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# If P and Q are circular regions, what is the sum of their areas?

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Math Expert
Joined: 02 Sep 2009
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If P and Q are circular regions, what is the sum of their areas?  [#permalink]

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17 Sep 2018, 22:06
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Difficulty:

65% (hard)

Question Stats:

47% (01:22) correct 53% (01:44) wrong based on 53 sessions

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If P and Q are circular regions, what is the sum of their areas?

(1) A sector of the smaller circle enclosing an angle of 120 degrees has the same area of a sector of the larger circle enclosing an angle of 30 degrees.

(2) The sum of the squares of the radii is 100.

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Re: If P and Q are circular regions, what is the sum of their areas?  [#permalink]

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18 Sep 2018, 14:28
to know the sum of the areas of 2 circles we need to know their respective radii otherwise there's an infinite number of posibilities.

from:
(1) A sector of the smaller circle enclosing an angle of 120 degrees has the same area of a sector of the larger circle enclosing an angle of 30 degrees.
this tells us that the total area of the smaller circle is 1/4th of the total area of the larger circle
(3x120 = 360 --> 3x30 = 90 = 1/4 of 360)
meaning PI*R1^2 = 4*PI*R2^2
however i have 2 variables and 1 equation, so i can find an infinite number of solutions since there is no limitations (other than that R has to be positive)
Not Suff

(2) The sum of the squares of the radii is 100.
this basically tells us that R1^2 + R2^2 = 100
a quick listing gives us only 2 possible combinations

R1 = 6 and R2 = 8
or
R1= 8 and R2 = 6

since the question is asking us about the total surface area, it doesn't matter which circle is largest since both cases result in the same total area.

don't get fooled into thinking you need (1) to limit the number of cases to 1

kudos if this was useful
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Re: If P and Q are circular regions, what is the sum of their areas?  [#permalink]

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28 Feb 2019, 12:06
A note on 2), you don't even need to determine what can = 100

If you write out the prompt π*Radius p²+π*Radius q² = combined Area
2) directly plugs into π(Rp²+Rq²)
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Re: If P and Q are circular regions, what is the sum of their areas?  [#permalink]

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28 Feb 2019, 19:00
aeon86 wrote:
(2) The sum of the squares of the radii is 100.
this basically tells us that R1^2 + R2^2 = 100
a quick listing gives us only 2 possible combinations

R1 = 6 and R2 = 8
or
R1= 8 and R2 = 6

since the question is asking us about the total surface area, it doesn't matter which circle is largest since both cases result in the same total area.

don't get fooled into thinking you need (1) to limit the number of cases to 1

Be careful with this reasoning. 6 and 8 / 8 and 6 definitely aren't the only possible combinations. That's because the problem text never says that the radii have to be integers. For instance, R1 = sqrt(50) and R2 = sqrt(50) is a perfectly valid combination.

However, no matter what, as long as the sum is 100 the sum of the areas will be the same. That's because the area of a circle is pi(r^2), so the area of both circles is pi(R1^2) + pi(R2^2) = pi(R1^2 + R2^2) = 100pi.
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Re: If P and Q are circular regions, what is the sum of their areas?   [#permalink] 28 Feb 2019, 19:00
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