Insect Race Challenge Problem

[ch339]

I hope this is the right forum. I had a question about the latest Challenge Problem (reproduced below). I don't quite get the arithmetic solution. I follow what you're doing but not why.

I get the numbers differently.

To make 16/5 look like 20/3, I have to multiply 16/5 by 1.25/.6 = 125/100*10/6 = 25/12. This corresponds to 25 revolutions for the small circle and 12 revolutions for the big circle. This results in 80 minutes for both. I was able to reverse-engineer this answer, but I'm having trouble coming up with a rule for this type of problem (work/rate problems are the bane of my existence). I know now that figuring out how to make equivalent fractions is the key. I also understand that 25 has to refer to the smaller circle, both because the rate is slower and the circumference is smaller (big circle requires fewer revolutions). However, what if the larger circle had had a slower rate? How would I use my method then?

Alternatively, can you provide an alternative explanation to the official solution posted? The arithmetic solution given is not intuitive for me. Thanks!

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Last Week's Problem: "The Insect Race"
Two circles, one with radius 10 inches and the other with radius 4 inches, are tangent at point Q. Two insects start crawling at the same time from point Q: one along the larger circle at 3Ï€ inches per minute, the other along the smaller circle at 2.5Ï€ inches per minute. How much time has elapsed when the two insects meet again at point Q?
Answer This Week's Question for Prizes

(A)

15 minutes
(B)

30 minutes
(C)

40 minutes
(D)

1 hour
(E)

1 hour, 20 minutes

[RonPurewal]

The whole point of this problem is that it's impossible to solve with algebra. So, if "an alternative solution" means algebra, then no fish are biting this time.
You can't set up an algebraic equation because it's not possible to tell when the insects will meet again. Instead, you have to think about common multiples of things.

In fact, this problem -- like a lot of GMAT problems -- is specifically designed as a smack in the face for students who think they need "rules" for everything. (Incidentally, that's also the entire reason why CR is on the test, but that's another topic for another forum.)
I honestly can't think of a "rule" that you can use here -- but, more importantly, such a thing would be useless on future problems anyway.

[RonPurewal]

Solution-wise:

I don't know what's in the official solution. (Please post it here!) So, what I'm about to write may be identical to what's already there -- I don't know.
It's probably different, because you're referring to the idea of "making the fractions look the same". Which scares me. Over my head.
o_O

But:

* The two insects take, respectively, 20/3 and 16/5 minutes to go around the circle. It appears you've figured this out already, so, no need to rehash the issue here.

* You're looking for a common multiple of 20/3 and 16/5.

* It should be clear that this common multiple is going to be an integer...
... because the non-integer values will always be xxxx/5 and xxxx/3, and so can't be the same,
and, MUCH more importantly,
... because the answer choices are all integers!

* The integer multiples of 20/3 are 20, 40, 60, etc.

* The integer multiples of 16/5 are 16, 32, 48, etc.

* The first integer that's in both of these lists is 80.

Done!

[RonPurewal]

Another very efficient way to solve this problem: Just take each answer choice and see whether it's a multiple of 16/5 and 20/3. First one that's both, wins.

[RonPurewal]

And finally ...

I hope this is the right forum.

It's not the right forum, actually. Although I appreciate that you're actually thinking about this, rather than just randomly posting in random places.

This problem is from MGMAT. It's not a CAT exam problem. So, it should be in the "MGMAT non-CAT math" folder.

I'm going to lock this thread. If you have further questions about the problem, please post them in the correct folder. Thank you.