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27 Mar 2019, 22:08
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B
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Difficulty:
95%
(hard)
Question Stats:
41% (02:51) correct
59%
(02:55)
wrong
based on 46
sessions
History
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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
(A) 1
(B) 2
(C) 3
(D) 4
(E) 5
28 Mar 2019, 07:43
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Bunuel
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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General Discussion
12 Mar 2024, 09:25
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Given: 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.
Asked: How many possibilities are there for the ordered pair (a,b)?
Total area = ab
Painted area = (a-2)(b-2) = ab/2
Unpainted area = ab - (a-2)(b-2) = ab/2
ab -2b-2a+4 = ab/2
ab - 4b - 4a + 8 = 0
(a-4)b = 4a-8
b = (4a-8)/(a-4)
(a,b) = {(5,12), (6,8)} : 2 ordered pairs: (8,6), (12,5) are rejected since b>a
IMO B
Asked: How many possibilities are there for the ordered pair (a,b)?
Total area = ab
Painted area = (a-2)(b-2) = ab/2
Unpainted area = ab - (a-2)(b-2) = ab/2
ab -2b-2a+4 = ab/2
ab - 4b - 4a + 8 = 0
(a-4)b = 4a-8
b = (4a-8)/(a-4)
(a,b) = {(5,12), (6,8)} : 2 ordered pairs: (8,6), (12,5) are rejected since b>a
IMO B
