Data Sufficiency

Don't know how to approach this problem. Started off by assuming basic values for r1 and r2, but got stuck. Please help.

TIA.

Question stem, rephrased:
What is A� : A₂ : A₃?

Statement 1: A₂ = A₃
Thus, A₂ : A₃ = 1:1.
No information about A�.
INSUFFICIENT.

Statement 2: A₂ + A₃ = 2A�
Case 1: A�=2, A₂=1, A₃=3
In this case, A� : A₂ : A₃ = 2:1:3.
Case 2: A�=2, A₂=2, A₃=2
In this case, A� : A₂ : A₃ = 2:2:2 = 1:1:1.
Since A� : A₂ : A₃ can be different values, INSUFFICIENT.

Statements combined:
Substituting A₂ = A₃ into A₂ + A₃ = 2A�, we get:
A₂ + A₂ = 2A�
2A₂ = 2A�
A₂ = A�.
Since A� = A₂ and A₂ = A₃, A� = A₂ = A₃.
Thus:
A� : A₂ : A₃ = 1:1:1.
SUFFICIENT.

The correct answer is C.

The answer to this seemed pretty straightforward, but I was wondering whether, since we are dealing with concentric circles, there is some way that one both of the statements could be sufficient on its or their own.

I proved that neither is sufficient by using the following ideas.

Statement 1: A₂ = A₃

Even if there is some reason that A�, A₂ and A₃ have to be related, A� could be 0, clearly. In that case A�/A₂ would be 0 and A₂/A₃ would be 1.

Meanwhile, A� does not have to be 0. If A� is not 0, then A�/A₂ is not 0. So already we have two different values for one of the ratios.

Insufficient.

Statement 2: A₂ + A₃ = 2A�

In this case, either A₂ or A₃ could be 0. If A₂ = 0, we get a different set of ratios from what we get if A₃ = 0.

Insufficient.

So, no, the fact that they are concentric circles does not make either statement sufficient on its own.

What a funny question, having all that stuff about circles, radii, cm and areas obscuring what is essentially the most basic ratio question ever. The question could have been simply, "A�, A₂ and A₃ are numbers. What is the ratio A� : A₂ : A₃?", but noooo. They give you all that other information, and you have to prove that it's useless. LOL

Circle 1, Circle 2 and Circle 3 have the same center, and have a radius r1<r2<r3. Let A1 be the area of circle 1, let A2 be the area of the region within Circle 2 and outside Circle 1, and let A3 be the area of the region within Circle 3 and outside Circle 2."

I found your explanation helpful, GMATGuruNY.

However, wanted to ask about the condition provided in the question stem that states, "r1<r2<r3."

Given the definition of a circle is πr^2, and substituting the definition of a circle into the ratio of the three circle's areas, I inferred that:

= A1 : A2 : A3
and replacing area with the definition of a circle:
= π(radius 1)^2: π(radius 2)^2 : π(radius 3)^2
results in the inequality below, given what we know about the relative sizes of the radii.
= A1 < A2 < A3

However, my logic is not correct given A2 = A3 in Statement 1.

How can A2 = A3 when the radii cannot be equal?

Thanks

Ovid wrote:Circle 1, Circle 2 and Circle 3 have the same center, and have a radius r1<r2<r3. --, let A2 be the area of the region within Circle 2 and outside Circle 1, and let A3 be the area of the region within Circle 3 and outside Circle 2."

I found your explanation helpful, GMATGuruNY.

However, wanted to ask about the condition provided in the question stem that states, "r1<r2<r3."

Given the definition of a circle is πr^2, and substituting the definition of a circle into the ratio of the three circle's areas, I inferred that:

= A1 : A2 : A3
and replacing area with the definition of a circle:
= π(radius 1)^2: π(radius 2)^2 : π(radius 3)^2
results in the inequality below, given what we know about the relative sizes of the radii.
= A1 < A2 < A3

However, my logic is not correct given A2 = A3 in Statement 1.

How can A2 = A3 when the radii cannot be equal?

Thanks

A₂ and A₃ do not represent the areas of the two larger circles.
Reread this portion of the prompt:
Let A� be the area of circle 1.
Let Aâ‚‚ be the area of the region within Circle 2 and outside Circle 1.
Let A₃ be the area of the region within Circle 3 and outside Circle 2.
These statements imply the following figure:

In accordance with the prompt:
A� = the green portion above.
Aâ‚‚ = the red portion above.
A₃ = the blue portion above.