Problem Solving

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

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

Let the wire from left to right be composed of 5 dashes, implying that the length of each dash is 1 meter:
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Case 1:
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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.

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

You can also look at this as if the cut will be somewhere in one side of the wire.

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.

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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.

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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.

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.