Find all School-related info fast with the new School-Specific MBA Forum

 It is currently 01 Aug 2015, 23:25

### GMAT Club Daily Prep

#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track
Your Progress

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

# Events & Promotions

###### Events & Promotions in June
Open Detailed Calendar

# A square is drawn by joining the midpoints of the sides of a

 Question banks Downloads My Bookmarks Reviews Important topics
Author Message
TAGS:
Manager
Joined: 07 Feb 2010
Posts: 160
Followers: 2

Kudos [?]: 221 [0], given: 101

A square is drawn by joining the midpoints of the sides of a [#permalink]  15 Oct 2010, 06:27
7
This post was
BOOKMARKED
00:00

Difficulty:

55% (hard)

Question Stats:

67% (02:36) correct 33% (01:37) wrong based on 311 sessions
A square is drawn by joining the midpoints of the sides of a given square. A third square is drawn inside the second square in the way and this process is continued indefinitely. If a side of the first square is 4 cm. determine the sum of areas of all squares?

A. 18
B. 32
C. 36
D. 64
E. None
[Reveal] Spoiler: OA
Intern
Joined: 03 Aug 2010
Posts: 4
Followers: 0

Kudos [?]: 0 [0], given: 0

Re: a square is drawn [#permalink]  15 Oct 2010, 07:00
B. 32

form GP s^2 + 2 (s/2)^2 + 4 (s/4)^2
Math Expert
Joined: 02 Sep 2009
Posts: 28781
Followers: 4594

Kudos [?]: 47475 [1] , given: 7123

Re: a square is drawn [#permalink]  15 Oct 2010, 07:21
1
This post received
KUDOS
Expert's post
2
This post was
BOOKMARKED
anilnandyala wrote:
a square is drawn by joining the midpoints of the sides of a given square . a third square is drawn inside the second square in the way and this process is continued indefinately . if a side of the first square is 4 cm. detemine the sum of areas of all squares?
a 18
b 32
c 36
d 64
e none

Let the side of the first square be $$a$$, so its area will be $$area_1=a^2$$;
Next square will have the diagonal equal to $$a$$, so its area will be $$area_2=\frac{d^2}{2}=\frac{a^2}{2}$$;
And so on.

So the areas of the squares will form infinite geometric progression: $$a^2$$, $$\frac{a^2}{2}$$, $$\frac{a^2}{4}$$, $$\frac{a^2}{8}$$, $$\frac{a^2}{16}$$, ... with common ration equal to $$\frac{1}{2}$$.

For geometric progression with common ratio $$|r|<1$$, the sum of the progression is $$sum=\frac{b}{1-r}$$, where $$b$$ is the first term.

So the sum of the areas will be $$sum=\frac{a^2}{1-\frac{1}{2}}=\frac{4^2}{\frac{1}{2}}=32$$.

Answer: B.
_________________
Manager
Status: ISB, Hyderabad
Joined: 25 Jul 2010
Posts: 176
WE 1: 4 years Software Product Development
WE 2: 3 years ERP Consulting
Followers: 6

Kudos [?]: 34 [1] , given: 15

Re: a square is drawn [#permalink]  17 Oct 2010, 19:17
1
This post received
KUDOS
1
This post was
BOOKMARKED
Straight 32. The Area of square created by joining the mid points of a square is half of the bigger square.

So it becomes 16+8+4+......
Apply Infinite GP formula a/(1-r). a = 16, r = 0.5
_________________

-AD

Math Expert
Joined: 02 Sep 2009
Posts: 28781
Followers: 4594

Kudos [?]: 47475 [0], given: 7123

Re: A square is drawn by joining the midpoints of the sides of a [#permalink]  04 Jun 2013, 04:54
Expert's post
Bumping for review and further discussion*. Get a kudos point for an alternative solution!

*New project from GMAT Club!!! Check HERE
_________________
Manager
Joined: 07 May 2013
Posts: 109
Followers: 0

Kudos [?]: 13 [0], given: 1

Re: A square is drawn by joining the midpoints of the sides of a [#permalink]  01 Dec 2013, 01:59
Buneul, I did it as below.
Firstly, construct the series:
16, 8, 4, 2, 1, 1/2(.5), 1/4(.25), 1/8(.125), 1/16(.06), 1/32(.03), 1/64(.01), 1/128(.00) and so on
Sum from 16 to 1/64 is 31.975. Further you observe that as we move forward in the series the value goes on becoming insignificant and adds infinitesimally small values to 31.975. So, answer will also be very close to 31.975. So, 32 is the answer. However, one of the drawbacks of the method I suggested is that one has to be very good in dealing with fractions.
Senior Manager
Joined: 13 May 2013
Posts: 475
Followers: 1

Kudos [?]: 87 [0], given: 134

Re: A square is drawn by joining the midpoints of the sides of a [#permalink]  12 Dec 2013, 05:20
Next square will have the diagonal equal to a, so its area will be area_2=\frac{d^2}{2}=\frac{a^2}{2};
And so on.

I don't quite follow. You say the diagonal is equal to a (the exterior length of the largest triangle) but isn't it equal to 1/2 the diagonal * √2?
Math Expert
Joined: 02 Sep 2009
Posts: 28781
Followers: 4594

Kudos [?]: 47475 [1] , given: 7123

Re: A square is drawn by joining the midpoints of the sides of a [#permalink]  12 Dec 2013, 05:36
1
This post received
KUDOS
Expert's post
WholeLottaLove wrote:
Next square will have the diagonal equal to a, so its area will be area_2=\frac{d^2}{2}=\frac{a^2}{2};
And so on.

I don't quite follow. You say the diagonal is equal to a (the exterior length of the largest triangle) but isn't it equal to 1/2 the diagonal * √2?

The area of a square is $$side^2$$ or $$\frac{diagonal^2}{2}$$.
Attachment:

Squares.png [ 2.4 KiB | Viewed 5900 times ]
The length of a diagonal of blue square is equal to the length of a side of black square. The area of black square is $$side^2=a^2$$ and the area of blue square is $$\frac{diagonal^2}{2}=\frac{a^2}{2}$$.

Hope it's clear.
_________________
GMAT Club Legend
Joined: 09 Sep 2013
Posts: 5711
Followers: 323

Kudos [?]: 63 [0], given: 0

Re: A square is drawn by joining the midpoints of the sides of a [#permalink]  27 Feb 2015, 05:02
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________
Re: A square is drawn by joining the midpoints of the sides of a   [#permalink] 27 Feb 2015, 05:02
Similar topics Replies Last post
Similar
Topics:
10 Four identical circles are drawn in a square such that each 7 24 Jul 2013, 10:14
1 A certain square is to be drawn on a coordinate plane. One 1 30 Mar 2012, 03:42
1 H,G,F and E are midpoints of the sides of square ABCD 3 10 Feb 2012, 23:48
56 A certain square is to be drawn on a coordinate plane 33 14 Sep 2010, 21:06
11 A certain square is to be drawn on a coordinate plane. One 75 09 Feb 2007, 00:01
Display posts from previous: Sort by

# A square is drawn by joining the midpoints of the sides of a

 Question banks Downloads My Bookmarks Reviews Important topics

 Powered by phpBB © phpBB Group and phpBB SEO Kindly note that the GMAT® test is a registered trademark of the Graduate Management Admission Council®, and this site has neither been reviewed nor endorsed by GMAC®.