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# A rectangular floor measures a by b feet, where a and b are positive i

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Math Expert
Joined: 02 Sep 2009
Posts: 54496
A rectangular floor measures a by b feet, where a and b are positive i  [#permalink]

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27 Mar 2019, 22:08
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Difficulty:

65% (hard)

Question Stats:

35% (02:52) correct 65% (03:02) wrong based on 20 sessions

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A rectangular floor measures a by b feet, where a and b are positive integers with b > a. An artist paints a rectangle on the floor with the sides of the rectangle parallel to the sides of the floor. The unpainted part of the floor forms a border of width 1 foot around the painted rectangle and occupies half of the area of the entire floor. How many possibilities are there for the ordered pair (a,b)?

(A) 1
(B) 2
(C) 3
(D) 4
(E) 5

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Math Expert
Joined: 02 Aug 2009
Posts: 7584
A rectangular floor measures a by b feet, where a and b are positive i  [#permalink]

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28 Mar 2019, 07:43
3
Bunuel wrote:
A rectangular floor measures a by b feet, where a and b are positive integers with b > a. An artist paints a rectangle on the floor with the sides of the rectangle parallel to the sides of the floor. The unpainted part of the floor forms a border of width 1 foot around the painted rectangle and occupies half of the area of the entire floor. How many possibilities are there for the ordered pair (a,b)?

(A) 1
(B) 2
(C) 3
(D) 4
(E) 5

Let us draw the two rectangles as below..

The area of bigger rectangle is a*b, and that of inner rectangle is (a-2)(b-2).
It is given that $$(a-2)(b-2)=\frac{ab}{2}$$

Let us solve this.
$$(a-2)(b-2)=\frac{ab}{2}..............2(ab-2(a+b)+4)=ab..........ab-4a-4b+8=0$$
$$..............ab-4a-4b+8+8-8=0.............ab-4a-4b+16-8=0.........a(b-4)-4(b-4)=8.$$

$$(a-4)(b-4)=8=1*8=2*4$$
Now a and b are positive integers, and b>a, so two cases
1) a-4=1 and b-4=8, that is a=5 and b=12
2) a-4=2 and b-4=4, that is a=6 and b=8

Thus 2 cases

B
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A rectangular floor measures a by b feet, where a and b are positive i   [#permalink] 28 Mar 2019, 07:43
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# A rectangular floor measures a by b feet, where a and b are positive i

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