Ratio of tiled area to untiled, on square countertop

[Jackie9988]

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 untiled area is 25 to 39, which of the following could be the width, in inches, of the strip?
I. 1
II. 3
III. 4

A) I only
B) II only
C) I and II only
D) I and III only
E) I, II, and III

Since the ratio of the tiled area to the untiled area is 25 to 39, the ratio of the tiled area to the total area of the countertop is 25/39+25 = 25/64.

Therefore, the ratio of the length of a side of the square tiled area to the length of a side of the square countertop is the square root 25/the square root of 64 = 5/8.

Let X be the length of a side of the countertop; let Y be the length of a side of the tiled area; and let W be the width of the untiled strip, as shown below.

Set up two equations to express the information that the length of the center tiled area is 5/8 the length of the countertop and that the width of the untiled strip is half the difference between X and Y: Y = 5/8X W = X-Y/2
Substitute 5/8X for the value of Y in the second equations, and solve for W:

W = X – 5/8X / 2 = W = 3/8X / 2 = 3/16 • X

This means that, for ANY positive value of W, there exists a countertop width that can be found using W = 3/16X. PLEASE EXPLAIN WHY all the answer choices are possible. (Please excuse the "all cap" words - I cannot under or use bold)
Answer: E

 

[Denis]

Re: Ratio

W = X – 5/8X / 2 = W = 3/8X / 2 = 3/16 • X
This means that, for ANY positive value of W, there exists a countertop width that can be found using W = 3/16X. PLEASE EXPLAIN WHY all the answer choices are possible.

True; however, appears that "they" mean an integer number of inches;
with x = 16w / 3, only w=3 provides an integer number of inches.

 

[Loren]

Re: Ratio

Maybe I'm not understanding the problem. If I were solving it as I understand it, I would draw a sketch of a square with another square in its center. I would call the side of the inside square "x". That would mean that the side of the larger square is "x+y+y" or "x+2y" where y is the width of the untiled strip around the central square. Then I would build a proportion based on that and the given ratio.
\(\displaystyle \frac{x^2}{(x+2y)^2}=\frac{25}{39}\).

I would then translate the proportion to a quadratic equation. I would then make three substitutions, one at a time.
I would let y = 1, plug that into my equation and solve for x. I would do the same for y=3 and y=4. In each case, I would figure out whether that value of y would lead to a possible result of x being a realistic value.

For instance if y=1, then I get the equation 7x^2-50x-50=0. Using the quadratic formula I get \(\displaystyle x=\frac{150\pm \sqrt{3900}}{14}\). If we use the + sign we get x = about 8.03 in. If we use the - sign we get a complex number as the result so that's out. We see that y=1 and x=8.03 might be a possibility. We check to see if \(\displaystyle \frac{8.03^2}{10.03^2}=\frac{25}{39}\). That works. So the width of the strip can be 1 inch and the ratio of the areas is 25/39.

Then you can do the same for the other two values of y and see which one(s) if any also work.

I'm sure there are simpler ways to do this, but this just came to mind so hope it helps.

 

[soroban]

Hello, Jackie!

I agree with Loren's approach . . .

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 untiled area is 25 to 39,
which of the following could be the width of the strip?
. . \(\displaystyle \text{(I)\;1}\qquad \text{(II)\;3} \qquad \text{(III)\;4}\)

\(\displaystyle (A)\;\text{ I only} \quad(B)\;\text{ II only} \quad(C)\;\text {I and II only} \quad (D)\:\text{ I and III only}\quad(E)\;\text{ I, II, and III}\)

\(\displaystyle \text{Let }x\text{ = side of the tiled square.}\)
\(\displaystyle \text{Let }y\text{ = width of the strip.}\)

      :  y  : -  x  - :  y  :
    - * - - - - - - - - - - *  -
    y |                     |  :
    - |     * - - - - *     |  :
    : |     |         |     |  :
    x |     |         |     | x+2y
    : |     |         |     |  :
    - |     * - - - - *     |  :
    y |                     |  :
    - * - - - - - - - - - - *  -
      : - - -  x+2y - - - - :

\(\displaystyle \text{The tiled area is: }\;x^2\)

\(\displaystyle \text{The untiled area is: }(x+2y)^2 - x^2 \:=\:4xy + 4y^2\)

\(\displaystyle \text{The ratio is: }\;\frac{x^2}{4xy + 4y^2} \:=\:\frac{25}{39}\)

\(\displaystyle \text{We have the quadratic: }\;39x^2 - 100xy - 100y^2 \:=\:0\)

. . \(\displaystyle \text{which factors: }\;(3x-10y)(13x+10y) \:=\:0\)

. . \(\displaystyle \text{and has the }positive\text{ root: }\;x \:=\:\frac{10}{3}y\)


\(\displaystyle \text{If all dimensions must be integers, then }y\text{ is a multiple of 3.}\)


\(\displaystyle \text{Since integers were }not\text{ stipulated, }y\text{ can be }any\text{ positive real number.}\)