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

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 --> \(\frac{10}{60}t+6=\frac{40}{60}t\) --> \(t=12\).

Answer: D.

Hope it's clear.
_________________

I used the following approach:

Circular P has 10rpm >> Therefore one rotation will be every 10/60 seconds

Circular Q has 40rpm >> Therefore one rotation will be every 40/60 seconds


Now we are asked after how many seconds will circular Q have made 6 more rotations than Circular P. Translating into algebra

4/6S-1/6(S)=6

Solve for S (where s is number of seconds)
S=12

(40-10)*t=6

t=1/5 mins

(1/5)*60=12 sec
_________________

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

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.

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.

----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

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.

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
Like my post Plz Send me Kudos

Here's a visual that should help. Notice that I turned 6 into \(\frac{360}{60}\) to preserve the common denominator.

_________________

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

Answer: D
_________________

Hi Bunuel,

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

when speed is already 10/60

then time is t

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

Thanks in advance

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) = ?
_________________

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

Final Answer:

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

Bunuel, can you share more such questions ?

Sure.

Similar questions to practice:
https://gmatclub.com/forum/a-circular-ri ... 65106.html
https://gmatclub.com/forum/two-wheels-ar ... 33697.html
https://gmatclub.com/forum/how-long-in-m ... 30026.html
https://gmatclub.com/forum/what-is-the-n ... 66668.html
https://gmatclub.com/forum/the-circumfer ... 90588.html
https://gmatclub.com/forum/a-point-on-th ... 65505.html
https://gmatclub.com/forum/at-the-same-t ... 99350.html
https://gmatclub.com/forum/m10-183873.html
https://gmatclub.com/forum/ben-is-drivi ... 04302.html
https://gmatclub.com/forum/a-bicycle-wh ... 45317.html
https://gmatclub.com/forum/two-intercon ... 22357.html
https://gmatclub.com/forum/a-bicycle-ti ... 63171.html

Hope it helps.
_________________

----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???

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

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

LalaB have you used the relative speed formula?