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Bunuel
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Two circles, one with radius 10 inches and the other with radius 4 inc 28 Oct 2021, 05:30
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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 3π inches per minute, the other along the smaller circle at 2.5π inches per minute. How much time has elapsed when the two insects meet again at point Q?

(A) 15 minutes
(B) 30 minutes
(C) 40 minutes
(D) 1 hour
(E) 1 hour, 20 minutes
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E
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Two circles, one with radius 10 inches and the other with radius 4 inc 28 Oct 2021, 06:32
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Larger circle circumference = 20 π
Smaller circle circumference = 8 π

Time taken by insect to travel one complete large circle = 20 π / 3 π mins = 20/3 mins
Time taken by insect to travel one complete small circle = 8 π / 2.5 π mins = 16/5 mins

Taking LCM of these times would give us the first time they would meet at Q again

LCM (20, 16) / HCF (3, 5) = 80/1 = 80 minutes = 1 hour and 20 mins

Option E
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Two circles, one with radius 10 inches and the other with radius 4 inc 16 Nov 2022, 00:14
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sanjitscorps18
Larger circle circumference = 20 π
Smaller circle circumference = 8 π

Time taken by insect to travel one complete large circle = 20 π / 3 π mins = 20/3 mins
Time taken by insect to travel one complete small circle = 8 π / 2.5 π mins = 16/5 mins

Taking LCM of these times would give us the first time they would meet at Q again

LCM (20, 16) / HCF (3, 5) = 80/1 = 80 minutes = 1 hour and 20 mins

Option E
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Can someone please explain why LCM of numerators and HCF of denominators is divided to get the answer?
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Two circles, one with radius 10 inches and the other with radius 4 inc 16 Nov 2022, 07:06
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sanjitscorps18
Larger circle circumference = 20 π
Smaller circle circumference = 8 π

Time taken by insect to travel one complete large circle = 20 π / 3 π mins = 20/3 mins
Time taken by insect to travel one complete small circle = 8 π / 2.5 π mins = 16/5 mins

Taking LCM of these times would give us the first time they would meet at Q again

LCM (20, 16) / HCF (3, 5) = 80/1 = 80 minutes = 1 hour and 20 mins

Option E

Can someone please explain why the LCM of numerators and HCF of denominators are divided to get the answer?
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To get the time required to meet at Q, we require the LCM of the times taken to complete a full circle by the insects i.e. LCM(20/3, 16/5)

Now when we take the LCM of fractions we calculate it by dividing the LCM of numerators by the HCF of denominators. Hope this clears up.
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Two circles, one with radius 10 inches and the other with radius 4 inc 02 Oct 2023, 08:34
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Bunuel
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 3π inches per minute, the other along the smaller circle at 2.5π inches per minute. How much time has elapsed when the two insects meet again at point Q?

(A) 15 minutes
(B) 30 minutes
(C) 40 minutes
(D) 1 hour
(E) 1 hour, 20 minutes
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KarishmaB, Bunuel, GMATNinjaTwo, GMATNinja

Can you please post a response here? I could the solve the question up to the point - Insect 1 completes the bigger circle in approx. 6.66 minutes and the insect 1 meanwhile will complete crawling around the smaller circle is 3.2 mins. I was plugging in the answer choices to find a number that is the LCM of both 6.67 and 3.2. I got stuck. How do I solve after that? How is the answer 1 hour 20 mins? Any other method is appreciated as well. Thanks for your response.
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Two circles, one with radius 10 inches and the other with radius 4 inc 02 Oct 2023, 09:29
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Bunuel
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 3π inches per minute, the other along the smaller circle at 2.5π inches per minute. How much time has elapsed when the two insects meet again at point Q?

(A) 15 minutes
(B) 30 minutes
(C) 40 minutes
(D) 1 hour
(E) 1 hour, 20 minutes
Show more

Time taken by insect 1 to cover one circle and come back to Q = Circumference of circle/Speed = 20π/3π = 20/3 min
Time taken by insect 2 to cover one circle and come back to Q = Circumference of circle/Speed = 8π/2.5π = 16/5 min

So insect 1 will come to Q every 20/3 min and insect 2 will come to Q every 16/5 mins.

Think about it. Had these numbers been simpler e.g. insect 1 comes to Q every 5 mins and insect 2 comes to Q every 4 mins, when will they meet at Q again? After 20 mins. Why? Because insect 1 will come at 5 min, 10 min, 15 min, 20 min etc marks and insect 2 will come at 4 min, 8 min, 12 min, 16 min, 20 min etc marks. So both will be at Q at 20 min mark because 20 is the LCM of 5 and 4. This question is different only because the time is in fractions. But we know how to find the LCM of fractions.

LCM = LCM of numerator/HCF of denominator
LCM of 20/3 and 16/5 = 80/1 = 80

So they will meet at Q together after 80 mins i.e. 1 hr 20 mins.

Check this post on my blog on how to find HCF and LCM of fractions: https://anaprep.com/number-properties-g ... fractions/
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Two circles, one with radius 10 inches and the other with radius 4 inc 31 May 2025, 12:18
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