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# The diagram above shows two wheels that drive a conveyor belt. The lar

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Math Expert
Joined: 02 Sep 2009
Posts: 56244
The diagram above shows two wheels that drive a conveyor belt. The lar  [#permalink]

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11 Jul 2017, 02:57
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Difficulty:

55% (hard)

Question Stats:

66% (02:38) correct 34% (02:30) wrong based on 121 sessions

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The diagram above shows two wheels that drive a conveyor belt. The larger wheel has a diameter of 40 centimeters; the smaller wheel has a diameter of 32 centimeters. In order for the conveyor belt to run smoothly, each wheel must rotate the exact same number of centimeters per minute. If the larger wheel makes r revolutions per minute, how many revolutions does the smaller wheel make per hour, in terms of r?

(A) 1280π/3

(B) 75r

(C) 48r

(D) 24r

(E) 64π/3

Attachment:

2017-07-11_1354.png [ 9.72 KiB | Viewed 2662 times ]

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The diagram above shows two wheels that drive a conveyor belt. The lar  [#permalink]

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11 Jul 2017, 04:20
2
1
If the larger wheel has a diameter of 40cm, the circumference is 40*pi.
Similarly, the smaller wheel(which has a diameter of 32 cm), will have 32*pi as circumference.

It has been said that the two wheels have the same number of rotations in a minute.
So, if the total number of revolutions the bigger wheel makes are r revolutions/minute, it would make 60r revolutions in an hour.
Let the revolution the smaller wheel makes in an hour be x.

40*pi*60*r = 32*pi*x

x = $$\frac{40*pi*60*r}{32*pi}$$ = $$\frac{5*60*r}{4}$$= 75r

The smaller wheel must make 75r revolutions in an hour(Option B)
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Re: The diagram above shows two wheels that drive a conveyor belt. The lar  [#permalink]

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11 Jul 2017, 07:26
Circumference of the larger wheel-2*pi*20--40pi cms
Circumference of the smaller wheel-2*pi*16--32pi cms

Now ,wheels must travel equal cms/sec to run smoothly.
Larger Wheel wheel makes 1 rotation/sec in 40 pi cms,while smaller wheel makes only 32pi cms by that time.

To tally the balance 8cms/sec ,smaller wheel now has to make an additional 1/4th of its rotation.

So while larger wheel makes 1R revolutions/sec,smaller wheel must make 1.25 rev/sec(extra 1/4th of 32pi-8 cms)

1.25R in 1 sec so 75R in one hr(1.25*60mins)R

Option B
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Re: The diagram above shows two wheels that drive a conveyor belt. The lar  [#permalink]

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11 Jul 2017, 11:28
1
distance covered by big wheel in 1 minute is 2*pi*20*r ( r revolutions per minute)
distance covered by the smaller wheel in 1 revolution is 2*pi*16

therefore the number of revolutions in terms of r by the smaller wheel in 1 minute (2*pi*20*r )/(2*pi*16)= 20/16 * r

so the number of revolutions in 1 hour is (20/16 * r) * 60 = 75 r

ans : B
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Re: The diagram above shows two wheels that drive a conveyor belt. The lar  [#permalink]

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11 Jul 2017, 13:35
Bunuel wrote:

The diagram above shows two wheels that drive a conveyor belt. The larger wheel has a diameter of 40 centimeters; the smaller wheel has a diameter of 32 centimeters. In order for the conveyor belt to run smoothly, each wheel must rotate the exact same number of centimeters per minute. If the larger wheel makes r revolutions per minute, how many revolutions does the smaller wheel make per hour, in terms of r?

(A) 1280π/3

(B) 75r

(C) 48r

(D) 24r

(E) 64π/3

Attachment:
2017-07-11_1354.png

Distance covered by the larger wheel in a minute = 2π*20*r
This has to be equal to the distance covered by the small wheel in a minute
Hence 2π*20*r = 2π*16*x

Revolutions by the small wheel in a minute, x = 5/4 r * 60 = 75r revolutions per hour.

Correct Option: B
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Re: The diagram above shows two wheels that drive a conveyor belt. The lar  [#permalink]

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01 Apr 2018, 11:39
Hi All,

This question can be solved in a number of different ways, including by TESTing VALUES. The prompt starts us off with the following information (which we can use to perform some initial calculations). Since the wheels are rotating, we'll need to use the CIRCUMFERENCE of each wheel in our calculation….

Large wheel:
Diameter = 40
Circumference = 40pi

Small wheel:
Diameter = 32
Circumference = 32pi

Large wheel makes R revolutions/MINUTE
Let's use R = 2
2 revolutions / minute
So it revolves 80pi / minute

We're told the small wheel has to revolve the SAME distance, so…

32pi(X) = 80pi
X = 80pi/32pi = 5/2
2.5 revolutions / minute

So the small wheel has to rotate 2.5 times to equal the large wheel rotating 2 times

The question asks for the number of revolutions that the small wheel makes PER HOUR.

(2.5/minute)(60 minutes) = 150 revolutions / hour

So we're looking for an answer that equals 150 when R=2...

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The diagram above shows two wheels that drive a conveyor belt. The lar  [#permalink]

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01 Apr 2018, 13:41
Bunuel wrote:

The diagram above shows two wheels that drive a conveyor belt. The larger wheel has a diameter of 40 centimeters; the smaller wheel has a diameter of 32 centimeters. In order for the conveyor belt to run smoothly, each wheel must rotate the exact same number of centimeters per minute. If the larger wheel makes r revolutions per minute, how many revolutions does the smaller wheel make per hour, in terms of r?

(A) 1280π/3

(B) 75r

(C) 48r

(D) 24r

(E) 64π/3

if smaller wheel/larger wheel rpm ratio inversely=40r/32, or 5r/4,
then smaller wheel rph=60*5r/4, or 75r
B
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Re: The diagram above shows two wheels that drive a conveyor belt. The lar  [#permalink]

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06 Mar 2019, 14:45
Official solution from Princeton:

Because the problem has variables in the answer choices, Plugging In is a great way to solve. Plug in an easy number, one that will make your calculations and answer checking simple. For the purposes of this explanation, let’s set r equal to 2. With each rotation, each wheel rotates the length of its circumference. Thus, the wheel with diameter 40 centimeters rotates 40π centimeters with each rotation; the wheel with diameter 32 centimeters rotates 32π centimeters with each rotation. The larger wheel makes r revolutions per minute; because we’ve set r equal to 2, it makes 2 revolutions per minute and thus rotates 80π centimeters per minute. According to the problem, the smaller wheel must rotate the same distance. Therefore, the smaller wheel also rotates 80π centimeters per minute, meaning it rotates 60 × 80π = 4,800π centimeters per hour. The smaller wheel covers 32π centimeters per rotation, so it must rotate 4,800π ÷ 32π = 150 times per hour. Plug 2 in for r in each of the answer
choices. The correct answer is (B), 75r.
Re: The diagram above shows two wheels that drive a conveyor belt. The lar   [#permalink] 06 Mar 2019, 14:45
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