. The number of white tiles is the same as the number of red tiles. A possible value of the number of tiles along one of the edges is
(a) 12
(b) 10
(c) 14
(d) 16

Answer 1

Hint: To begin with, we will assume a variable for the row and column of the inner rectangle with red tiles. So, the number of red tiles will be the product of number of times in row and number of tiles in column. Then, we will find the number of tiles on the edges using the variables assigned to the number of rows and columns and thus we can find an expression for the number of white tiles. It is given that the number of white tiles is equal to the number of red tiles. Thus, we will equate the expression for the number of red tiles and the number of white tiles and get the numerical value for the number of tiles.

Complete step by step answer:
Let the number of rows of red tiles in the inner rectangle be y and the number of columns of red tiles in the inner rectangle is x.
Thus, the number of red tiles in the inner rectangle will be the product of rows of inner rectangle and columns of inner rectangle.
Thus, number of red tiles = xy
Now, in a rectangle, there are 2 horizontal edges and 2 vertical edges.
Thus, the number of tiles on the horizontal edge will be 2 greater than the number of columns of inner tiles and the number of tiles on the vertical edges will be equal to the number of rows of the inner tiles.
Therefore, number of white tiles = (x + 2)2 + 2y = 2x + 2y + 4
Now, it is given that number of white tiles = number of red tiles
\[\Rightarrow \] 2x + 2y + 4 = xy
\[\Rightarrow \] 2x + 4 = xy – 2y
\[\Rightarrow \] 2x + 4 = y(x – 2)
\[\Rightarrow \] y = $\dfrac{2x+4}{x-2}$
Now, y is an integer. So, we will find a value of x for which y is an integer and xy is the same as 2x + 4 + 2y.
If x = 10,
y = $\dfrac{2\left( 10 \right)+4}{10-2}$
y = $\dfrac{20+4}{8}$
y = $\dfrac{24}{8}$
y = 3.
Now, Number of white tiles 2(10) + 4 + 2(3) = 30 tiles
And Number of red tiles = 10(3) = 30 tiles
Thus, x = 10 and y = 3 satisfies the condition that the number of white tiles is equal to the number of red tiles.
The horizontal edge has (x + 2) tiles.
Therefore, x + 2 = 10 + 2 = 12

So, the correct answer is “Option A”.

Note: We are looking for values of x that will make y an integer if y = $\dfrac{2x+4}{x-2}$. If we put x = 4, then we get y = 6. Thus, the number of tiles on one of the edges should be x + 2 = 4 + 2 = 6. But none of the options is 6, even though x = 4 and y = 6 satisfy all the conditions. Therefore, we put x = 10.