Circle 1, Circle 2 and Circle 3 have the same center, and have a radiu

Question

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. What are the values A1/A2 and A2/A3?

(1) A2=A3

(2) A2+A3 = 2 A1

(1) A2=A3

(2) A2+A3 = 2 A1

BrushMyQuant · Accepted Answer

↧↧↧ Video Solution to the Problem ↧↧↧

https://www.youtube.com/watch?v=S88alz37GF4

Explaining the question:  
Area of Circle 1 =  A_1  =  Π (r_1)^2 
Area of Circle 2 =  A_1  +  A_2  =  Π (r_2)^2 
Area of Circlet 3 =  A_1  +  A_2  +  A_3  =  Π (r_3)^2

Solution  
To get  \frac{A_1}{A_2}  and  \frac{A_2}{A_3}  we need to express all of these values in terms of one value ( A_1  or  A_2  or  A_3 )
STAT1  A_2  =  A_3  alone is not sufficient as I don't know relation of  A_1  with  A_2  to find  \frac{A_1}{A_2} 
STAT2  A_2 + A_3  = 2  A_1  alone is not sufficient as I don't know the individual relationship between  A_1 , A_2  and  A_3

Combining both of them we have 
 A_2  =  A_3 
 A_2 + A_3  = 2  A_1 
=> 2 A_3  = 2 A_1  or  A_1  =  A_3  =  A_2 
So  \frac{A_1}{A_2}  and  \frac{A_2}{A_3}  both will be 1 each
So, both together are  sufficient .

So,  Answer will be C 
Hope it helps!

Watch the following video to Learn Basics of Circles

https://www.youtube.com/watch?v=vDtyMOcDlCU

BrentGMATPrepNow · Answer

Target question :    What are the values A1/A2 and A2/A3?

Statement 1 : A2=A3  
This will help us determine the value of A2/A3 (it equals 1) but, since we have no information about A1, statement 1 is NOT SUFFICIENT

Statement 2 : A2+A3 = 2 A1  
This tells us that the COMBINED areas of A2 and A3 are twice the area of A1
So, for example, we COULD have a scenario that looks like this...
 https://i.imgur.com/uRlMM7o.png 
..... in which the green area is twice the area of the A1.

Or we COULD have a scenario that looks like this...
 https://i.imgur.com/TY2iXIT.png 
Notice that the green area is still twice the area of the A1.

As you can see, the value of A2/A3 is different for each of the above cases. 
Since we cannot answer the  target question  with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined    
Statement 1 tells us that A2=A3, so let's let x = the area of each part
Statement 2 tells us that A2+A3 = 2 A1, which means we can now write: x + x = 2(A1)
In other words, 2x = 2(A1), which means A1 = x

So, all 3 regions have area x:
 https://i.imgur.com/rbm0XuY.png

This means  A1/A2 = x/x = 1, and A2/A3 = x/x = 1 
Since we can answer the  target question  with certainty, the combined statements are SUFFICIENT

Answer: C

Cheers,
Brent

ravisinghal · Answer

area of circle 2 would be :- 3.14*r2^2 - 3.14 * r1^2
similarly for circle 3 would be :- 3.14*r3^2 - 3.14 * r2^2

however could not establish the relation between A1,A2,A3?

Can someone please explain ? Bunnel can u help here

above explanation,according to me has provided a wrong explanation.

Thanks,

GinGMAT · Answer

Could someone please help with this one? How do you disprove statement 2 does not fit? I understand why C is correct but seems too obvious and I spent time trying to understand statement 2.

keats · Answer

Bunuel : Can you please help us here. Thanks.

PS: I believe I am over analysing statement II

abhimahna · Answer

Ok. Let me try helping you out.

See, Statement 2 says A3 + A2 = 2A1.

Even if you go with the approach  ravisinghal  has given above,

you will not get the values of both the fractions asked.

As per him,

A3 = pi r3^2 - pi r2^2

A2 = pi r2^2 - pi r1^2

A1 = pi r1^2

Substitute these values in the equation given in statement 2.

We will have pi r3^2 - pi r2^2 + pi r2^2 - pi r1^2 = 2(pi r1^2)

or pi r3^2 - pi r1^2 = 2(pi r1^2)

or pi r3^2 = 3(pi r1^2)

or r3=3r1 ( radius cannot be negative)

so, now I give the ball to you. Try substituting these values in A1/A2 and A2/A3, you will NEVER get the real values(independent of variables). Hence, B is insufficient.

(Hint : We donot know what is the relation between r2 and r1 or r3 and r1)

Therefore, with the statement 1, we can find a relation between r2 and r3 and then we can get the relation between r2 and r1, and on substituting we will get the value of each fraction =1.

Hence, C.

keats · Answer

I have noted the point you made  abhimahna  Since there is no relation established between r1 and r3 and r2 and r3 it is impossible to find real value for A1/A2 and A2/A3!

Nevernevergiveup · Answer

Please have a look and let me know if I'm wrong.

I understand that after all the math, we get A1 = A3 = A2 after combining both the statements 1 & 2.

In other words, the solution says areas of all three circles are equal.

But

As per the data mentioned in the question, we have Circles having the same center and distinct radii.
In other words we say that circles having common center and radii are called concentric circles.

https://images.tutorvista.com/cms/images/131/3-Concentric-Circles1.png

My point is  when radii are different, it is not possible that areas of 3 circles are equal. 
Since radius are r1<r2<r3, we get areas of circles must be A1<A2<A3.
The question must be incorrect for sure.
I want some experts to validate my point

thapliya · Answer

>> In other words, the solution says areas of all three circles are equal.
 Nevernevergiveup : A2 and A3 are not area of circle with radii r2 and r3.
A2 is area of circle A2 - Area of Circle A1
A3 is area of circle A3 - Area of circle A2

Nevernevergiveup · Answer

Thanks for the clarification. thapliya . I really got mistaken. Possibly a bit overconfident. 8-)

GMATsamurai · Answer

**************

1) A2 = A3 
 A1/A2 = ? (we do not have any information about A1)
 A2/A3 = A3/A3 = 1
 --> INSUFFICIENT

2) A2+A3 = 2*A1 
 A1 = (A2+A3)/2
 Substitute in A1/A2: (A2+A3)/2/A2 = (A2+A3)/(2*A2) = A2/(2*A2) + A3/(2*A2) = 1/2 + A3/(2*A2) 
 We could try to get A2/A3 but considering A1/A2 did not give a definite answer we can stop and test 1) and 2) simultaneously
 --> INSUFFICIENT

1)+2) A2 = A3 AND A2+A3 = 2*A1 
 A2/A3 = 1 (from 1))
 A1/A2 = 1/2 + A3/(2*A2) (from 2))
 A2 = A3 so substitution gives us A1/A2 = 1/2 + A3/(2*A3) = 1/2 + 1/2 = 1 
 --> SUFFICIENT

ANSWER C

Nikkb · Answer

Given : r1<r2<r3

Also: pi*r1^2 =A1
pi*r2^2 -pi*r1^2 =A2
pi*r3^2 - pi*r2^2 =A3

Find Part 1: A1/A2 and Part 2: A2/A3

1) A2=A3

So A2/A3 = 1 .  We got answer of 2nd part

Now A1/A2 =? We dont have any additional information to solve this. So No we are not able to find 1st part.

So option 1 not Sufficient

2) A2+A3=2A1

substitute equations here:

A2+A3=2A1
(pi*r2^2 -pi*r1^2) + (pi*r3^2 - pi*r2^2) =2*pi*r1^2
pi*r3^2 - pi*r1^2 = 2*pi*r1^2

pi*r3^2 = 3*pi*r1^2

No information we got from here

so option 2 is Not sufficient

1+2
So we already have answer for 2nd part. We have to find 1st part A1/A2

so as A2+A3=2A1 and A2=A3

A2+A2=2A1 => 2A2=2A1

So A1/A2 =1 . Got answer for 1st Part

So sufficient

Answer : C

strongcat · Answer

Can this one not be simplified as follows?

Just call A1 = X; A2 = Y; A3 = Z

We are looking for 2 ratios.
X / Y and Y / Z

1) Y = Z
Obviously this gives us one of the ratios, but tells us nothing about X. INSUFF

2) Y + Z = 2X
I don't see anyway to get the ratios with that. INSUFF

1 + 2)
We have the Y / Z ratio, but need the X / Y ratio.

Substituting the information in 1 (Y = Z) into the 2nd equation, we get:

Y + Y = 2X
Or 2Y = 2X

So now we can find the other ratio

Forget about the circles. Any flaws here?

neeleshravi · Answer

I really think this question is trying to distract the test taker and overcomplicate things with all the details about the circles and radii. That information is unnecessary.

Let me know if anything is wrong in my approach here. Admittedly, I got this one wrong while taking the test because I got caught up in all the additional details about circles and areas. After analysing for a bit here's how I would do it instead:
 
Question asks us to find values of A1/A2 and A2/A3.

Jump to answer choices.

Statement 1. No info about A1. Insuff

Statement 2. Gives us 3 variables, one equation. No further info. Insuff

Combined. Plug in info from St 1 into St 2. We should finally get A1=A2+A3

Now we can find A1/A2 and A2/A3.

oasis90 · Answer

Ya I agree. The same thing happened to me. I was trying to link Radiis with areas and make all sorts of connections. After 3 min, I ended up guessing and moving on.

tann412 · Answer

I did write on the test, A1 = pie r1^2... 
Then I saw A2=A3 and simply got ok, we know one of the ratios is 1. What about 2nd?
 
Looking at B, it also looked clearly insufficient.

Combining both, its simply 2A1=2A2, Hence the second Ratio is also 1.

This particular questions is nothing but 2 equations in 2 variables. And NO INFO from the questions stem, on radii and stuff is required to solve it. Sometimes, we do complicate things unnecessarily, this is an example. Have noticed more of this happening in IR though.

sumert · Answer

This question got me!
Just like others here, I kept banging my head against the wall with the areas (pi (r1 square)) etc.
All you have to do is look at A1/A2 and A2/A3. See if the values given in the options makes it true. Had I done that, I would've got this one right. Alas...

amandesai17 · Answer

GMATPrepNow  Can you help me out with this question please?

-Aman

energetics · Answer

(1) A2=A3 
We are asked for values, so we need to know the definite ratios of A1,A2,A3
From (1) we know that A2/A3 = 1, but no info is given about A1, not sufficient

(2) A2+A3 = 2 A1 
We can't determine a value for either A1/A2 or A2/A3 from this

Together:
We know A2=A3 and A2+A3=2A1
A2/A3 = 1 as above
Substitute A2 for A3 --> 2A2 = 2A1
So A1/A2 = 1 
Sufficient

Circle 1, Circle 2 and Circle 3 have the same center, and have a radiu

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GMAT Data Sufficiency (DS) Questions

Circle 1, Circle 2 and Circle 3 have the same center, and have a radiu

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