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# Circular gears P and Q start rotating at the same time at

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Re: Circular gears P and Q start rotating at the same time at [#permalink]
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(40-10)*t=6

t=1/5 mins

(1/5)*60=12 sec
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enigma123 wrote:
Circular gears P and Q start rotating at the same time at constant speeds. Gear P makes 10 revolutions per minute and Gear Q makes 40 revolutions per minute. How many seconds after the gears start rotating will gear Q have made exactly 6 more revolutions than gear P ?

A. 6
B. 8
C. 10
D. 12
E. 15

Work done by Faster gear Q is 6 more than slower gear P in same time

i.e W(q) = W(p) + 6
Rq X t = Rp X t + 6
40t=10t + 6
30t = 6
t =1/5 hrs
t = 12 mins
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Re: Circular gears P and Q start rotating at the same time at [#permalink]
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My approach was the following (similar to what LalaB wrote):

Difference between gears Q and P is 30 revolutions per min (40-10).
To make 6 more rotations, the faster gear needs (6 rev / 30 rev/min = 1/5 min). Translate to seconds: 1/5 x 60 = 12.
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Re: Circular gears P and Q start rotating at the same time at [#permalink]
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enigma123 wrote:
Circular gears P and Q start rotating at the same time at constant speeds. Gear P makes 10 revolutions per minute and Gear Q makes 40 revolutions per minute. How many seconds after the gears start rotating will gear Q have made exactly 6 more revolutions than gear P ?

A. 6
B. 8
C. 10
D. 12
E. 15

All the above answers are excellent and I would like to one little tweak that has helped me in quite a few number of Quant problems. This is applicable for people who love to find answers by plugging in the options given

As soon as you arrive at the 2 numbers "6 seconds and 1.5 seconds" (Time taken to complete one rotation) , look at the options given. The best options(9 out of 10 times) are the ones that are a multiple of the above two numbers. Hence you will possibly start with option A and then move to option B and voila.

NOTE: Use this method in scenarios such as this example wherein the number of options that fall under the "multiple umbrella"are quite low. If all the options are multiples it is better to switch on to one of the above strategies.
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Re: Circular gears P and Q start rotating at the same time at [#permalink]
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----RATE-----time--------distance
P:---10 -----x/10------------ x
Q:---40------(x+6)/40-------x+6

The time should be equal, so we solve x/10 = (x+6)/40
x=2

so time = 2/10 min = 12 sec
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Re: Circular gears P and Q start rotating at the same time at [#permalink]
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enigma123 wrote:
Circular gears P and Q start rotating at the same time at constant speeds. Gear P makes 10 revolutions per minute and Gear Q makes 40 revolutions per minute. How many seconds after the gears start rotating will gear Q have made exactly 6 more revolutions than gear P ?

A. 6
B. 8
C. 10
D. 12
E. 15

Another approach: P makes 10 R(evolutions) in 60 secs, so 1 R in 6 Sec. Similarly, as Q makes 40 R in 60 Sec or 4 R in 6 Sec. We can extrapolate the above details to state that P would make 2 R in 12 Sec and Q would make 8 R in 12 sec. Hence Q needs additional 12 Sec to overtake P by 6 Revolutions.
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Re: Circular gears P and Q start rotating at the same time at [#permalink]
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We can resolve this by as following

Gear P

Speed of gear P per second = 10revolutions/60 secs
=1/6 revolutions per second

Speed of gear Q per second = 40revolutions/60 secs
=4/6 revolutions per second

Distance travelled P Q
1sec 1/6 revolutions 4/6 revolutions
6 sec 1 revolution 4 revolution
12 sec 2 revolution 8 revolutions

Hence in 12 seconds Gear P completes 2 rev and Gear Q completes 8 rev
Difference =8-2= 6 revolutions
Therefore in 12 sec Gear Q revolves 6 times ahead than Gear P

Hope this makes it
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Circular gears P and Q start rotating at the same time at [#permalink]
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Here's a visual that should help. Notice that I turned 6 into $$\frac{360}{60}$$ to preserve the common denominator.
Attachments

Screen Shot 2016-03-23 at 5.49.06 PM.png [ 61.57 KiB | Viewed 138058 times ]

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Re: Circular gears P and Q start rotating at the same time at [#permalink]
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enigma123 wrote:
Circular gears P and Q start rotating at the same time at constant speeds. Gear P makes 10 revolutions per minute and Gear Q makes 40 revolutions per minute. How many seconds after the gears start rotating will gear Q have made exactly 6 more revolutions than gear P ?

A. 6
B. 8
C. 10
D. 12
E. 15

As we can see, Gear Q makes 30 more revolutions per minute than Gear P does. So, we can set up the following proportion to solve for the number of seconds for Gear Q to make 6 more revolutions than Gear P does:

30 revolutions/1 minute = 6 revolutions/x seconds

We need the units of time to be the same in the denominator. Since 1 minute = 60 seconds, we can say:

30/60 = 6/x

Cross multiply and we have:

30x = 360

x = 12

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Re: Circular gears P and Q start rotating at the same time at [#permalink]
Bunuel wrote:
enigma123 wrote:
Circular gears P & Q start rotating at the same time at constant speeds. Gear P makes 10 revolutions per minute and Gear Q makes 40 revolutions per minute. How many seconds after the gears start rotating will gear Q have made exactly 6 more revolutions than gear P ?

A. 6
B. 8
C. 10
D. 12
E. 15

Aany idea how to solve this?

Note that we are given revolutions per minute and asked about revolutions in seconds. So we should transform per minute to per second.

Gear P makes 10 revolutions per minute --> gear P makes 10/60 revolutions per second;
Gear Q makes 40 revolutions per minute --> gear Q makes 40/60 revolutions per second.

Let $$t$$ be the time in seconds needed for Q to make exactly 6 more revolutions than gear P --> $$\frac{10}{60}t+6=\frac{40}{60}t$$ --> $$t=12$$.

Hope it's clear.

Hi Bunuel,

How did u arrive at (10/60)t?

then time is t

are we calculating the distance here? Could u pls explain the logic to me?

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Re: Circular gears P and Q start rotating at the same time at [#permalink]
zanaik89 wrote:
Bunuel wrote:
enigma123 wrote:
Circular gears P & Q start rotating at the same time at constant speeds. Gear P makes 10 revolutions per minute and Gear Q makes 40 revolutions per minute. How many seconds after the gears start rotating will gear Q have made exactly 6 more revolutions than gear P ?

A. 6
B. 8
C. 10
D. 12
E. 15

Aany idea how to solve this?

Note that we are given revolutions per minute and asked about revolutions in seconds. So we should transform per minute to per second.

Gear P makes 10 revolutions per minute --> gear P makes 10/60 revolutions per second;
Gear Q makes 40 revolutions per minute --> gear Q makes 40/60 revolutions per second.

Let $$t$$ be the time in seconds needed for Q to make exactly 6 more revolutions than gear P --> $$\frac{10}{60}t+6=\frac{40}{60}t$$ --> $$t=12$$.

Hope it's clear.

Hi Bunuel,

How did u arrive at (10/60)t?

then time is t

are we calculating the distance here? Could u pls explain the logic to me?

10/60 is the number of revolutions per second and t is the time in seconds. What would you get when you multiply those two? (revolutions per second)(time in seconds) = ?
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Re: Circular gears P and Q start rotating at the same time at [#permalink]
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Hi All,

Once you convert each rate into revolutions/second, TESTing THE ANSWERS is a remarkably easy way to get to the correct answer. Algebraically, you can treat it as a "combined rate" question though:

Distance = (Rate) x (Time)

6 revolutions = (difference in rates) x (Time)

6 revolutions = (30 revolutions/min) x (Time)

6/30 = Time in minutes

1/5 minute = Time

Since the question asks for an answer in SECONDS, we have to convert....

1/5 minute = 12 seconds

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Re: Circular gears P and Q start rotating at the same time at [#permalink]
Bunuel, can you share more such questions ?
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Re: Circular gears P and Q start rotating at the same time at [#permalink]
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Re: Circular gears P and Q start rotating at the same time at [#permalink]
----RATE-----time--------distance
P:---10 -----x/10------------ x
Q:---40------(x+6)/40-------x+6

The time should be equal, so we solve x/10 = (x+6)/40
x=2

so time = 2/10 min = 12 sec

Can anyone explain me how we get from x=2 ----> to 12 sec???
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Re: Circular gears P and Q start rotating at the same time at [#permalink]
Hi hsn81960,

You did not label the "units" in your table, but based on the work that you presented, I think that you know that time is in MINUTES.

You solved for X and got X=2... when you plug that back in for X you get:

2/10 = 1/5 of a minute for Gear P
8/40 = 1/5 of a minute for Gear Q

Thus, it clearly takes 1/5 of a 1 minute for the revolutions to match what the question asked for. Since 1 minute = 60 seconds, then (1/5)(60) = 12 seconds

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Circular gears P and Q start rotating at the same time at [#permalink]
LalaB have you used the relative speed formula?
Circular gears P and Q start rotating at the same time at [#permalink]
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