M14-17

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?

A. \(\frac{1}{6}\)

B. \(\frac{1}{5}\)

C. \(\frac{3}{10}\)

D. \(\frac{1}{3}\)

E. \(\frac{2}{5}\)

Show Answer

Official Answer

E

A.  \frac{1}{6}

B.  \frac{1}{5}

C.  \frac{3}{10}

D.  \frac{1}{3}

E.  \frac{2}{5}

Bunuel · Accepted Answer

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. \(\frac{2}{5}\)

Show Answer

Official Answer

E

A.  \frac{1}{6}

B.  \frac{1}{5}

C.  \frac{3}{10}

D.  \frac{1}{3}

E.  \frac{2}{5}

To have an area greater than 1, the side of the square must be greater than 1, and thus the perimeter must be greater than 4, which means that the longer piece must be more than 4 meters. Check the diagram shown below:
 
 https://gmatclub.com/tests-beta/resources/import/img/m14-17.png 
 
If the wire is cut anywhere in the red region, then the longer piece will be more than 4 meters long. The probability of the cut being in the red region is  \frac{2}{5} , as there are 2 red regions out of 5 total regions. Therefore, the probability that the area of the square will be more than 1 is  \frac{2}{5} .

Answer: E

lastshot · Answer

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

Bunuel · Answer

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.

samarthpawar · Answer

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.

BigSwingingDuck · Answer

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

ranaazad · Answer

Excellent question!

nitsrj · Answer

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. \(\frac{2}{5}\)

Show Answer

Official Answer

E

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.

Shiprasingh1100 · Answer

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

RK007 · Answer

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

nishantxn · Answer

Was trying to explain this question to someone on the YouTube community section. When I realised this

Think of this situation as if you were playing darts with this wire and the wire is cut at where ever the dart lands ( the question stem says the wire is cut at an arbitrary point )
 
L ————————————-———— R
 |1cm|_______|3cm|_________|1cm|

As long as the dart lands on either of the 1cm lengths we get our required result

So now we have 2cm of the total 5cm where making a cut will yield the required result

Hence 2/5 (e)

Posted from my mobile device

Bunuel · Answer

I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.

Chloee · Answer

I'm still confused because, there was no "integer" clue.
So I thought the longer piece could be 4≤x<5(like 4.2, 4,4..).

Can anyone answer my question?

Thanks.

Bunuel · Answer

Consider a number line: 'total' is a 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. You can check other solutions here: https://gmatclub.com/forum/a-5-meter-lo ... 06448.html Hope this helps.

MAN11 · Answer

when we cut line in two part, why there are 5 parts why not 2. on longer piece and one smaller piece what's the concept behind this type of question?

Bunuel · Answer

When you cut the wire, you obviously get only two pieces. The image is just a visual representation, splitting the 5-meter wire into 5 equal 1-meter sections (6 dots = 5 segments). Check the discussion above for more detail/insights.

Jayant9090 · Answer

I have got 1/5.

Logic: length of larger piece have to be >4 mt at any cost to make area>1 mt^2.

Why symmetry is playing a role here, i can't understand?

whether i cut from first red block or last red block in both of the case larger piece will >4 mt and smaller will be (<1)mt.

whole thing rest on larger piece being greater than >4mt.

Please make me understand!

thanks!

Bunuel · Answer

You're overthinking it. The red regions mark where the cut must be for the longer piece to be >4, not where the longer piece itself is. It’s symmetric: a cut at 0.3 or 4.7 both give a longer piece >4. That’s why symmetry matters and why total red length is 2 meters out of 5, so probability is 2/5. It’s a hard question, review the full discussion carefully. Also check this thread for more explanations: https://gmatclub.com/forum/a-5-meter-lo ... 06448.html

akshatv45 · Answer

Let the lengths are x and 5-x, x is the longer part of the wire. So x forms the perimeter of a square. All sides of a square are equal. side = x/4.

Area of square = x^2/16 >1 so x >4 or x<-4. Discard, x <-4.

Working with x>4 --> This basically means that the rope should not be cut at 1m more length on either side.

so, only in 40% of the area of rope/wire, there can be a cut. Hence, E is the answer.

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