Expert: Ahmed Salami - 10/23/2008

Question
Simon is covering a wall with equal size tiles that cannot be cut into smaller pieces.  The wall is 66" high by 72" wide.  What is the area of the largest square tile that Simon can use?

Answer
Hi,
By simple reasoning. If the tiles have length l(inches) and are
arranged in quantity x horizontally and y vertically, then
lx = 72
and
ly = 66
But l has to be a factor of both 72 and 66 (we're assuming here that
the tiles have a whole number unit of length). Hence, we seek the
highest common factor of 72 and 66.
72 = 2 . 2 . 2 . 3 . 3
66 = 2 . 3 . 11
2.3 = 6 is common to both and so is the highest common factor here.
The area of the largest square tile is therefore
6 . 6 = 36 square inches

Another way to consider it is by showing the ratio of the number of
tiles to be arranged horizontally to vertically. If the tiles have
the same arrangement as before, we still have
lx = 72
ly = 66
The ratio of x to y is therefore
72 : 66 = 12 : 11
which means that for every 12 tiles arranged horizontally we have to
arrange exactly 11 tiles vertically. The options of arrangement we
have for this ratio to be satisfied in terms of number of tiles in our
situation are therefore
12 by 11
24 by 22
36 by 33
72 by 66
(Notice that, for example, we cant have 48 by 44 because 48 is not a
factor of 72 and neither is 44 a factor of 66. This consideration is
due to the fact that we're assuming whole number lengths for the tiles)
And so for each option the length of tiles to be used are
6
3
2
1
respectively with the length of 6 inches being the largest. The area
of the largest square tile is therefore 6 . 6 = 36 square inches.

Regards.