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

Question

https://gmatclub.com/forum/download/file.php?id=61564 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 2017-07-11_1354.png

SVaidyaraman · Accepted Answer

1. Let the number of revolutions made by the smaller wheel in 1 hr be s
2. Given: Number of centimeters rotated by both the wheels in 1 hr is the same

3. Number of centimeters rotated by the larger wheel in one revolution = its circumference=2*pi*20=40Pi
4. Number of centimeters rotated by the larger wheel in 1 hr = 40*pi*60*r

5. Number of centimeters rotated by the smaller wheel in one revolution=its circumference=2*pi*16=32*pi
6.Number of centimeters rotated by the smaller wheel in 1 hr= 32*pi*s

7. Equating (4) and (6) based on (2),

40*pi*60*r = 32*pi*s 
i.e., s= 75r

Bunuel · Answer

https://gmatclub.com/forum/download/file.php?id=61564 
 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

The radius of the larger wheel is 20cm so, its circumference is  2\pi{R}=40\pi . Since the larger wheel makes r revolution per minute or 60r revolutions per hour then it rotates  60r*40\pi  centimeters per hour.

The radius of the smaller wheel is 16cm so, its circumference is  2\pi{R'}=32\pi . Now, in order the smaller wheel to rotate the same centimeters per time interval it should make  \frac{60r*40\pi}{32\pi}=75r  revolutions per hour.

Answer: B.

Priyanka2011 · Answer

Hai guys..

I just tried this way..

40P/no of centimeters = R per minute

So no of centi = 40P/R

since both the wheels move at the same rate i.e in terms of centi 
Revol for the smaller one = 32P*R/40P
=4/5R per minute and 48R per hour...

What am I missing ?

Bunuel · Answer

Check this: two-wheels-are-connected-via-a-conveyor-belt-the-larger-133697.html#p1092048 The circumference of the larger wheel is 2\pi{R}=40\pi . Since the larger wheel makes r revolution per minute or 60r revolutions per hour, then it rotates 60r*40\pi centimeters per hour. The radius of the smaller wheel is 16cm so, its circumference is 2\pi{R'}=32\pi . Now, in order the smaller wheel to rotate the same centimeters per time interval it should make \frac{60r*40\pi}{32\pi}=75r revolutions per hour. Answer: B. Hope it's clear.

TGC · Answer

Visualization is the key in such problems.

To work smoothly

The belt left by big wheel should be taken up by small wheel. Implicitly,

Length left by big wheel /min = length accepted by small wheel/min

40 *pi * r = 32 * pi * x

x = (5/4) * r in minute

x= (5/4) * r * 60 in hour
 Hence B

gmatprav · Answer

There is a simple formula for this kind of problems. Given two pulleys/wheels are connected tightly with a conveyor belt, each wheels completes same distance in an hour as the other wheel unless there is slippage. only their revolutions per min or hour differ due to their difference is size.

If r1 and r2 are revolutions per hour of each wheel and
R1 and R2 are radius of each wheel.

equating distance of each wheel

2pi*R1*r1 = 2pi*R2*r2

R1*r1 = R2*r2

\frac{R1}{R2} = \frac{r2}{r1}

given 3 of the values we can find the 4th one.

WholeLottaLove · Answer

Two wheels are connected via a conveyor belt. The larger wheel has a 40cm diameter and the smaller wheel has a 32cm diameter. In order for the conveyor belt to work smoothly, each wheel must rotate the exact same number of centimetres per minute. If the larger wheel makes r revolution per minute, how many revolutions does the smaller wheel make per hour in terms of r?

A. 1280π
B. 75r
C. 48r
D. 24r
E. (64π)/3

Interesting to note that the larger wheel has a diameter of 40 (8*5) while the smaller one has a diameter of 32 (8*4)...

If the large wheel has a diameter of 40 and the small wheel, 32, then their circumferences are 40pi and 32pi respectively. In order for them to move the conveyor belt at the same rate, the smaller wheel would need to rotate 1.25 times as fast as the larger wheel. Lets say the large wheel makes 10 revolutions per minute, the smaller wheel would need to make 10*1.25 = 12.5 revolutions per minute. If the large wheel makes 10 revolutions per minute it makes 600 per hour. Therefore, the smaller wheel would need to make 600*1.25 = 750 revolutions per hour.

If r = 10, then the answer choice must be b.

B. 75r

pushpitkc · Answer

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)

brs1cob · Answer

B is the answer.
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

OptimusPrepJanielle · Answer

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

EMPOWERgmatRichC · Answer

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 
Radius = 20 
Circumference = 40pi

Small wheel: 
Diameter = 32 
Radius = 16 
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...

Final Answer:  B

GMAT assassins aren't born, they're made, 
Rich

energetics · Answer

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.

Gmatstudent2017 · Answer

Hi Bunuel,

I tried to solve this problem by picking numbers. So what I did was found the LCM between 40 and 32 since both wheels cover the same distance. 40*32 = 1280 so that means the bigger wheel must rotate 1280/40 = 32 times per minutes. That means in 60 minutes it'll rotate 1,920 times. Since the smaller wheel also rotates the same distance that is 1,920/32 = 60. So now the the target I'm looking for in the answer choice is 60 when r equals 32. When I plug in r equals to 32, I don't get the target answer of 60 anywhere. Can you tell me what I did wrong please?

Thank you!

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

Show Spoiler

Attachment:

2017-07-11_1354.png

(A) 1280π/3

(B) 75r

(C) 48r

(D) 24r

(E) 64π/3

EMPOWERgmatRichC · Answer

Hi Gmatstudent2017,

In your approach, you lost track of the 'units' that each number represented. To start, the LCM of 40 and 32 is actually 160 (not 1280), but we can use 1280 when solving this question. To be mathematically accurate though, it's actually 1280pi (since we're dealing with the circumference of each of the circles - and they're 40pi and 32pi).

Since the larger circle has a circumference of 40pi, then it will travel 1280pi/40pi = 32 REVOLUTIONS per MINUTE, so R = 32.

The smaller circle will also travel that SAME distance, so it will travel 1280pi/32pi = 40 REVOLUTIONS per MINUTE.

We're asked how many revolutions the smaller circle will make in 1 HOUR though, so that is (60 minutes)(40 revolutions/min) = 2400 revolutions per HOUR.

In this example, we are looking for an answer that equals 2400 when R = 32. Only one answer matches.
 
GMAT assassins aren't born, they're made,
Rich

shivangibh · Answer

The explanation to the question is added in the picture.

Fdambro294 · Answer

For each wheel, because the 2 wheels are connected by a conveyor belt and the (cm) / (min) must be the same, for any One minute that passes:

(Distance Large Wheel Travels) = (Distance Smaller Wheel Travels)

Distance any given wheel travels in 1 min = (No. of Revolutions in 1 min.) * (Circumference)

D-large = (R) * (40(pi))

D-small: let the no. of revolutions that the smaller wheel makes be X (cm) / (min)

D-small = (X) * (32 (pi))

(R) * 40 (pi) = (X) * 32 (pi)

X = (5/4) R

But this is measured in (revolutions) per (MINUTE). We need to confer to a per-hour unit

(5R / 4) * (60 min)/(1 hour) =

75R

Or

You can answer directly using proportionality theory. Given a constant distance that 2 wheels travel, the number of revolutions a circle makes is inversely proportional to the Diametes/radii/Circumference of the 2 circles

D of Large : D of small = 40 : 32 = 5 : 4

the ratio of Revolutions each makes per minute will be inversely proportional to the above ratio

# Rev of Large : # of Rev of Small = 4 : 5

For every 4 revolutions the large wheel makes ————> smaller wheel will make 5 revolutions

If the large wheel makes R revolutions per minute ———> the smaller wheel will make (5/4) R revolutions per minute

Then convert to (rev) per (hour) ———> 75R

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