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Manager  Joined: 25 Aug 2008
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The figure is made up of a series of inscribed equilateral triangles.  [#permalink]

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5
23 00:00

Difficulty:   95% (hard)

Question Stats: 49% (02:37) correct 51% (02:18) wrong based on 170 sessions

### HideShow timer Statistics The figure is made up of a series of inscribed equilateral triangles. If the pattern continues until the length of a side of the largest triangle (i.e. the entire figure) is exactly 128 times that of the smallest triangle, what fraction of the total figure will be shaded?

A. $$\frac{1}{4}(2^0 + 2^{-4} + 2^{-8} + 2^{-12})$$

B. $$\frac{1}{4}(2^0 + 2^{-2} + 2^{-4} + 2^{-6})$$

C. $$\frac{3}{4}(2^0 + 2^{-4} + 2^{-8} + 2^{-12})$$

D. $$\frac{3}{4}(2^0 + 2^{-2} + 2^{-4} + 2^{-6})$$

E. $$\frac{3}{4}(2^0 + 2^{-1} + 2^{-2} + 2^{-3})$$

Attachment:
File comment: Inscribed Traingles InscribedTraingles.JPG [ 6.08 KiB | Viewed 12407 times ]

Originally posted by varunmaheshwari on 17 Sep 2011, 20:45.
Last edited by Bunuel on 02 Oct 2017, 11:38, edited 1 time in total.
Renamed the topic and edited the question.
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Re: The figure is made up of a series of inscribed equilateral triangles.  [#permalink]

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8
4
varunmaheshwari wrote:
Attachment:
InscribedTraingles.JPG
The figure is made up of a series of inscribed equilateral triangles. If the pattern continues until the length of a side of the largest triangle (i.e. the entire figure) is exactly 128 times that of the smallest triangle, what fraction of the total figure will be shaded?

A. 1\4(2^0 + 2^-4 + 2^-8 + 2^-12)

B. 1\4(2^0 + 2^-2 + 2^-4 + 2^-6)

C. 3\4(2^0 + 2^-4 + 2^-8 + 2^-12)

D. 3\4(2^0 + 2^-2 + 2^-4 + 2^-6)

E. 3\4(2^0 + 2^-1 + 2^-2 + 2^-3)

PS: I am sorry for the formatting of the answer choices like this.

We can arrive at the answer quickly by estimating. The outermost triangle is split into 4 equal triangles 3 of which are completely shaded. Hence the fraction shaded is more than 3/4. So we are probably looking at options (C), (D) and (E).

We are left with the center triangle which occupies 1/4th of the area. 3/4th of this 1/4th is not shaded while a part of 1/4th of 1/4th is shaded. How much of this is completely shaded? 3/4th since the 1/4th of 1/4th triangle is also split into 4 equal areas and 3 of those areas are shaded. Hence the area of the shaded region would be something like this:

3/4 + (3/4)*(1/4)*(1/4) + ... = (3/4)(1 + 1/16 + ...)

We see that this matches option (C) which is 3/4(1 + 1/16 + ...)

Note that none of the other options can give an area close to this one.
D. 3\4(2^0 + 2^-2 + 2^-4 + 2^-6) = (3\4)(1 + 1/4 + ...) Too much
E. 3\4(2^0 + 2^-1 + 2^-2 + 2^-3) = (3\4)(1 + 1/2 + ...) Too much again
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Re: The figure is made up of a series of inscribed equilateral triangles.  [#permalink]

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varunmaheshwari wrote:
Attachment:
InscribedTraingles.JPG
The figure is made up of a series of inscribed equilateral triangles. If the pattern continues until the length of
a side of the largest triangle (i.e. the entire figure) is exactly 128 times that of the smallest triangle, what
fraction of the total figure will be shaded?

A. 1\4(2^0 + 2^-4 + 2^-8 + 2^-12)

B. 1\4(2^0 + 2^-2 + 2^-4 + 2^-6)

C. 3\4(2^0 + 2^-4 + 2^-8 + 2^-12)

D. 3\4(2^0 + 2^-2 + 2^-4 + 2^-6)

E. 3\4(2^0 + 2^-1 + 2^-2 + 2^-3)

PS: I am sorry for the formatting of the answer choices like this.

$$Area \hspace{3} of \hspace{3} equilateral \hspace{3} triangle = \frac{\sqrt{3}}{4}*(side)^2$$

$$Total \hspace{3} Area = \frac{\sqrt{3}}{4}(128)^2=\frac{\sqrt{3}}{4}(2^7)^2=\frac{\sqrt{3}}{4}*2^{14}$$

$$Shaded \hspace{3} portion \hspace{3} of \hspace{3} outer \hspace{3} most= \frac{3}{4}*\frac{\sqrt{3}}{4}*(2^7)^2$$ {:Side=128=2^7}
Skip the next: {:Side=64}
$$Shaded \hspace{3} portion \hspace{3} of 3^{rd}= \frac{3}{4}*\frac{\sqrt{3}}{4}*(2^5)^2$$ {:Side=32=2^5}
$$Shaded \hspace{3} portion \hspace{3} of 5^{th}= \frac{3}{4}*\frac{\sqrt{3}}{4}*(2^3)^2$$ {:Side=8=2^3}
$$Shaded \hspace{3} portion \hspace{3} of 7^{th}= \frac{3}{4}*\frac{\sqrt{3}}{4}*(2^1)^2$$ {:Side=2=2^1}

Take the fraction:
$$\frac{\frac{3}{4}*\frac{\sqrt{3}}{4}(2^{14}+2^{10}+2^{6}+2^{2})}{\frac{\sqrt{3}}{4}*2^{14}}$$

$$=\frac{\frac{3}{4}(2^{14}+2^{10}+2^{6}+2^{2})}{2^{14}}$$

$$=\frac{3}{4}(\frac{2^{14}}{2^{14}}+\frac{2^{10}}{2^{14}}+\frac{2^{6}}{2^{14}}+\frac{2^{2}}{2^{14}})$$

$$=\frac{3}{4}(2^{(14-14)}+2^{(10-14)}+2^{(6-14)}+2^{(2-14)})$$

$$=\frac{3}{4}(2^0+2^{-4}+2^{-8}+2^{-12})$$

Ans: "C"
##### General Discussion
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Re: The figure is made up of a series of inscribed equilateral triangles.  [#permalink]

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1

To get a triangle 128 times(2^7) smaller than the main triangle, you will have to partition it 7 times.
For each Odd partition you will get 3 Shaded regions.
For each Even partition you will get 1 shaded region the will again be partitions into regions by subsequent odd partition.So, we will no consider these partitions.

Partition 1 2 3 4 5 6 7
Shaded Regions 3 1 3 1 3 1 3

Consider the area of main triangle be A.
So the shaded region will be (3)*(1/4^7)A + (3)*(1/4^5)A + (3)*(1/4^3)A + (3)*(1/4^1)A

= 3/4 (2^-12 + 2^-8 + 2^-4 + 2^0 ) A

To get the fraction of shaded region , (3/4 (2^-12 + 2^-8 + 2^-4 + 2^0 ) A) / A
= 3/4 (2^-12 + 2^-8 + 2^-4 + 2^0 )
Manager  Joined: 18 Oct 2010
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Re: The figure is made up of a series of inscribed equilateral triangles.  [#permalink]

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may i ask which source it is from? i dont think real test will have such kinda of question
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Re: The figure is made up of a series of inscribed equilateral triangles.  [#permalink]

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1
IMO C.

I did not calculate it completely. Just calculated the area of first two partitions and established a pattern and then used the pattern to guess the answer. That's what I am going to do if something like this comes in real test as well Manager  Joined: 03 Dec 2012
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Re: The figure is made up of a series of inscribed equilateral triangles.  [#permalink]

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1
fluke wrote:
varunmaheshwari wrote:
Attachment:
InscribedTraingles.JPG
The figure is made up of a series of inscribed equilateral triangles. If the pattern continues until the length of
a side of the largest triangle (i.e. the entire figure) is exactly 128 times that of the smallest triangle, what
fraction of the total figure will be shaded?

A. 1\4(2^0 + 2^-4 + 2^-8 + 2^-12)

B. 1\4(2^0 + 2^-2 + 2^-4 + 2^-6)

C. 3\4(2^0 + 2^-4 + 2^-8 + 2^-12)

D. 3\4(2^0 + 2^-2 + 2^-4 + 2^-6)

E. 3\4(2^0 + 2^-1 + 2^-2 + 2^-3)

PS: I am sorry for the formatting of the answer choices like this.

$$Area \hspace{3} of \hspace{3} equilateral \hspace{3} triangle = \frac{\sqrt{3}}{4}*(side)^2$$

$$Total \hspace{3} Area = \frac{\sqrt{3}}{4}(128)^2=\frac{\sqrt{3}}{4}(2^7)^2=\frac{\sqrt{3}}{4}*2^{14}$$

$$Shaded \hspace{3} portion \hspace{3} of \hspace{3} outer \hspace{3} most= \frac{3}{4}*\frac{\sqrt{3}}{4}*(2^7)^2$$ {:Side=128=2^7}
Skip the next: {:Side=64}
$$Shaded \hspace{3} portion \hspace{3} of 3^{rd}= \frac{3}{4}*\frac{\sqrt{3}}{4}*(2^5)^2$$ {:Side=32=2^5}
$$Shaded \hspace{3} portion \hspace{3} of 5^{th}= \frac{3}{4}*\frac{\sqrt{3}}{4}*(2^3)^2$$ {:Side=8=2^3}
$$Shaded \hspace{3} portion \hspace{3} of 7^{th}= \frac{3}{4}*\frac{\sqrt{3}}{4}*(2^1)^2$$ {:Side=2=2^1}

Take the fraction:
$$\frac{\frac{3}{4}*\frac{\sqrt{3}}{4}(2^{14}+2^{10}+2^{6}+2^{2})}{\frac{\sqrt{3}}{4}*2^{14}}$$

$$=\frac{\frac{3}{4}(2^{14}+2^{10}+2^{6}+2^{2})}{2^{14}}$$

$$=\frac{3}{4}(\frac{2^{14}}{2^{14}}+\frac{2^{10}}{2^{14}}+\frac{2^{6}}{2^{14}}+\frac{2^{2}}{2^{14}})$$

$$=\frac{3}{4}(2^{(14-14)}+2^{(10-14)}+2^{(6-14)}+2^{(2-14)})$$

$$=\frac{3}{4}(2^0+2^{-4}+2^{-8}+2^{-12})$$

Ans: "C"

Can you please explain why u skipped certain sides?
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Re: The figure is made up of a series of inscribed equilateral triangles.  [#permalink]

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vinodmallapu wrote:

To get a triangle 128 times(2^7) smaller than the main triangle, you will have to partition it 7 times.
For each Odd partition you will get 3 Shaded regions.
For each Even partition you will get 1 shaded region the will again be partitions into regions by subsequent odd partition.So, we will no consider these partitions.

Partition 1 2 3 4 5 6 7
Shaded Regions 3 1 3 1 3 1 3

Consider the area of main triangle be A.
So the shaded region will be (3)*(1/4^7)A + (3)*(1/4^5)A + (3)*(1/4^3)A + (3)*(1/4^1)A

= 3/4 (2^-12 + 2^-8 + 2^-4 + 2^0 ) A

To get the fraction of shaded region , (3/4 (2^-12 + 2^-8 + 2^-4 + 2^0 ) A) / A
= 3/4 (2^-12 + 2^-8 + 2^-4 + 2^0 )

Excuse me but would you explain why you skipped the even partitions?

Cheers!
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Re: The figure is made up of a series of inscribed equilateral triangles.  [#permalink]

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I believe the even portions are skipped because those are considered shaded in the next odd portions.
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Re: The figure is made up of a series of inscribed equilateral triangles.  [#permalink]

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varunmaheshwari wrote:
Attachment:
InscribedTraingles.JPG
The figure is made up of a series of inscribed equilateral triangles. If the pattern continues until the length of a side of the largest triangle (i.e. the entire figure) is exactly 128 times that of the smallest triangle, what fraction of the total figure will be shaded?

A. 1\4(2^0 + 2^-4 + 2^-8 + 2^-12)

B. 1\4(2^0 + 2^-2 + 2^-4 + 2^-6)

C. 3\4(2^0 + 2^-4 + 2^-8 + 2^-12)

D. 3\4(2^0 + 2^-2 + 2^-4 + 2^-6)

E. 3\4(2^0 + 2^-1 + 2^-2 + 2^-3)

PS: I am sorry for the formatting of the answer choices like this.

I didn't get into details, and solved it with logic.
we have 4 equal triangles. we know for sure that the shaded area will be greater than 3/4 of the total figure.
A and B are right away out.

i started with E.
3/4 *(1+1/2 + 1/4 + 1/8) = 3/4 * (8+4+2+1)/8 = 3/4 * 15/8 ->result is greater than 1. so out.

D
3/4 * (1+1/4 +1/16 + 1/64) = 3/4* (64+16+4+1)/64 = 3/4 * 85/64 = 255/256 - clearly no!

only left - C
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Re: The figure is made up of a series of inscribed equilateral triangles.  [#permalink]

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varunmaheshwari wrote:
Attachment:
InscribedTraingles.JPG
The figure is made up of a series of inscribed equilateral triangles. If the pattern continues until the length of a side of the largest triangle (i.e. the entire figure) is exactly 128 times that of the smallest triangle, what fraction of the total figure will be shaded?

A. 1\4(2^0 + 2^-4 + 2^-8 + 2^-12)

B. 1\4(2^0 + 2^-2 + 2^-4 + 2^-6)

C. 3\4(2^0 + 2^-4 + 2^-8 + 2^-12)

D. 3\4(2^0 + 2^-2 + 2^-4 + 2^-6)

E. 3\4(2^0 + 2^-1 + 2^-2 + 2^-3)

PS: I am sorry for the formatting of the answer choices like this.

Just a glance at the image shows that the shaded area would be around 75%. Calculating all the options till two terms only as rest are too small, we get C as the answer.
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Re: The figure is made up of a series of inscribed equilateral triangles.  [#permalink]

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varunmaheshwari wrote: The figure is made up of a series of inscribed equilateral triangles. If the pattern continues until the length of a side of the largest triangle (i.e. the entire figure) is exactly 128 times that of the smallest triangle, what fraction of the total figure will be shaded?

A. $$\frac{1}{4}(2^0 + 2^{-4} + 2^{-8} + 2^{-12})$$

B. $$\frac{1}{4}(2^0 + 2^{-2} + 2^{-4} + 2^{-6})$$

C. $$\frac{3}{4}(2^0 + 2^{-4} + 2^{-8} + 2^{-12})$$

D. $$\frac{3}{4}(2^0 + 2^{-2} + 2^{-4} + 2^{-6})$$

E. $$\frac{3}{4}(2^0 + 2^{-1} + 2^{-2} + 2^{-3})$$

Attachment:
InscribedTraingles.JPG

Though i marked the wrong answer, Lets try to solve easy way..
1/128 = 1/2^7 . So center triangle will go upto 8th triangle in the center and will be unshaded.
Let A be the area of bigger triangle. Lets number biggest triangle as 1 and then smaller center triangles as 2 and next smaller center triangle as 3 and so on.

So, First focusing on 1st bigger triangle no. 1..

Now , In 2nd smaller triangle no. 2 , we have only shaded region from 1/4th of the part i.e from triangle no.3. So we will consider it in the next calculation in triangle no.3

In triangle 3,
Shaded region = 3/4 * triangle 3 = 3/4 *(1/4)^2 A

In triangle no. 4 , we have only shaded region from 1/4th of the part i.e from triangle no.5. So we will consider it in the next calculation in triangle no.5

In triangle 5,
Shaded region = 3/4 * triangle 5 = 3/4* 1/4 *1/4 * triangle 3 = 3/4 *(1/4)^2 * (1/4)^2 A

In triangle 6, we have only shaded region from 1/4th of the part i.e from triangle no.7. So we will consider it in the next calculation in triangle no.7
Shaded region = 3/4 * (1/4)^2 * (1/4)^2 * (1/4)^2 A

Sum of shaded regions = 3/4 A + 3/4 *(1/4)^2 A + 3/4 *(1/4)^2 * (1/4)^2 A + 3/4 * (1/4)^2 * (1/4)^2 * (1/4)^2 A
= 3/4 (1/2^0 + 1/2^4 + 1/2^8 + 1/2^12) A

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