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

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.

The original question is improperly framed and hence is not representative of a proper GMAT question. It is however helpful towards understanding the many nuances of this class of problems.

The distance between any two points on a line has to be a non zero value and hence there can only be finite number of points on a line

This is an official problem (from an early software edition of the GMATPrep tests) which tests the same concept as the question in this thread:

https://gmatclub.com/forum/a-point-is-a ... 51246.html

There isn't any ambiguity about how to interpret a question like this (on the GMAT, or in mathematics generally). A line (or 'wire') is continuous; it is not a finite collection of points, and if someone solves a problem like this by thinking of the line as a finite set of points, they'll only be approximating the answer.

This is not an official GMAT problem. This is a problem from the GMATCLUB tests collection. The linkage you have provided is to some different problem that has a different formulation. Both appear to be from the GMATCLUB tests collection.

The question in this thread is from the gmatclub tests. The question I linked to certainly is not; it's from the 2007 edition of the GMATPrep software test. You can believe that or not, as you like.

I agree with you that there are very often issues with wording in prep company questions. I frequently point out those issues on this forum. But I'm not able to understand what ambiguity you detect in this particular problem, nor do I agree with your characterization of what makes a question ambiguous. A GMAT question would be considered unambiguous if mathematicians agreed on its meaning. I'm not sure if you find ambiguous the wording in the question I linked to (which is essentially the same as the one in this thread, but simpler), but the wording in that link is the wording they use on the real test, so it would not be considered ambiguous on the GMAT.

Your conclusion does not follow. The distance between any two distinct integers is nonzero, but they are infinite in number. Or if you want to take a bounded set, the distance between any two distinct fractions 1/n, where n is a positive integer, is nonzero, but they are also infinite in number.

A line is a mathematical construct. It is defined to be an infinite collection of points (unless you're working in niche areas of finite geometry that you'd only ever encounter in very advanced math classes, and which are irrelevant on the GMAT). You can find the definition of a line counterintuitive if you like, but that is the definition, both on the GMAT and in standard mathematics. I don't understand how you think you might be helping test takers reading this thread by insisting a line is something other than what the GMAT defines it to be.

Hi All,

This question is vaguely worded in spots, but I've offered an explanation that matches the 'intent' of the question. There's an aspect to this question that many Test Takers would miss: regardless of where on the wire the 'cut' is made, the 'longer' piece is the one that's used to make the square. Thus, unless you cut the wire exactly in the middle, there are will always be two versions of each measure.

For example...
if you cut the wire at the 1m "mark", you'll have a 1m piece and a 4m piece
if you cut the wire at the 4m "mark", you'll also have a 1m piece and a 4m piece

Since area of a square is (side)^2, for the area to be GREATER than 1 m^2, the side lengths have to be GREATER than 1. By extension, this means that the perimeter would have to be GREATER than 4. Thus, any cut that produces a longer piece that is greater than 4m will satisfy what this question is asking for...

Cutting LESS than 1m will do it.
Cutting MORE than 4m will also do it.

That's approximately 2m of a 5m wire... 2/5

Final Answer:

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

OK went through a good chunk of the posts in this thread. Clearly lots of deep discussion. For simplicity, the idea here is we get 40% because the rope can be cut in 2 places.

A vast oversimplification?

When you assume x is the longer piece of rope, x must be between 2.5 and 5 meters. So your denominator should be 2.5, and not 5. Otherwise your solution is good.

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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

Area of square, \(a^2\in(\frac{25}{64},\frac{25}{16})\)

hence, probability of \(a^2\) greater than \(1\) will be \(\frac{\frac{25}{16}-1}{\frac{25}{16}-\frac{25}{64}}\)

Can you please point out the mistake?

It's when you squared things that the solution becomes incorrect, though it's quite a subtle error. We're choosing a random length, not, as your solution assumes, a random area, and lengths and areas are not distributed the same way.

You can see why this is a problem with a much simpler example: say you choose a random real number from between 0 and 10, and you use that number as the length of the side of a square. Say you want to know the probability the area of this square will be less than 25. That will happen if we pick a number less than 5, so it will happen 5/10 = 1/2 of the time. But if you square things first, and conclude "the area is something random between 0 and 100", you'd think the answer was 25/100 = 1/4. The problem is that the distance between squares grows the bigger the numbers you're squaring. The big squares, in this example, are rarer than the small ones (more dramatically - an area in my simpler example between 0 and 1 isn't all that unlikely, we'll get that 1/10 of the time, but an area between 99 and 100 is very unlikely - the probability is close to 1/200).

If you complete your solution without squaring anything, then it will become correct (if you just stop once you find lengths, and ask the probability the piece of rope is long enough, your solution will be right).