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LMP
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A rectangular floor is fully covered with square tiles of identical si Updated on: 09 Aug 2018, 10:36
Originally posted by LMP on 09 Aug 2018, 07:52.
Last edited by chetan2u on 09 Aug 2018, 10:36, edited 1 time in total.
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95% (hard)

Question Stats:

40% (02:58) correct 60% (02:40) wrong based on 91 sessions

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A rectangular floor is fully covered with square tiles of identical size. The tiles on the edges are white and the tiles in the interior are red. The number of white tiles is the same as the number of red tiles. A possible value of the number of tiles along one edge of the floor is :

(a) 10
(b) 12
(c) 14
(d) 16
(e) 18
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B
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chetan2u
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A rectangular floor is fully covered with square tiles of identical si 09 Aug 2018, 10:23
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Bulusuchaitanya
A rectangular floor is fully covered with square tiles of identical size. The tiles on the edges are white and the tiles in the interior are red. The number of white tiles is the same as the number of red tiles. A possible value of the number of tiles along one edge of the floor is :

(a) 10
(b) 12
(c) 14
(d) 16
(e) 18
Show more

Let the inner rectangle have x tiles in column and y tiles in row that is rectangle of x*y, so red tiles =xy
The white tiles=(x+2)*1*2+y*1*2)=2x+4+2y
So 2x+2y+4=xy.........
2x+4=xy-2y=y(x-2)..........
\(y=\frac{2x+4}{x-2}\)..
Now x will be 2 less than the value we are looking for and y should be an integer..
Let us see..
A) 10......X=8, so y=(2*8+4)/(8-2)=20/6...NO
B) 12.....x=10, so y=(2*10+4)/(10-2)=24/8=3....YES
C) 14....y=28/10...NO
D) 16...y=32/12...NO
E) 18....y=40/16...NO

B
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A rectangular floor is fully covered with square tiles of identical si 09 Aug 2018, 11:29
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chetan2u
Bulusuchaitanya
A rectangular floor is fully covered with square tiles of identical size. The tiles on the edges are white and the tiles in the interior are red. The number of white tiles is the same as the number of red tiles. A possible value of the number of tiles along one edge of the floor is :

(a) 10
(b) 12
(c) 14
(d) 16
(e) 18

Let the inner rectangle be in column and row x*y, so red tiles =xy
The white tiles=(x+2)*1*2+y*1*2)=2x+4+2y
So 2x+2y+4=xy.........
2x+4=xy-2y=y(x-2)..........
\(y=\frac{2x+4}{x-2}\)..
Now x will be 2 less than the value we are looking for and y should be an integer..
Let us see..
A) 10......X=8, so y=(2*8+4)/(8-2)=20/6...NO
B) 12.....x=10, so y=(2*10+4)/(10-2)=24/8=3....YES
C) 14....y=28/10...NO
D) 16...y=32/12...NO
E) 18....y=40/16...NO

B
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Hi
I lost you here itself : Let the inner rectangle be in column and row x*y, so red tiles =xy
The white tiles=(x+2)*1*2+y*1*2)=2x+4+2y
Please elaborate
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A rectangular floor is fully covered with square tiles of identical si 09 Aug 2018, 18:45
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Bulusuchaitanya
chetan2u
Bulusuchaitanya
A rectangular floor is fully covered with square tiles of identical size. The tiles on the edges are white and the tiles in the interior are red. The number of white tiles is the same as the number of red tiles. A possible value of the number of tiles along one edge of the floor is :

(a) 10
(b) 12
(c) 14
(d) 16
(e) 18

Let the inner rectangle be in column and row x*y, so red tiles =xy
The white tiles=(x+2)*1*2+y*1*2)=2x+4+2y
So 2x+2y+4=xy.........
2x+4=xy-2y=y(x-2)..........
\(y=\frac{2x+4}{x-2}\)..
Now x will be 2 less than the value we are looking for and y should be an integer..
Let us see..
A) 10......X=8, so y=(2*8+4)/(8-2)=20/6...NO
B) 12.....x=10, so y=(2*10+4)/(10-2)=24/8=3....YES
C) 14....y=28/10...NO
D) 16...y=32/12...NO
E) 18....y=40/16...NO

B
Hi
I lost you here itself : Let the inner rectangle be in column and row x*y, so red tiles =xy
The white tiles=(x+2)*1*2+y*1*2)=2x+4+2y
Please elaborate
Show more

I have added a pic and attached a figure.
Since the tiles are square, I take y*X tiles of red colour placed in a rectangular form.
The outer layer will have one more on each side.
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A rectangular floor is fully covered with square tiles of identical si 09 Nov 2018, 07:04
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Bunuel could you please explain this? Thank you
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A rectangular floor is fully covered with square tiles of identical si 05 Oct 2019, 08:50
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LMP
A rectangular floor is fully covered with square tiles of identical size. The tiles on the edges are white and the tiles in the interior are red. The number of white tiles is the same as the number of red tiles. A possible value of the number of tiles along one edge of the floor is :

(a) 10
(b) 12
(c) 14
(d) 16
(e) 18
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\(area: xy = white + red\)
\(white: 2(x+y)-corners=2(x+y)-4\)
\(red: xy-white=xy-(2(x+y)-4)=xy-2(x+y)+4\)
\(white=red…2(x+y)-4=xy-2(x+y)+4…4(x+y)-xy-8=0…4x+4y-xy-8=0\)
\((x,y)=intergers…4x+4y-xy-8=0…4x-xy=8-4y…x=(8-4y)/(4-y)=integer\)

\(y=10…(8-4y)/(4-y)=8-40/4-10=-32/-6≠integer\)
\(y=12…(8-4y)/(4-y)=8-48/4-12=-40/-8=integer\)
\(y=14…(8-4y)/(4-y)=8-4(14)/4-14≠integer\)
\(y=16…(8-4y)/(4-y)=8-4(16)/4-16≠integer\)
\(y=18…(8-4y)/(4-y)=8-4(18)/4-18≠integer\)

Answer (B)
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A rectangular floor is fully covered with square tiles of identical si 16 Dec 2024, 06:22
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