Problem Solving

A Thin piece of wire 40 meters long is cut into 2 pieces. One piece is used to form a circle with radius r, and other is used to form a square. No wire is left over . Which of the following represents the Total area, in square meters, of the circular region and square regions in terms of r?

circumference of circle = 2pi*r

so permimeter of square = (40 - 2pi*r)

hence side of square is (40 -2pi*r)/4 = 10 - 1/2 pi*r

so totoal area is pi*r^2 + (10 - 1/2 pi*r)^2

But here's a question....why does it not work when you do it this way?

perimeter of square: 20
side of square: 5
area of square: 25

radius of circle: 10/pi
area of circle: 100/pi

then I converted the area of the square into terms based on r:
((pi*r)^2)/4=25

area of the circle:
pi*r^2
area of the square:
((pi*r)^2)/4

area of the two: answer c

I definitely see how the explanation that samir put up works, but I don't understand what in this methodology doesn't work. Obviously I'm getting caught by a trap answer, but I thought the GMAT was generally a test where there are multiple methods you can use for something to work.....

Thanks!

Anonymous wrote:A Thin piece of wire 40 meters long is cut into 2 pieces. One piece is used to form a circle with radius r, and other is used to form a square. No wire is left over . Which of the following represents the Total area, in square meters, of the circular region and square regions in terms of r?

A) (Pi) r^2
B) (Pi) r^2+10
C) (Pi) r^2 +1/4(pi)^2*r^2
D) (Pi) r^2 + (40-2(Pi)* r)^2
E) (Pi) r ^2 + (10-1/2(pi)*r)^2

The OA is E. Can someone please explain this.

Thanks

ALWAYS LOOK AT THE ANSWERS.
The first term in each answer choice (�r²) clearly represents the area of the circle.
Our only concern is the second term, which represents the area of the square.

Plug in r = 4.
Area of circle = �r² = �4² = 16� ≈ 48
Circumference of circle = 2�r = 8� ≈ 24.
Perimeter of square = remaining wire = 40-24 = 16.
Side of square = 16/4 = 4.
Area of square = 4² = 16. This is our target.

Now we scan the answers to see which second term will yield our target of 16 when r=4.

Only answer choice E works:
(10 - 1/2 pi r)^2 ≈ (10 - (1/2)*3*4))² = 16.

The correct answer is E.

I see what I was doing....misreading the question stem. I assumed that they were two equal pieces of wire, reading what I wanted to read. So lesson learned: read what's actually written, not what you want to see!

And thanks GMAT Guru for a different approach!

samirpandeyit62 wrote: circumference of circle = 2pi*r

so perimeter of square = (40 - 2pi*r)

hence side of square is (40 -2pi*r)/4 = 10 - 1/2 pi*r

so total area is pi*r^2 + (10 - 1/2 pi*r)^2

I have seen this question before. It's definitely a 700 level question, one of the more tricksy ones that in reality, "regular" people like me have a high likelihood of getting wrong. Even though I have a much deeper appreciation for all the math and the varying techniques I've learned thus far, my advice (and I am NO EXPERT) is that if you get a question like this on test day, unless you're a math superstar and know what to do, I wouldn't plug in. Truth is that if you know you're stuff, you won't need to.

I know this is the opposite of what I was taught in PR, but the problem is that I'm not a math genius, and by the time I'm done plugging in on a hard question like this (which is not the same as plugging in for medium questions, which I do all the time and highly recommend), assuming I haven't made any stupid errors along the way, it's been over 3 minutes and by the time we'd be getting this question it would be well into the middle of the math section, - and timing is everything.

Personally, I ripped my brain apart getting into the guts of the geometry, so @ this point I can see right away that A and B are not the answer, because it only accounts for the area of the circle, and then just farts in a 10 (who knows what error they predict we'll make to include this, but anyone that truly studied geometry will know that A and B are out. We've just gone from a 20% chance to a 33% chance at guessing correctly. Hooray.

Samir - you're quick way, utilizing the perimeter of a square to get the final answer is the point of what I wanted to say here: One needs a DEEP understanding of the math in order to properly solve a question like this.

If you can't look @ this problem and intuitively "know" which is wrong (like me), then my advice (and I practice what I preach) is to do a thorough review of geometry and break your brain for a week getting to the guts of it. Because really, I believe that what this problem is asking is really: "do you really know all the geometry formulas and know how to apply them in order to work backwards - and, can you do it really quickly?"

Samir knows that the perimeter of the square is all four sides added together. He also knows that the area of a square is BxH. But you and I (regular people) also know that. We also know the area of a circle. But the test makers know that you and I may not be able to realize we need to apply the perimeter and circumference formulas here and work backwards, and also presume that while we may realize we need to account for the difference in the area of the circle (which they already put in every answer choice) we won't quite know how to do it. And even if we do, hopefully, they figure, we'll mess it up along the way.

So, back to the question... we know that the perimeter of the square - (again - you have to know you need to do this in order to apply it) is going to be the total perimeter of the circle (also known as circumference) subtracted from the total wire length. So they give us 40 as the total. So it's going to be 40-(pi)D, and notice that to complicate it further they use 2pi*r (so now we also have to know when to apply each of these, and also not confuse it with (pi)r^2, (area of a circle).

So now, as Samir explained, "so perimeter of square = (40 - 2pi*r)"

So now what? Even when we get this far, we have to know that all we've gotten to is the perimeter of the square, and what we really need in order to get the area of the square is BxH, and in order to get that we need to know the side of a square, so we have to divide this sucker by 4.

But look @ the answer choices. See how D "kinda" looks like what we just got? That's a trap. Expletive. And now we've realized that we have to divide this by 4. Look at answer choice C. That one kinda looks like they've done some calculation where they get the quarter of what we're trying to get to. Another trap. Double expletive.

But since you and I are going to be geometry machines by the time we take the test, not only will we know that these might be traps and not go right to them, but we'll also have done a zillion practice questions from the official guide, and I know this sounds weird, but you kind of get a feel sometimes for what to expect in your 700+ question trap answer choice. But, because we're not experts like Mitch and Samir, we also won't know to eliminate them for sure. That's ok, because with a solid understanding of the fundamentals and the formulas, we can get there.

My PR GMAT teacher drilled into my head a whole bunch of rules:

1. DO NOT SKIP STEPS
2. DO NOT DO ANYTHING IN YOUR HEAD.
3. KNOW YOUR TARGET
4. BITE SIZE PIECES.

So we follow through instead of taking the bait. We do the math. As Samir said below...

hence side of square is (40 -2pi*r)/4 = 10 - 1/2 pi*r

A quarter of 40 is 10, and one quarter of 2 is 1/2. So then we try to put it all together and we get:

(pi)r^2 + (10 - 1/2 pi*r)

Now @ this point we're scanning the answer choices and we see that E kind of looks like it, but is it the same thing? Almost. Luckily, when we look @ it we remember that we forget to account for the actual area formula of a square which is BxH, so they're squaring it for us.

I wondered about this question as to why they didn't give us another answer choice that looks almost like this but not quite, and then I realized that they must have figured that if they did so, a person who didn't know all of the above may get the answer correct by using the answer choices and put it together, so they figured that if you got this far, you're going to remember to square it anyway. But note that the traps try to get us to pick a quicky answer, and then they try to mess with us by giving us tempting answer choices if we were able to get to the midpoint to steer us away from picking the real answer.

So as Samir said, the " total area is pi*r^2 + (10 - 1/2 pi*r)^2 "

I know that I didn't add a new technique here or somehow get you there using a different method, this was more to reach out because sometimes when I see answers to my questions from experts, I get discouraged that, not only was I not able to see it without help, but that I can't utilize these methods because I'm just not as quick as them. But be discouraged not!

My advice here simply is that for a really hard level math question like this one (@ least, for me it's hard), I'd advise not to "only" use plugging in on test day. It could take to long, you could make a mistake along the way, you may end up choosing a trap answer - just too many variables to account for. If properly prepared, you should be able to apply whatever technique works quickest to get to the answer - I believe here it's applying the formulas, carefully, to get to the right answer. However, if you find yourself in front of a problem you know that you won't be able to properly solve within the time allotted, it's best to use quickly POE to get rid of the ones you know for sure are not it, and allow yourself under 45 seconds"ish" to make your final selection. You're not losing anything if you you know you weren't going to solve it anyway, you gain time for other questions that you can properly solve, and hey - maybe it was a guinea pig question which you're not even getting graded on anyway.

Hope this helped.

Anonymous wrote:A Thin piece of wire 40 meters long is cut into 2 pieces. One piece is used to form a circle with radius r, and other is used to form a square. No wire is left over . Which of the following represents the Total area, in square meters, of the circular region and square regions in terms of r?

A) (Pi) r^2
B) (Pi) r^2+10
C) (Pi) r^2 +1/4(pi)^2*r^2
D) (Pi) r^2 + (40-2(Pi)* r)^2
E) (Pi) r ^2 + (10-1/2(pi)*r)^2

The OA is E. Can someone please explain this.

Thanks

It is worth noting that by increasing the radius of the circle (and thus its area) you decrease the remaining area of the square. The resulting sum of both areas has to account for this offsetting relationship.

Answer choice A omits the square area. Eliminate A.

All other answer choices add both individual areas up, so the second term has to decrease when the first term increases. Eliminate B and C.

How were you misreading the question stem?? I did the exact same thing - the methodology looks right to me

brightwinds wrote:But here's a question....why does it not work when you do it this way?

perimeter of square: 20
side of square: 5
area of square: 25

radius of circle: 10/pi
area of circle: 100/pi

then I converted the area of the square into terms based on r:
((pi*r)^2)/4=25

area of the circle:
pi*r^2
area of the square:
((pi*r)^2)/4

area of the two: answer c

I definitely see how the explanation that samir put up works, but I don't understand what in this methodology doesn't work. Obviously I'm getting caught by a trap answer, but I thought the GMAT was generally a test where there are multiple methods you can use for something to work.....

Thanks!

why does it matter if they are equal pieces or not - it didnt say they werent equal pieces?

brightwinds wrote:I see what I was doing....misreading the question stem. I assumed that they were two equal pieces of wire, reading what I wanted to read. So lesson learned: read what's actually written, not what you want to see!

And thanks GMAT Guru for a different approach!