A square countertop has a square tile inlay in the center, leaving an

Question

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

honneeey · Accepted Answer

Ratio of squares is 9:16 that means the ratio of sides of squares is 3:4.
Now moving ahead any number multiplied to 3 and 4 shall result in answer.
For example-3*3=9 and 4*3=12 therefore difference is 3.
Similarly, 3*5=15 and 4*5=20 difference is 5.
Next, 3*6=18 and 4*6=24 difference is 6.
So we saw all the 3 options can be derived.thus answer is E.

Please press kudos is it helped!

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adstudy · Answer

Let uniform width be "x"

Let side of the square countertop be equal to "a"

So,
total area of countertop =  a^2 
total area of tiled part =  (a-2x)^2

Now as per question,
 \frac{total area of tiled part }{ total area of countertop}  =  \frac{(a-2x)^2}{a^2}  =  \frac{9}{16} 
=>  \frac{(a-2x)}{a}  =  \frac{3}{4} 
=>  4a - 8x = 3a 
=>  a = 8x 
=>  x = \frac{a}{8}

So as we need to find "x", we see that it is dependent on the value of "a" and hence can be anything as we have no restriction for value of "a"

Hence any real number will be possible as value of "x"

Hence E

soapbolt · Answer

Great Explanation is given by @Bunnel in the below similar question.

Check this out for better understanding
https://gmatclub.com/forum/a-square-wooden-plaque-has-a-square-brass-inlay-in-the-89215.html

Hope it helps

sandip29 · Answer

9 and 12
15 and 20
18 and 24
This is length of Inner square and outer square . Width will be = 12-9= 3/2 
........= 5/2
........= 6/2

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abannore · Answer

Shouldn't the width of the strip be difference/2?
In that case, how is E the answer?

CEdward · Answer

Hi guys, I am wondering if I approached this correctly.

Let A be the inlay, and B be the area of the entire square

1. W = 3

This implies 3 + 3 + 3x = 4x --> x = 6

2. W = 5

5 + 5 + 3x = 4x --> x = 10

3. W = 6

6 + 6 + 3x = 4x --> x = 12

So basically all of these widths qualify because we end up with an integer.

E.

Bunuel  is this correct or is there a better way?

Aannie · Answer

Bunuel can you please explain this question , did not get how it is related to https://gmatclub.com/forum/a-square-woo ... 89215.html . In this question it does not say total area !

Shikhar22 · Answer

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

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

We are only dealing with ratios here as no absolute values of the sides of the squares are given. Since we are asked about possible values of the width, we must decipher the value of width in the given ratio of 9:16 or 3^2/4^2. The width is essentially the change in the value of side that is 3 to the side that is 4. The width in the given ratio is 0.5 and not 1, as the width will be added on each sides of the tiled (smaller) square. Now this 0.5 is nothing but the value of the width in the given ratio. Each given values of 3,4 and 5 is a multiple of 0.5; hence E is the correct answer. This question is at large nothing but a play with ratios and what values fit the given ratio.

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egmat · Answer

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)

https://i.imgur.com/YVwzsZ1.png

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)

https://i.imgur.com/trsw6N0.png

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.   
  Thus,  q = 4√k . -----(4)

Finding x:  
So, from the figure drawn above, it can be easily inferred that: 
 q = p + x + x

Using (3) and (4), we get 4√k = 3√k + 2x. 
  √k = 2x  
  x = √k /2

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

vivekbhansali93 · Answer

Hello can you please explain the significance of the differences here??

NamanB · Answer

in your numbers the width of the strip is coming as 1.5, 2.5 and 3. 
Not 3, 5 and 6!

Double the numbers you took to multiply with the ratio 3:4.

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