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by me_1234 » Tue Dec 09, 2014 5:57 pm
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

I understand for this problem that P rate = 10/1 (in minutes) and 1/6 (in seconds, and that Q = 40/1 (in minutes) and 2/3 (in seconds). You can write out up to 12 steps counting to see when they are 6 rotations apart, but what is the algebraic expression to solve this?

Would it be 1/6x = 2/3x + 6
1/6x - 2/3x = 6
- 3/6x =6
x = -12
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by [email protected] » Tue Dec 09, 2014 6:09 pm
Hi melanie.espeland,

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: D

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by me_1234 » Tue Dec 09, 2014 6:13 pm
That was very helpful, thank you!
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by GMATGuruNY » Tue Dec 09, 2014 8:22 pm
melanie.espeland 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
When elements COMPETE, determine the DIFFERENCE between the rates.
Q's rate - P's rate = 40-10 = 30 revolutions per minute.
Time for Q to make 6 more revolutions = (number of revolutions)/(rate difference) = 6/30 = 1/5 of a minute = 12 seconds.

The correct answer is D.

Other problems that can be solved by determining the CATCH-UP rate:

https://www.beatthegmat.com/rates-work-t119850.html
https://www.beatthegmat.com/dune-buggie-t108778.html
https://www.beatthegmat.com/interesting- ... 16571.html
https://www.beatthegmat.com/rate-problem-t111575.html
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by Brent@GMATPrepNow » Wed Dec 10, 2014 3:27 pm
melanie.espeland 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
I like Mitch's approach a lot, but here's one more (longer) approach for your math toolbox:

First rewrite speeds as revolutions per second (since the question uses these units)

Gear P makes 10 revolution per minute, in other words 10 revolutions per 60 seconds.
To determine the number of revolutions per 1 second, divide 10 by 60, to get 10/60 revolutions per second (a.k.a. 1/6 revolutions per second)

Gear Q makes 40 revolution per minute (or 40 revolutions per 60 seconds).
To determine the number of revolutions per 1 second, divide 40 by 60, to get 40/60 revolutions per second (a.k.a. 2/3 revolutions per second)

Now let t = the time in seconds

The number of revolutions gear P makes in t seconds = (1/6)t
The number of revolutions gear Q makes in t seconds = (2/3)t

We need to determine the number of seconds it takes such that gear Q makes exactly 6 more revolutions than gear P.

So, we want to know the value of t such that:
(Q's revolutions) - (P's revolutions) = 6
Or . . . (2/3)t - (1/6)t = 6
To solve, first multiply both sides by 6 to get: 4t - t = 36
3t = 36
t = 12

It will take 12 seconds, so the answer is D

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Brent
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by me_1234 » Wed Dec 10, 2014 4:33 pm
Thanks all for the help
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by Gmatdecoder » Sun Dec 14, 2014 1:51 am
Back solving...
we know gear P in one sec completes 1/6 rev and gear Q 2/3 rev.

starting from choice D which is multiple of factors in both fractions, P would have completed 2 revs and Q 8 revs. The difference is Q-P =6.
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by Mathsbuddy » Tue Dec 16, 2014 8:03 am
Comparing ratios is very simple -
P:Q:difference
1:4:3
2:8:6
2 revolutions takes 2/10 minutes = 12 seconds
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by me_1234 » Tue Dec 16, 2014 9:22 am
@ Mathsbuddy

I'm not clear on how you derived these ratios, can you please explain?
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by mbawisdom » Tue Dec 16, 2014 9:57 am
Relative speed: Q rotates 30 revolutions more per minute (60 seconds) that P.
To find out how long it takes Q to do 6 more revolutions than P we simply need to multiply 60 seconds by 1/5 (which is obtained by dividing 6/30).
Answer: 12 seconds.
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