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

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Math Expert
Joined: 02 Sep 2009
Posts: 50610
Two cars are racing along two concentric circular paths. What is the  [#permalink]

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14 Sep 2018, 00:58
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Difficulty:

45% (medium)

Question Stats:

72% (01:42) correct 28% (01:41) wrong based on 18 sessions

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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.

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Re: Two cars are racing along two concentric circular paths. What is the  [#permalink]

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14 Sep 2018, 23:14
Bunuel wrote:
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.

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

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Two cars are racing along two concentric circular paths. What is the  [#permalink]

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15 Sep 2018, 01:11
Bunuel wrote:
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.

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...

Two cars are racing along two concentric circular paths. What is the &nbs [#permalink] 15 Sep 2018, 01:11
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# Two cars are racing along two concentric circular paths. What is the

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