A 5 meter long wire is cut into two pieces. If the longer piece is the

Hi preetamsaha,

The wording of the prompt is clearly vague in spots - and thankfully the questions that you'll face on the Official GMAT will be written in a far more precise "style." That having been said, I've provided an explanation for what the writer was likely trying to ask us to solve (in a post that appears immediately above yours in the thread). The fact that there are an infinite number of ways to cut the wire is irrelevant; we're trying to define the fraction of the possibilities that will end up giving us a "longer piece" that is greater than 4 meters.... and there are 2 ways to do that:

-With a 'cut' that is made somewhere between the 0 meter and 1 meter mark or
-With a 'cut' that is made somewhere between the 4 meter and 5 meter mark

Those two options account for approximately (1+1)/5 = 2/5 of the possible spots that the wire can be cut to produce the outcome that we're looking for.

GMAT assassins aren't born, they're made,
Rich

EMPOWERgmatRichC thanks.

When I say pick a number between 0 and 5, you could pick 0.4 or 1.987 or 4.672091 etc. Now what is the probability that it will lie between 4 and 5?
Well, it could lie between 0 and 1 or between 1 and 2 or between 2 and 3 or between 3 and 4 or between 4 and 5. Each of the five intervals is equally likely and hence the probability that it will lie between 4 and 5 is 1/5.

Now imagine the wire has markings of 1, 2, 3, 4 and 5. If you cut somewhere between 0 and 1 or somewhere between 4 and 5, you will get a bigger piece which is more than 4 in length. Since both these intervals are acceptable, the probability is 2/5.

VeritasKarishma thanks for the explanation.

there are 5 subdivisions you can cut the wire
x>1| 1<x<2 | 2<x<3 | 3<x<4 | 4<x<5 |

For area greater or equal to 1, the longer piece must be
x/4 = 1
x=4 meters long

if you cut a x>=1, longer piece is at least 5-1 = 4 long
if you cut at x>=4, longer piece is at least 5-1 = 4 long

2/5

I think this problem could be interpreted by defining x as a continuous random variable which represents the length of wire (not neseccarily the longer nor smaller piece) and I'm assuming as well that x is uniformly distributed from (0,5) meaning the probability of every point being cut in the wire has an equal probability. So we could also define a probability density function in which the x values are only from 0 to 5, with each of the continuous x values being plotted into a similar y. Since the area under the curve of any pdf must be one, so we could observe a rectangle with height a and breath 5. So 5a=1, a=1/5. So if the event we desire is that the area of the rectangle must be more than 1, than the x satisfying that event will be from 4 to 5. so the area below the curve of y=1/5 from x=4 to x=5 is 1/5. But since the wire could be cut from either side of the wire such that the longer piece will be more than or equal to 4, the probability will be twice as much, or 2*1/5=2/5. I might be wrong here but this is my interpretation and I will accept any critiques.

Not able to get the answer.

How can we decide the number of places where wire can be cut?

Shouldn't the question be suppported with a diagram?

Hi IanStewart sir,

In the question under discussion, to obtain a ratio of 4:1 we can make the cut at either 1m mark or the 4m mark. The favourable outcomes thus equate to 2.
Suppose, we had to obtain a ratio of 3:2, then we would have to cut the rope at 2m mark or 3m mark. Again the favourable outcomes equate to 2.

Thus, the probability of obtaining 4:1 ratio and 3:2 ratio will be the same.

Please correct if my understanding is wrong.

Regards
Lipun

I think you might be misunderstanding the question. It sounds as if you're thinking we must cut the rope in integer lengths, and that's not the case -- we might cut the rope into two pieces of lengths 3.77 and 1.23 meters, for example.

Wherever we cut this rope, the longer piece will be somewhere between 2.5 and 5 meters long, and each length in that range is equally likely. To make a square of area 1 or greater, we need the longer piece to be between 4 and 5 meters long. So we're picking a random length from a range of 2.5 meters, and we need to pick a length from a range of 1 meter to get our square, and the answer is thus 1/2.5 = 2/5.

Given: A 5 meter long wire is cut into two pieces.
Asked: 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?

Let the longer piece of wire be of length x meters
Shorter piece's length = (5- x) meters

Perimeter of square = x meters
Side of square = \(\frac{x}{4}\) meters
Area of the square = \(\frac{x^2}{16}\) square meters

\(\frac{x^2}{16} > 1\)
\(x^2 > 16\)
\(x > 4\)

For the length of longer piece x to be >4, wire may be cut at < 1 meters or > 4 meters

Probability = \(\frac{2}{5}\)

IMO E

Thank you for the quick response.
Perhaps I didn't put forward my point clearly. I do understand that it is not necessary to cut the rope in integer lengths.

A------B(2.5)----X(4)--C(5) {fig 1}
A--X(1)----B(2.5)------C(5) {fig 2}

My understanding prior to your explanation:
To obtain a min length of 4m, we can cut the rope in the range XC (fig 1) or AX (fig 2). Since, we have two favourable ranges (XC and AX) and the probability of each range is 1/5, the total probability is 2/5.

From your explanation, I understand the above reasoning mayn't be correct.

Can you please explain for the scenario where we need a rope of at-least 3m? Will it be (2.5-0.5)/2.5 = 4/5 ?

Regards
Lipun

we know that length of wire is 5 meters
which when cut is formed a perimeter of square
target find the the probability that the area of the square will be more than 1 if the original wire was cut at an arbitrary point?

let the wire be marked as
0;.1;.2;.3;.3.....1;1.1;1.2;1.3,1.4.......2;2.1..... so on till 4.8'4.9;5.0
so total values of divisions b/w each length integer mark is 10 points ; so for 5 such points 50
now the wire can be cut either at points from 0 to 1 (0.1 to 1) and 4 to 5(4.1 to 5.0) i.e total 20 points
so P of getting wire such that area >1 ; 20/50 ; 2/5
option E

that the area of the square will be more than 1

Please note: it doesn't say equal or more than 1.

More than 4 means= ( 4.1, 4.2,4.3,4.4,4.5,4.6,4.7,4.8,4.9 and 0.1….0.9 ) = 18
Total markings ( 0.1 to 4.9) = 49
18/49 = 36.7%


A. 16.7%
B. 20%
C. 30%
D. 33.33%
E. 40%

Should not I select D in this way?

IanStewart

Lipun - I thought i might have misunderstood what you were asking. Yes, your understanding is correct, and the answer to your 3- meter question is indeed 4/5.

For itsSKR and firas, this is a probability question where we have an infinite number of possible choices. We could cut the rope so the longer piece is 2.767676.... meters long, or 4.9993 meters long, or π meters long. The GMAT does occasionally test this, but only ever (as far as I've seen) using geometry. Because you have an infinite number of choices, you can't precisely answer such questions by looking only at a finite set of possibilities (like 4.1, 4.2, 4.3, etc). If you do that, you'll just be making an estimate (which might be enough, depending on the answer choices).

These questions can deal with one of two concepts: lengths, or areas. If you are asked the probability, when choosing a random length, that your chosen length meets some requirement, the answer is equal to

the size of the set of lengths that meets your requirement / the total length you're choosing from

So if you were asked this question, very similar to the one in this thread: a 10 meter rope is cut at a random point. What is the probability the shorter piece is less than 1.23 meters long? The shorter piece is anywhere from 0 to 5 meters long, so we're picking from a total length of 5 meters. The lengths that "work" are all of those between 0 and 1.23 meters. So the answer is 1.23/5 = 0.246

This concept is tested slightly more often using area. If you choose a random point from some area, and want to know the probability the point belongs to some smaller region, the answer will be

area of the smaller region / total area you are choosing from

So, for example, say you had a circular dartboard of radius 4, with a small "bullseye" circle of radius 0.2 in the middle. If you throw a dart at the board, and it hits a random point on the board, the probability the dart will end up in the bullseye circle is

area of bullseye / total area of the board = π (0.2)^2 / π (4)^2 = 0.04/16 = 1/400

In any infinite probability question of this type, the probability of making any specific selection is always zero. So in the original question, with the 5 meter rope, the probability the longest piece is exactly 4 meters long, or exactly 3.13131313... meters long, or any other specific value, is zero. That's because we're trying to pick 1 thing from an infinite set, and 1/∞ = 0. So in the original question, it doesn't matter if you answer "what is the probability the area of the square is 1 or greater?" or "what is the probability the area of the square is greater than 1?" The answer to those two questions is the same since the probability the area is exactly 1 is zero. That's why I didn't bother to check the original wording of the problem to see which language it used, because it doesn't affect the solution.

Thanks IanStewart for lucid explanation.

To me this seems counterintuitive.

The theory that the distance between two points can be arbitrarily small defies common sense logic.

Consider points A and B with a distance of 5m between them. Consider an ant which occupies only a point region. It travels from B to A and moves slower and slower that its speed keeps reducing by one tenth of the previous sec and so travels one tenth of the distance per sec compared to the previous second. So it would never reach A though it is actually moving as the distance it moves forward tends to zero . I believe there is something seriously wrong here.

A point region is defined and exists in space. The point region next to it is undefined but in reality there should be one next to it. Does not the understanding that there are infinite number of points between them go against reality?

The math may be correct but what about the logic? Ultimately math depends on logic.

I think this is a fundamental problem beyond the level of even math.

I believe space does not exist as points. It is more as stretches and any two points in the stretch are not distinguishable and are one and the same. Thus i believe the reality is that, there are finite number of stretches and not infinite number of points.

One thing that's true of math (and of physics, for that matter) is that it doesn't matter what one "believes" or finds "intuitive". Mathematics studies what you can logically deduce from a set of definitions and axioms, nothing more. So if you want to take as an axiom that there's a finite number of points on a line, or that some distances cannot be subdivided, you can do that -- then you get into the well-developed field of mathematics known as "Finite Geometry". That kind of geometry barely resembles what we see on the GMAT, which is Euclidean Geometry, and in Euclidean Geometry, lines are continuous and distances can be infinitely subdivided.

It's easy to see why the geometry we're all familiar with (GMAT geometry) becomes self-contradictory (and thus nonsensical) if you assume there is a smallest possible distance, or that points have nonzero length. I'll prove that below, but this is completely irrelevant to GMAT test takers:

- if you can draw a length of 1, you can make a 1x1 square
- you can draw the diagonal of that square, so you can make a length of √2
- you can then subdivide that diagonal into a length of 1 and a length of √2 - 1
- if you can do that, you can make any length of the form (√2)(c) + d, where c and d are integers. For example, to make a length of 4√2 - 3, you can divide the diagonal of a 4x4 square into a length of 3 and a length of 4√2 - 3
- if you think there is a length that cannot be further subdivided, you must think that there is a smallest length we can make in this way, so a smallest positive number that can be written (√2)(a) + b, where a and b are integers
- that number is between 0 and 1, so if we square it, it gets smaller. But if we square that number we get (√2)(2ab) + 2a^2 + b^2. That's another number in the form (√2)(c) + d, so it's another distance we can make, and it's smaller than (√2)(a) + b.
- So if we assume there is a smallest possible distance, we reach a contradiction -- we can make an even smaller distance. So there can be no such thing, in ordinary geometry, as a "smallest distance"; distances can be infinitely subdivided.

So if you pick a random point from a line -- say, for ease of illustration, from the numbers on the number line between 0 and 1 -- the probability you pick a specific point or number, say 0.565656.... is zero, because you're trying to pick one specific thing from an infinitude of possibilities.

I'd add that the reason "paradoxes" like Zeno's paradox or the Hilbert Hotel Paradox exist is because most people do not find the concept of infinity "intuitive". But if you were to use whatever conception of infinity I gather you do find intuitive, and develop mathematics from the postulate that some distances cannot be subdivided, then you'd have to discard all of calculus, advanced probability theory, advanced statistics, analytic number theory -- most of advanced math besides discrete math.