A 5 meter long wire is cut into two pieces. If the longer piece is then used to form a perimeter of a square, what is the probability that the area of the square will be more than 1 if the original wire was cut at an arbitrary point?
A) 1/6
B) 1/5
C) 3/10
D) 1/3
E) 2/5
OA:E
Source: GMAT club tests
Hard combined Geometry and Probability question
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Let the wire from left to right be composed of 5 dashes, implying that the length of each dash is 1 meter:Mo2men wrote:A 5 meter long wire is cut into two pieces. If the longer piece is then used to form a perimeter of a square, what is the probability that the area of the square will be more than 1 if the original wire was cut at an arbitrary point?
A) 1/6
B) 1/5
C) 3/10
D) 1/3
E) 2/5
-----
Case 1:
-----
If a cut is made anywhere in the blue portion, the length of the remaining wire will be greater than 4 meters, allowing us to form a square with an area greater than 1.
Since the blue portion constitutes 1/5 of the wire, P(cut in the blue portion) = 1/5.
Case 2:
-----
If a cut is made anywhere in the red portion, the length of the remaining wire will be greater than 4 meters, allowing us to form a square with an area greater than 1.
Since the red portion constitutes 1/5 of the wire, P(cut in the red portion) = 1/5.
Since a favorable outcome will be yielded by Case 1 OR Case 2, we ADD the probabilities:
P(square with an area greater than 1) = 1/5 + 1/5 = 2/5.
The correct answer is E.
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Hi Mo2men,
As complex as this question might look, it's actually simpler than you realize.
Since we're using one of the wire pieces (the longer piece) to form a square, the only way for the area to be greater than 1 is if the sides of the square are ALL greater than 1 (so the perimeter would have to be greater than 4). The 'quirky' part of the logic is that you have to recognize that there are 2 'options' that will create that type of 'long piece' - cutting at ANY point between 0 to <1 meter OR >4 to 5 meters will create a "long piece" that is greater than 4 (and thus would create a square with an area > 1). Those options account for a little less than 2 meters out of the 5 total meters.
Final Answer: [spoiler]2/5 - E[/spoiler]
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As complex as this question might look, it's actually simpler than you realize.
Since we're using one of the wire pieces (the longer piece) to form a square, the only way for the area to be greater than 1 is if the sides of the square are ALL greater than 1 (so the perimeter would have to be greater than 4). The 'quirky' part of the logic is that you have to recognize that there are 2 'options' that will create that type of 'long piece' - cutting at ANY point between 0 to <1 meter OR >4 to 5 meters will create a "long piece" that is greater than 4 (and thus would create a square with an area > 1). Those options account for a little less than 2 meters out of the 5 total meters.
Final Answer: [spoiler]2/5 - E[/spoiler]
GMAT assassins aren't born, they're made,
Rich
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You can also look at this as if the cut will be somewhere in one side of the wire.Mo2men wrote:A 5 meter long wire is cut into two pieces. If the longer piece is then used to form a perimeter of a square, what is the probability that the area of the square will be more than 1 if the original wire was cut at an arbitrary point?
A) 1/6
B) 1/5
C) 3/10
D) 1/3
E) 2/5
If the wire is 5 meters long, in order for there to be a longer piece, the cut must be anywhere but the middle of the wire. Since the middle of the wire is at 2.5 meters, the cut can be made anywhere between 2.5 meters and the end of the wire. See the red section below.
--------------------
If the cut is made anywhere in the red section, the piece of the wire to the left will be longer than the piece of the wire to the right.
The red section is 1/2 of the wire, or 2.5 meters of wire.
In order for the area of the square made with the piece on the left to be 1 or greater, the four sides of the square must be length 1 or greater. So in order for it to be used to make a big enough square, the piece on the left must be 4 meters long or longer, meaning that the cut has to be made in the blue section below.
--------------------
The blue section is 1 meter long, because for the long piece to be at least 4 meters long, the short piece has to be one meter long or less.
So to create a longer piece, we can cut anywhere in a 2.5 meter range.
To create a longer piece that is at least 4 meters long, we can cut anywhere in a 1 meter range out of the 2.5 meter range.
1/2.5 = 2/5
The correct answer is E.
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Hi Mo2men,Mo2men wrote:A 5 meter long wire is cut into two pieces. If the longer piece is then used to form a perimeter of a square, what is the probability that the area of the square will be more than 1 if the original wire was cut at an arbitrary point?
A) 1/6
B) 1/5
C) 3/10
D) 1/3
E) 2/5
OA:E
Source: GMAT club tests
Solving it Algebraically...
Say the cut-wire is x meter long such that x > 5/2, and the side of the square is 'a' meter.
Thus,
Perimeter of a square = 4a = x => a = x/4
=> Area of square = x^2/16
We have to find out the probability that x^2/16 > 1.
=> x^2/16 > 1 => x^2 > 16 => x > 4
To make sure that the probability that the area of the square will be more than '1,' the range of the cut wire (5 > x > 4) would be 5 - 4 = 1 meter.
We know that the range of the cut wire (5 > x > 2.5) is 5 - 2.5 = 2.5 meter.
Thus, the required probability = 1/2.5 = [spoiler]2/5[/spoiler].
The correct answer: E
Hope this helps!
Relevant book: Manhattan Review GMAT Combinatorics and Probability Guide
-Jay
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Let's suppose that the piece cut to form the square has length x, and that the remainder is (5 - x).
We know that the square's area is then (x/4)². We need to solve
(x/4)² > 1
x² > 16
x > 4
So we need to have cut the wire so that one piece has at least length 4. The overall length is 5, so we need to have made the cut in the first fifth or the last fifth of the string. That gives us (1/5) + (1/5), or E.
We know that the square's area is then (x/4)². We need to solve
(x/4)² > 1
x² > 16
x > 4
So we need to have cut the wire so that one piece has at least length 4. The overall length is 5, so we need to have made the cut in the first fifth or the last fifth of the string. That gives us (1/5) + (1/5), or E.