Conveyor Belt problem

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Sak32: Senior | Next Rank: 100 Posts; Posts: 30; Joined: Mon Oct 14, 2013 11:06 am

by Sak32 » Thu Oct 31, 2013 10:00 am

Please take the time to provide me with a clear explanation for the problem below:

A diagram 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 revoultions per minute, how many revolutions does the smaller wheel make per hour, in terms of r?

(A) 1280Ï€/3
(B) 75r
(C) 48r
(D) 24r
(E) 64Ï€/3

The answer is B.

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Source: Beat The GMAT — Problem Solving | Thu Oct 31, 2013 10:00 am

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by [email protected] » Thu Oct 31, 2013 11:34 am

Hi Sak32,

This question can be solved in a number of different ways. Here's a method that uses TESTing values:

Large wheel:
Diameter = 40
Radius = 20
Circumference = 40pi

Small wheel:
Diameter = 32
Radius = 16
Circumference = 32pi

Since the wheels are rotating, we'll need to use the circumference of each wheel in our calculation....

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 shell makes PER HOUR.

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

Using R = 2, the only answer that matches is B

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Rich

Contact Rich at [email protected]

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GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

by GMATGuruNY » Thu Oct 31, 2013 12:32 pm

Sak32 wrote:Please take the time to provide me with a clear explanation for the problem below:

A diagram 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 revoultions per minute, how many revolutions does the smaller wheel make per hour, in terms of r?

(A) 1280Ï€/3
(B) 75r
(C) 48r
(D) 24r
(E) 64Ï€/3

Revolutions per minute = (distance rotated each minute)/(circumference).

Big diameter : small diameter = 40:32 = 5:4.
Since circumference = Ï€d, we get:
Big circumference : small circumference = 5Ï€ : 4Ï€.

To make the math easier, we can use these smaller values -- 5Ï€ and 4Ï€ -- to determine how many revolutions are made by the small wheel for every revolution made by the big wheel.
Let the distance rotated each minute = the LCM of 5Ï€ and 4Ï€ = 20Ï€ centimeters.

Big wheel's revolutions per minute = r = (20Ï€)/(5Ï€) = 4.
Small wheel's revolutions per minute = (20Ï€)/(4Ï€) = 5.
Thus:
Small wheel's revolutions per 60 minutes = 60*5 = 300. This is our target.

Now plug r=4 into the answers to see which yields our target of 300.
Only B works:
75r = 75*4 = 300.

The correct answer is B.

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Mathsbuddy: Master | Next Rank: 500 Posts; Posts: 447; Joined: Fri Nov 08, 2013 7:25 amFollowed by: 1 member; Thanked: 25 times

by Mathsbuddy » Sat Nov 16, 2013 3:17 pm

Sak32 wrote:Please take the time to provide me with a clear explanation for the problem below:

A diagram 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 revoultions per minute, how many revolutions does the smaller wheel make per hour, in terms of r?

(A) 1280Ï€/3
(B) 75r
(C) 48r
(D) 24r
(E) 64Ï€/3

The answer is B.

Here's an alternative approach, if it helps:

Using the formula
v = f * c
where v = velocity, f = frequency, c = circumference

The question states that v must be constant (the same number of cm per minute)

so f1 * c1 = f2 * c2

Also, circumference, c, is proportional to diameter, d, so we can write

f1 * d1 = f2 * d2
r * 40 = f * 32
f = 5r/4 per minute
so f = 60 x 5r/4 per hour
f = 75r per hour

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