Official Solution:

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. \(\frac{1}{6}\)
B. \(\frac{1}{5}\)
C. \(\frac{3}{10}\)
D. \(\frac{1}{3}\)
E. \(\frac{2}{5}\)


In order the area of a square to be more than 1, its side must be more than 1, or the perimeter must be more than 4. Which means that the longer piece must be more than 4. Look at the diagram below:



If the wire will be cut anywhere at the bolded region, then the rest of the wire (longer piece) will be more than 4 meter long. The probability of that is \(\frac{2}{5}\) (2 red pieces out of 5).


Answer: E
_________________

Bunnel,

I am confused in your explanation as , I got to the logic that in order for area of square to be greater than 1, its perimeter must be greater than 4.

Now, Point of confusion is , That there are n ( no. of points ) in between point 4-5 of rope,

How you came up with the idea of region ?

Also in case of there are more similar question & you have link, pls share ...

Regards
LS
_________________

Don't understand what you mean by the red part above. There are infinite number of points from 4 and 5...

Consider a number line: {total} is line segment of 5 units (from 0 to 5) and {favorable} is a line segment of 2 units (from 0 to 1 and 4 to 5), thus P = {favorable}/{total} = 2/5.

Check Probability and Geometry Questions to practice from our Special Questions Directory.

Hope it helps.
_________________

the question says area>1 hence the cut must be beyond the red dots shown and there can infinite potints satistfying that condition eg. 4.27 ,4.2899 etc. so how can u say that the facourable cases are 2/5

Check here: a-5-meter-long-wire-is-cut-into-two-pieces-if-the-longer-106448.html#p836781

Hope it helps.
_________________

Maybe this can simplify the question:

Main concept: Probability = (desired outcomes)/(possible outcomes)

Possible outcomes for the longer out of the 2 wires = 5-2.5 = 2.5
Desired outcomes for the longer wire to be greater than 4 = 5-4=1

Therefore the probability will be = 1/2.5 = 2/5.

I think this is a high-quality question and I agree with explanation. nice question.. its tricky

Hie Bunuel,

I see a flaw in the reasoning of the answer.
Probability = fav outcome / total outcome
if fav. outcome in this case = 2 ( 1 from each side), then total outcome = 10 (i.e. 5 from each side)

correct me if I am wrong

No, that's not right.

Again: we have
If the wire will be cut anywhere at the bolded region, then the rest of the wire (longer piece) will be more than 4 meter long. The probability of that is \(\frac{2}{5}\) (2 red pieces out of 5).


If it's still not clear you can check longer discussion of rthis question here: https://gmatclub.com/forum/a-5-meter-lo ... 06448.html

P.S. All quant questions from GMAT Club tests are correct, so if you are getting different answer it means that you went wrong somewhere.
_________________

Excellent question!
_________________

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?

Since the question says longer piece is used to form a perimeter of a square:
We have Perimeter between 2.5 meter and 5 meter
It implies: side of a square will be between 2.5/4 equals 5/8 and 5/4, On a number line it will be represented as : 5/8__________1__favourable___5/4

Probability= favourable/Total
Favourable= 5/4-1=1/4
Total= 5/4-5/8= 5/8

Probability= 1/4 divided by 5/8 which equals to 2/5 Ans.

I think this is a high-quality question and I agree with explanation.

I think this is a high-quality question and I agree with explanation.

I solved it like this. For Area to be greather than 1, the longer wire must be > 4 metre

So when a cut is made, 3 possibilites arise. wire can be <4 =4 or >4 lenth

thus prob = favourable/total = 1/3

What is wrong with this ?

can someone help me with another approach here. I cant seem to understand the red one

Check here: https://gmatclub.com/forum/a-5-meter-lo ... 06448.html Hope it helps.
_________________

Isn't the wire uniform? So, it should be 1/5? In either case whether the cut is made in the first 1/5th or the 4/5th section, the remaining part is the same. Please help

Hi,

Can someone help with why my approach is incorrect?

I assumed one side to be x and the other to be 5-x. (x is assumed to be the longer side)

The side of the square is therefore x/4 and area is \( x2\)/16.

For area to be >1, \( x2\)/16 should be greater than 1 or |x|>4.

Since 5-x>0, x<5. Therefore, favorable values of x are between 4 and 5. The whole rope is 5 units. Therefore ans = 1/5

Hi Bunuel,

Could you please help with why my approach is incorrect?

I assumed one side to be x and the other to be 5-x. (x is assumed to be the longer side)

The side of the square is therefore x/4 and area is \( x2\)/16.

For area to be >1, \( x2\)/16 should be greater than 1 or |x|>4.

Since 5-x>0, x<5. Therefore, favorable values of x are between 4 and 5. The whole rope is 5 units. Therefore ans = 1/5

Also the wire is uniform, so why do we need to consider where the wire was cut? I understand the logic to the answer provided by you but why am I not getting it mathematically?