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Circles

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Circles  [#permalink]

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New post 25 Jun 2018, 10:08
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A circular gear with a diameter of 24 centimeters is mounted directly on another circular gear with a diameter of 96 centimeters. Both gears turn on the same axle at their exact centers and each gear has a single notch, at the 12 o'clock position. At the same moment, the gears begin to turn at the same rate, with the larger gear moving clockwise and the smaller gear counterclockwise. How far, in centimeters, will the notch on the larger gear have traveled the second time the notches pass each other?

(A) 32.2Pi
(B) 35.6Pi
(C) 38.4Pi
(D) 39.2Pi
(E) 40.8Pi

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Re: Circles  [#permalink]

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New post 25 Jun 2018, 10:11
Since the notches start in the same position and move in opposite directions towards each other, they will trace a circle together when they pass for the first time, having covered a joint total of . When the notches meet for the second time, they will have traced two full circles together for a total of .

Since the circumference of the large gear is 4 times greater than that of the small gear, the large notch will cover only 1/4 of the number of degrees that the small notch does. We can represent this as an equation (where x is the number of degrees covered by the large notch):



So the large notch will have covered when the notches pass for the second time. Since the circumference of the large gear is , we can set up the following proportion to solve for the linear distance (call it d) covered by the large notch:



The correct answer is C.

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Re: Circles  [#permalink]

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New post 25 Jun 2018, 10:13
MrJglass wrote:
Since the notches start in the same position and move in opposite directions towards each other, they will trace a circle together when they pass for the first time, having covered a joint total of . When the notches meet for the second time, they will have traced two full circles together for a total of .

Since the circumference of the large gear is 4 times greater than that of the small gear, the large notch will cover only 1/4 of the number of degrees that the small notch does. We can represent this as an equation (where x is the number of degrees covered by the large notch):



So the large notch will have covered when the notches pass for the second time. Since the circumference of the large gear is , we can set up the following proportion to solve for the linear distance (call it d) covered by the large notch:



The correct answer is C.

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This is the official Manhattan solution but I don't understand it.
Anyone who understands it or have a more intelligible solution should help please.

Thanks!
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Re: Circles  [#permalink]

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New post 25 Jun 2018, 10:52
MrJglass wrote:
A circular gear with a diameter of 24 centimeters is mounted directly on another circular gear with a diameter of 96 centimeters. Both gears turn on the same axle at their exact centers and each gear has a single notch, at the 12 o'clock position. At the same moment, the gears begin to turn at the same rate, with the larger gear moving clockwise and the smaller gear counterclockwise. How far, in centimeters, will the notch on the larger gear have traveled the second time the notches pass each other?

(A) 32.2Pi
(B) 35.6Pi
(C) 38.4Pi
(D) 39.2Pi
(E) 40.8Pi

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Re: Circles &nbs [#permalink] 25 Jun 2018, 10:52
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