Bunuel wrote:
A square countertop has a square tile inlay in the center, leaving an untiled strip of uniform width around the tile. If the ratio of the tiled area to the total area of the countertop is 9 to 16, which of the following could be the width, in inches, of the strip?
I. 3
II. 5
III. 6
A. I only
B. III only
C. I and II only
D. II and III only
E. I, II and III
GIVEN: - Square tiled area at the center of a square countertop.
- An untiled strip of uniform width around the tile.
- \(\frac{Tiled Area}{Area of countertop}= \frac{9}{16}\)
TO FIND: - Possible width(s) of the strip
SOLUTION: To get a clear picture of what all is happening, let’s first
visualize the situation in a figure.
Visualization: We have two squares, a smaller one (
tiled area) lying at the
center of a bigger square (full
countertop). Let’s name the smaller square ABCD with side ‘
p’ and the bigger one EFGH with side ‘
q’. ---(1)
Now, the question says that the untiled strip (that is the area between the smaller and the bigger squares) is of
uniform width, say x. (Observe the figure below)
Finding p and q (sides of the small and big squares, respectively): We’re given that \(\frac{Tiled Area}{Area of countertop}=\frac{9}{16}\)
Let Tiled area (ABCD) = 9k and Total area of countertop (EFGH) = 16k for some positive number k. ----- (2)
- Tiled area (ABCD)
- From (2), tiled area (ABCD) = 9k
- And from (1), tiled area (ABCD) = \(p^2\)
- Hence, \(p^2\) = 9k, or p = 3√k.
- Note: √k is defined since k is positive. ------ (3)
- Similarly, countertop area (EFGH) =\( q^2\) = 16 k. [From (1) and (2)]
Finding x: So, from the figure drawn above, it can be easily inferred that:
Using (3) and (4), we get 4√k = 3√k + 2x.
Now, let’s see what all values, out of the given options, can x take. So, if we check the three given values:
- x = 3
- 3 = √k /2 or √k = 6.
- So, x = 3 is POSSIBLE for k = 36.
- x = 5: POSSIBLE when k is 100.
- x = 6: POSSIBLE when k is 144.
Note: From this, we can say that all positive ‘x’ is possible for the width of untiled strip, depending on the value of ‘k’.
So, the correct answer is
option E.
TAKEAWAYS: - Visualization is a key skill for almost all Geometry questions. If you can visualize a given problem correctly, you will find it way easier to solve it further.
- Specific to this question - Suppose a figure F is surrounded by a border/strip of uniform width (x), to form a bigger figure B. Then, each dimension 'a' of figure F will correspond to dimension (F + 2x) of B.
Best,
Aditi Gupta
Quant expert,
e-GMAT _________________