Two cars are racing along two concentric circular paths. What is the r

Question

Two cars are racing along two concentric circular paths. What is the ratio of the speed of the car on the outer circle to that on the inner circle, if they were to complete a lap in the same amount of time?
 
(1) The difference in the area between the outer circle and the inner circle is 192π.
(2) The difference in radius between the outer circle and the inner circle is 8.

chetan2u · Answer

Hi
Firstly what are we looking at
The time taken is same, so the speed or ratio of speed will depend on the ratio of perimeter and perimeter would depend upon the radius of each circle..

So lets see if statements tell us about radius

1) it gives the difference of areas
So  π(r_1^2-r_2^2)= 192π 
So r1 and r2 cannot be found 
Insuff..

2)  r_1-r_2=8 ..
Again nothing individually..
Insuff

Combined 
 r_1²-r_2²=192 ...(1)
 r_1-r_2=8 ...(2)
Divide 1 by 2..  r_1+r_2=\frac{192}{8}=24.. (3)
Add (2) and (3)..
r _1-r_2+r_1+r_2=8+24....2*r_1=32......r_1=16........
r_2=16-8=8.. 
So the ratio of speed= 8/16=1/2
Suff
C

CrackverbalGMAT · Answer

Hi duahsolo,

The best approach to apply on DS word problems is to  deconstruct the question  before moving on to the statements.

Let the speed of the car on the outer circle be  S_o  and the speed of the car on the inner circle be  S_i . We need to ratio of the speed of the car on the outer circle to that on the inner circle, so we need  S_o / S_i . We have also been told that the times that both the cars take is a constant. Since Speed and Distance are directly proportional to each other and since the time is constant  D_o / D_i , where  D_o  is the distance traveled by the car on the outer circle and  D_i  is the distance traveled by the car on the inner circle.

The distance traveled by the car on the outer circle = Circumference of the outer circle = 2π *  R_o  ;  R_o  being the outer circle radius.
The distance traveled by the car on the inner circle = Circumference of the inner circle = 2π *  R_i  ;  R_i  being the inner circle radius.

D_o / D_i = (2π *  R_o ) / (2π *  R_i ) =  R_o / R_i .

So here if the statements give us the ratio of radii, we will be able to find the ratio of the distances and subsequently the ratio of speeds.

Statement 1 : The difference in the area between the outer circle and the inner circle is 192π

π( R_o )^2 - π( R_i )^2 = 192π.
Simplifying we get (R_o)^2 - (R_i)^2 = 192 -----> ( R_o  +  R_i )( R_o  -  R_i ) = 192.
This does not give us enough information about the ratio of radii as there can be multiple values of  R_o  and  R_i .  Insufficient .

Statement 2 : The difference in radius between the outer circle and the inner circle is 8

R_o  -  R_i = 8. This again does not give us enough information to determine the ratio. Insufficient.

Combining Statements 1 and 2

From Statement 1 we have ( R_o  +  R_i )( R_o  -  R_i ) = 192 and from Statement 2 we have  R_o  -  R_i = 8
Solving we get  R_o  +  R_i  = 24. Now since we have  R_o  -  R_i = 8 and  R_o  +  R_i  = 24, we can solve to get definite values of radii which gives us a definite ratio.  Sufficient .

OA : C

CrackVerbal Academics Team

GMATinsight · Answer

CONCEPT : Ratio of speeds = Ratio of distances when the time is constant

Question : What is the ratio of the speed of the car on the outer circle to that on the inner circle,

Statement 1 : The difference in the area between the outer circle and the inner circle is 192π 
i.e. π(R^2-r^2) = 192π 
This doesn't give us the ratio fo distance hence
NOT SUFFICIENT

Statement 2 : The difference in radius between the outer circle and the inner circle is 8 
i.e.  R-r =8 
but different values of R and r are possible giving us different ratios of the distance covered by two cars hence
NOT SUFFICIENT

COmbining the two statements  
 π(R^2-r^2) = 192π 
i.e.  π*(R-r)*(R+r) = 192π  and also  R-r =8 
hence R and r can be calculated and ratio of distances may be calculated hence
SUFFICIENT

Answer: Option C

Bigpic · Answer

Speed = Distance/Time.
For the same time, the ratio of speed = ratio of distance =  \frac{diameter of outercircle}{diameter of innercircle} = \frac{R}{r} 
where  R  is the radius of outer circle and  r  is the radius of inner circle

a) Area of outer circle - Area of inner circle =  192\pi 
 i.e  \pi R^2  -  \pi r^2  =  192\pi . 
 Not sufficient

b) Radius of outer circle R - radius of inner circle r = 8
 Not sufficient

Combining both
We have two equations, a quadratic equation and a linear equation and two unknowns. So we can jump straight to answer C

Wait..
Since it is a 600-level, I jumped straight to answer C.
However if it had been a 700-level problem, I would have gone one more step further, as the quadratic equation may have two solutions.
So the two equations are
 \pi R^2  -  \pi r^2  =  192\pi 
 R  -  r  =  8

substituting second equation in the first
 (r+8)^2  -  r^2  =  192 
 16r  +  64  =  192  and thus we can solve for r since  r^2  cancels off.
However sometimes  r^2  might not cancel and we may get two solutions for  r  and in that case we might not be able to solve for unique solution for r.
Ironically, I find that the difficulty level of each question is making my approach to the problem different. I have to change it.
Anyway sometimes better to go one more step in solving equations, just to confirm...
 
So answer is C here

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Two cars are racing along two concentric circular paths. What is the r

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Two cars are racing along two concentric circular paths. What is the r

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