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Math Expert V
Joined: 02 Sep 2009
Posts: 58322

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11 00:00

Difficulty:   55% (hard)

Question Stats: 61% (02:20) correct 39% (02:28) wrong based on 241 sessions

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There are two concentric circles with radii 10 and 8. If the radius of the outer circle is increased by 10% and the radius of the inner circle decreased by 50%, approximately by what percent does the area between the circles grow?

A. 140%
B. 141%
C. 190%
D. 192%
E. 292%

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Math Expert V
Joined: 02 Sep 2009
Posts: 58322

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Official Solution:

There are two concentric circles with radii 10 and 8. If the radius of the outer circle is increased by 10% and the radius of the inner circle decreased by 50%, approximately by what percent does the area between the circles grow?

A. 140%
B. 141%
C. 190%
D. 192%
E. 292%

The area between the circles is the difference between their areas: $$\pi R^2 - \pi r^2$$.

Find the increase:

$$\frac{\pi R_2^2 - \pi r_1^2}{\pi R_1^2 - \pi r_1^2} =$$

$$\frac{R_2^2 - r_1^2}{R_1^2 - r_1^2} =$$

$$\frac{11^2 - 4^2}{10^2 - 8^2} =$$

$$\frac{121 - 16}{100 - 64} =$$

$$\frac{105}{36} = 2\frac{33}{36} =$$

$$3 - \frac{1}{12} \approx 2.92$$

The area between the circles grew by 192%.

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Intern  Joined: 09 Dec 2013
Posts: 4
Concentration: Social Entrepreneurship, Finance
GMAT 1: 560 Q31 V36 GMAT 2: 680 Q44 V39 Show Tags

Bunuel wrote:
Official Solution:

There are two concentric circles with radii 10 and 8. If the radius of the outer circle is increased by 10% and the radius of the inner circle decreased by 50%, approximately by what percent does the area between the circles grow?

A. 140%
B. 141%
C. 190%
D. 192%
E. 292%

The area between the circles is the difference between their areas: $$\pi R^2 - \pi r^2$$.

Find the increase:
$$\frac{\pi R_2^2 - \pi r_1^2}{\pi R_1^2 - \pi r_1^2} =$$
$$\frac{R_2^2 - r_1^2}{R_1^2 - r_1^2} =$$
$$\frac{11^2 - 4^2}{10^2 - 8^2} =$$
$$\frac{121 - 16}{100 - 64} =$$
$$\frac{105}{36} = 2\frac{33}{36} =$$
$$3 - \frac{1}{12} \approx 2.92$$

The area between the circles grew by 192%.

Hi Bunuel. What formula did you use to solve for the increase? Since you divided the new difference by the old difference, this doesn't look like the percent change formula. Thanks!
Intern  Joined: 02 Jun 2012
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i think we are taking difference as 36 instead of 36pie.

the old difference if 36pie , apprx 108 and new difference is 105pie, approx 315.

difference between 315-108 = 207.

using the percentage difference formula, 207/108*100 ~ 192%

is the above right?
Intern  Joined: 08 Jan 2015
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Hello,

In the above answer it received ~2.92.
in percentages it will be 292%. Why answer D is true and not E?
Math Expert V
Joined: 02 Sep 2009
Posts: 58322

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samners wrote:
Bunuel wrote:
Official Solution:

There are two concentric circles with radii 10 and 8. If the radius of the outer circle is increased by 10% and the radius of the inner circle decreased by 50%, approximately by what percent does the area between the circles grow?

A. 140%
B. 141%
C. 190%
D. 192%
E. 292%

The area between the circles is the difference between their areas: $$\pi R^2 - \pi r^2$$.

Find the increase:
$$\frac{\pi R_2^2 - \pi r_1^2}{\pi R_1^2 - \pi r_1^2} =$$
$$\frac{R_2^2 - r_1^2}{R_1^2 - r_1^2} =$$
$$\frac{11^2 - 4^2}{10^2 - 8^2} =$$
$$\frac{121 - 16}{100 - 64} =$$
$$\frac{105}{36} = 2\frac{33}{36} =$$
$$3 - \frac{1}{12} \approx 2.92$$

The area between the circles grew by 192%.

Hi Bunuel. What formula did you use to solve for the increase? Since you divided the new difference by the old difference, this doesn't look like the percent change formula. Thanks!

It gives a fraction/ratio. If the ratio is 2.92, then it means that the growth was by 192%. For example, the growth from x to 2,92x means the growth by 192%: x + 1.92x = 2.92x.
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4
Change in big circle: 10^2-8^2 = 36
Change in small circle: 11^2-4^2 = 105

105 is barely less than 3 times bigger. 3 times bigger is 200% increase so D is the closest.
Intern  Joined: 22 Jun 2015
Posts: 1

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I think this is a high-quality question and I agree with explanation.
Manager  B
Joined: 23 Jun 2009
Posts: 173
Location: Brazil
GMAT 1: 470 Q30 V20 GMAT 2: 620 Q42 V33 Show Tags

Can I simply assume the extra 1 radius point area and take out 1 (from the 100% of the previous area)?

Is it a good approximation?  πr^2=
22/7*1^2-1
≈ 200%

So, D
Current Student B
Joined: 08 Jan 2015
Posts: 74

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I got to a point when I had to choose between 190% and 192%, then I saw that among answers there is 292%, which clearly indicated that the correct answer is 192%. GMAT sometimes tries to trick those, who don't subtract 1, and sometime this trick in turn helps to arrive to the correct answer.
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1
105 is new area and 36 is old area

(105-36)/36 = 1.92
Current Student B
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What is 1/12 in '3−1/12≈2.92'??
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Manager  B
Joined: 01 Nov 2016
Posts: 58
Concentration: Technology, Operations

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3
I did this a little differently. The official answer makes it very easy to make a careless mistake. I think this is easier to follow:

r1 = 8
R1 = 10
a1 = 64pi
A1 = 100 pi
difference1 = 100pi - 64pi = 36pi

r2 = 4pi
R2 = 11pi
a2 = 16pi
A2 = 121pi
difference2 = 121pi - 16pi = 105 pi

growth = 105pi - 36pi = 69pi
%growth = 69pi/36pi * 100 = 23*25/3 = 575/3 = 191.66666667 which is about 192.
Intern  B
Joined: 23 Jan 2018
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Intial Spacing = 36pi
Final Spacing = 105pi

36pi corresponds to ----->100%
Then, 105pi corresponds to----->(100%/36pi)*105pi ~ 292%

Growth Percentage = 292%-100% = 192%
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irinabarvi wrote:
I think this is a high-quality question and I agree with explanation.

I agree as well.

I was tempted to chose option E which was deceptive..If X was orginal areas and 2X is increases area means Ratio=2X/X but actual increase in area from X ..its is X
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I think this is a high-quality question and I agree with explanation.
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Bunuel wrote:
There are two concentric circles with radii 10 and 8. If the radius of the outer circle is increased by 10% and the radius of the inner circle decreased by 50%, approximately by what percent does the area between the circles grow?

A. 140%
B. 141%
C. 190%
D. 192%
E. 292%

B = big circle, s = small circle

Original:
rB (Big) = 10, Area B = 100
rs (small) = 8, Area s = 64
Area A - Area s = 100-64 = 36

Change:
rB*11/10 = 11
rs*1/2 = 4

New:
Area B = 121
Area s = 16
Area B - Area s = 121 - 16 = 105

Difference in new area / Difference in original area = Percent new over original
105 / 36 = 2 & 33/36 = 2 & 11/12, 1/12 is .083 so it's about 2.92.
We have to remember that "by what percent does the area between the circles grow?" means what is the INCREASE rather than PERCENT OF. So, 2.92 - 1 = 1.92
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I think this is a high-quality question and I agree with explanation. Re M01-37   [#permalink] 25 Jul 2019, 08:58
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