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 are is 25 to 39, which of the following could be the width, in inches, of the strip?
I. 1
II. 2
III. 4
(A) I only
(B) II only
(C) I and II only
(D) I and III only
(E) I, II, and III
Here is the explanation, i understand how it works, the only thing im confused about is how they figure the width of the untiled strip is half the difference of the area of the untiled minus the tiled divided by 2. If anyone can explain this I will greatly appreciate it. Thank you for your time. Explanation below
Explanation
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 square countertop is Square Root of 25/64 = 5/8. Lex 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.
Set up the two equations to express the information that the length of the center tiled area is 5/8 the length of the coutertop and that the width of the untiled strip is half the difference between x and y
y = 5x/8
w = x - y/2 (I DONT UNDERSTAND THIS RELATIONSHIP-CAN SOMEONE P
PLEASE EXPLAIN)
Substitute 5x/8 for the value of y in the second equation, and solve for w;
w = x - (5x/8) / 2
w = (3x/8) / 2 = 3x/16
This means that, for ANY positive value of w, there exists a countertop width that can be found using w = 3x/16. Therefore, all the answer chioces are possible
E
I. 1
II. 2
III. 4
(A) I only
(B) II only
(C) I and II only
(D) I and III only
(E) I, II, and III
Here is the explanation, i understand how it works, the only thing im confused about is how they figure the width of the untiled strip is half the difference of the area of the untiled minus the tiled divided by 2. If anyone can explain this I will greatly appreciate it. Thank you for your time. Explanation below
Explanation
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 square countertop is Square Root of 25/64 = 5/8. Lex 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.
Set up the two equations to express the information that the length of the center tiled area is 5/8 the length of the coutertop and that the width of the untiled strip is half the difference between x and y
y = 5x/8
w = x - y/2 (I DONT UNDERSTAND THIS RELATIONSHIP-CAN SOMEONE P
PLEASE EXPLAIN)
Substitute 5x/8 for the value of y in the second equation, and solve for w;
w = x - (5x/8) / 2
w = (3x/8) / 2 = 3x/16
This means that, for ANY positive value of w, there exists a countertop width that can be found using w = 3x/16. Therefore, all the answer chioces are possible
E
