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"A" and "B" run around a circular track starting from the same point s 25 Jul 2016, 05:08
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"A" and "B" run around a circular track starting from the same point simultaneously in the same direction. "A" meets "B" for the first time when "A" is exactly in the middle of his 5th round. If "A" is faster than "B" and take 70 seconds to complete 1 lap, how long will B take to complete 1 lap?

A) 90 seconds
B) 54.44 seconds
C) 110 seconds
D) 63 seconds
E) 77 seconds

Show SpoilerSOLUTION
A is faster than B. Every second he increases distance between A and B. They will meet for the first time again when he increases distance between them by one full circle. (Say the lap is of 100m. He keeps increasing distance between them. When he is 90 m ahead of him, it is the same as 10 m behind him because they are moving in a circle. Finally when he is 100 m ahead of him, he is exactly 1 lap ahead and hence both are at the same point).

A takes 70 secs for one full lap. So he covers 4.5 laps in exactly 70*4.5 = 315 seconds.

In 315 secs, B completes 3.5 laps. So for each lap, he takes 315/3.5 = 90 secs

Answer (A)
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"A" and "B" run around a circular track starting from the same point s 28 Jul 2016, 14:32
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Draw a circle with 2 ends A' and B'(Diametric) if they both start at A', then mid of 5th lap would mean A is at point B' That means 4.5 laps. Total time : 315 {4.5*70}seconds. B should also be at the same point. Start using options. If we take 90 seconds, a quick glimpse will tell us that 90+90+90+45 (3.5 laps) is also 315 and also the same position i.e. B' Don't need to look further. Hence A is the answer .
A quick way of avoiding calculations!



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"A" and "B" run around a circular track starting from the same point s 28 Jul 2016, 10:20
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VeritasPrepKarishma
Responding to a pm:

"A" and "B" run around a circular track starting from the same point simultaneously in the same direction. "A" meets "B" for the first time when "A" is exactly in the middle of his 5th round. If "A" is faster than "B" and take 70 seconds to complete 1 lap, how long will B take to complete 1 lap?

A) 90 seconds
B) 54.44 seconds
C) 110 seconds
D) 63 seconds
E) 77 seconds

A is faster than B. Every second he increases distance between A and B. They will meet for the first time again when he increases distance between them by one full circle. (Say the lap is of 100m. He keeps increasing distance between them. When he is 90 m ahead of him, it is the same as 10 m behind him because they are moving in a circle. Finally when he is 100 m ahead of him, he is exactly 1 lap ahead and hence both are at the same point).

A takes 70 secs for one full lap. So he covers 4.5 laps in exactly 70*4.5 = 315 seconds.

In 315 secs, B completes 3.5 laps. So for each lap, he takes 315/3.5 = 90 secs

Answer (A)
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How did you get "In 315 secs, B completes 3.5 laps."? Please elaborate.
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"A" and "B" run around a circular track starting from the same point s 28 Jul 2016, 13:56
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VeritasPrepKarishma
Responding to a pm:

"A" and "B" run around a circular track starting from the same point simultaneously in the same direction. "A" meets "B" for the first time when "A" is exactly in the middle of his 5th round. If "A" is faster than "B" and take 70 seconds to complete 1 lap, how long will B take to complete 1 lap?

A) 90 seconds
B) 54.44 seconds
C) 110 seconds
D) 63 seconds
E) 77 seconds

A is faster than B. Every second he increases distance between A and B. They will meet for the first time again when he increases distance between them by one full circle. (Say the lap is of 100m. He keeps increasing distance between them. When he is 90 m ahead of him, it is the same as 10 m behind him because they are moving in a circle. Finally when he is 100 m ahead of him, he is exactly 1 lap ahead and hence both are at the same point).

A takes 70 secs for one full lap. So he covers 4.5 laps in exactly 70*4.5 = 315 seconds.

In 315 secs, B completes 3.5 laps. So for each lap, he takes 315/3.5 = 90 secs

Answer (A)
Show more

Hi! VeritasPrepKarishma,

Can't there be a possibility that B is in its first lap when A meets B.

Also, I am confused that how can A meets B for the first time in the middle of 5th lap since its a circle. Please enlighten.
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"A" and "B" run around a circular track starting from the same point s 28 Jul 2016, 21:48
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abhimahna
VeritasPrepKarishma
Responding to a pm:

"A" and "B" run around a circular track starting from the same point simultaneously in the same direction. "A" meets "B" for the first time when "A" is exactly in the middle of his 5th round. If "A" is faster than "B" and take 70 seconds to complete 1 lap, how long will B take to complete 1 lap?

A) 90 seconds
B) 54.44 seconds
C) 110 seconds
D) 63 seconds
E) 77 seconds

A is faster than B. Every second he increases distance between A and B. They will meet for the first time again when he increases distance between them by one full circle. (Say the lap is of 100m. He keeps increasing distance between them. When he is 90 m ahead of him, it is the same as 10 m behind him because they are moving in a circle. Finally when he is 100 m ahead of him, he is exactly 1 lap ahead and hence both are at the same point).

A takes 70 secs for one full lap. So he covers 4.5 laps in exactly 70*4.5 = 315 seconds.

In 315 secs, B completes 3.5 laps. So for each lap, he takes 315/3.5 = 90 secs

Answer (A)

How did you get "In 315 secs, B completes 3.5 laps."? Please elaborate.
Show more
Divyadisha

A covers 4.5 laps in 315 seconds. This is when they meet. So B has also been running for 315 seconds. How much distance will he cover in this time?
Think about it: If B was still on his first lap, A would have met him in his second lap. Why? Because B would be somewhere on the circular path when A takes his second lap. So he would have met B then. But A met him for the first time after 4.5 laps.
Since they are both running in the same direction, till the time they cover equal number of laps (perhaps A is very slightly faster than B), they don't meet. B lags behind A. But when A covers a distance of 1 lap more than B, he has to cross B again (since they are running on the same circular path).

Try to draw a diagram to imagine it. They meet for the first time when A covers exactly 1 lap more than B.
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"A" and "B" run around a circular track starting from the same point s 22 Aug 2016, 05:04
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VeritasPrepKarishma
Responding to a pm:

"A" and "B" run around a circular track starting from the same point simultaneously in the same direction. "A" meets "B" for the first time when "A" is exactly in the middle of his 5th round. If "A" is faster than "B" and take 70 seconds to complete 1 lap, how long will B take to complete 1 lap?

A) 90 seconds
B) 54.44 seconds
C) 110 seconds
D) 63 seconds
E) 77 seconds

A is faster than B. Every second he increases distance between A and B. They will meet for the first time again when he increases distance between them by one full circle. (Say the lap is of 100m. He keeps increasing distance between them. When he is 90 m ahead of him, it is the same as 10 m behind him because they are moving in a circle. Finally when he is 100 m ahead of him, he is exactly 1 lap ahead and hence both are at the same point).

A takes 70 secs for one full lap. So he covers 4.5 laps in exactly 70*4.5 = 315 seconds.

In 315 secs, B completes 3.5 laps. So for each lap, he takes 315/3.5 = 90 secs

Answer (A)
Show more

Dear Karishma

I have a question on this - when it says A catches B when A is exactly in the middle of the 5 th round does not this mean that 5.5 laps for A and 4.5 laps for B.
As I have solved the problem with 5.5 laps in 385 secs (70*5.5) for A and 4.5 Laps for B in the same time frame.
And result is 85.5 secs.
Can you help where I am doing wrong?

Thanks
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"A" and "B" run around a circular track starting from the same point s 22 Aug 2016, 05:16
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Rasul_Rasul
VeritasPrepKarishma
Responding to a pm:

"A" and "B" run around a circular track starting from the same point simultaneously in the same direction. "A" meets "B" for the first time when "A" is exactly in the middle of his 5th round. If "A" is faster than "B" and take 70 seconds to complete 1 lap, how long will B take to complete 1 lap?

A) 90 seconds
B) 54.44 seconds
C) 110 seconds
D) 63 seconds
E) 77 seconds

A is faster than B. Every second he increases distance between A and B. They will meet for the first time again when he increases distance between them by one full circle. (Say the lap is of 100m. He keeps increasing distance between them. When he is 90 m ahead of him, it is the same as 10 m behind him because they are moving in a circle. Finally when he is 100 m ahead of him, he is exactly 1 lap ahead and hence both are at the same point).

A takes 70 secs for one full lap. So he covers 4.5 laps in exactly 70*4.5 = 315 seconds.

In 315 secs, B completes 3.5 laps. So for each lap, he takes 315/3.5 = 90 secs

Answer (A)

Dear Karishma

I have a question on this - when it says A catches B when A is exactly in the middle of the 5 th round does not this mean that 5.5 laps for A and 4.5 laps for B.
As I have solved the problem with 5.5 laps in 385 secs (70*5.5) for A and 4.5 Laps for B in the same time frame.
And result is 85.5 secs.
Can you help where I am doing wrong?

Thanks
Show more

That's a common mistake. Think about it - when I say, A caught B in the middle of the first round, what does that mean? Does it mean 0.5 or 1.5? The middle of the first round is when A has covered only half a round. So it will be 0.5.
Similarly, middle of 5th round is 4 complete rounds and a half.
It's the way we think about age.
Middle of first year means the baby is only 6 months old. The fifth year is the year that starts after the child completes 4 yrs.
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"A" and "B" run around a circular track starting from the same point s 29 Oct 2018, 04:14
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Let the time taken by B to complete 1 lap = x seconds.
Time taken by A = 70 seconds.
Given ,
A meets B for the first time at the mid point of 5th round = 4.5 * 70 seconds.
Therefore , he will me B for the first time at the starting point after 2 * 4.5 * 70 = 630 seconds.

LCM of x and 70 should be 630.

Only A fits
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Hi RK84 -how are you so sure that 90 is the answer just by using 90+90+90+45? The way I did it was rule out choices (B) and (D), since they contradict the fact that A is faster than B (these would mean B is faster than A). Then, I was down to 90, 77, and 110 seconds as my 3 possible answer options. I was not sure how to go from there. I see based off of your reasoning that 315 seconds is a multiple of 90 (90*3.5=315), but I'm not sure how you're using this as reasoning to solve this problem.

KarishmaB - also struggling to see your reasoning a bit here. If you could elaborate a bit more on how to calculate the lap time for B beyond simply askin us to visualize the gap between A and B and how it changes, that would help.­
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"A" and "B" run around a circular track starting from the same point s 17 Apr 2024, 22:46
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kkannan2
Hi RK84 -how are you so sure that 90 is the answer just by using 90+90+90+45? The way I did it was rule out choices (B) and (D), since they contradict the fact that A is faster than B (these would mean B is faster than A). Then, I was down to 90, 77, and 110 seconds as my 3 possible answer options. I was not sure how to go from there. I see based off of your reasoning that 315 seconds is a multiple of 90 (90*3.5=315), but I'm not sure how you're using this as reasoning to solve this problem.

KarishmaB - also struggling to see your reasoning a bit here. If you could elaborate a bit more on how to calculate the lap time for B beyond simply askin us to visualize the gap between A and B and how it changes, that would help.­
Show more
­
It uses the concepts of ratios, time speed distance and circular motion.
When A and B start from the same point at the same time and move in the same direction around a circle, they meet for the first time when the faster covers exactly one lap more than the slower.
Since A has covered 4.5 laps when they meet, it means B has covered 3.5 laps in that time. 
So ratio of their speeds is 4.5:3.5 = 9:7
Hence ratio of time taken by them to cover the same distance (say 1 lap) will be 7:9 (inverse of speed)
Since A takes 70 secs for 1 lap, B will take 90 secs. 

You can check out Ratios, TSD and circular motion in my content for free on Sunday through Super Sundays. Details here:
https://youtu.be/gN_vlDpUflo
 
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"A" and "B" run around a circular track starting from the same point s 18 Apr 2024, 04:34
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Best way:
At same time A covers 4.5D (D is circumference)
, B covers 0.5D, or 1.5D , or 2.5D, 3.5D

Ratio of speeds of A&B=9/1,9/3,9/5,9/7

Time taken by B= 70* ratio=only last ratio works= 90sec

A)

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"A" and "B" run around a circular track starting from the same point s 18 Apr 2024, 15:43
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Thanks KarishmaB for the explanation! The point here is that with circular motion, the faster person will meet the slower person if and only if the faster person has "lapped" the slower person, right? Meaning that the faster person will be exactly 1 lap ahead. Visualizing it that way (as if two people were running on a track) and thinking about when they would meet helped me understand your explanation clearer.
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"A" and "B" run around a circular track starting from the same point s 18 Apr 2024, 22:19
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Given: "A" and "B" run around a circular track starting from the same point simultaneously in the same direction. "A" meets "B" for the first time when "A" is exactly in the middle of his 5th round.

Asked: If "A" is faster than "B" and take 70 seconds to complete 1 lap, how long will B take to complete 1 lap?

When A complete 4.5 rounds, B complete 4.5-1 =3.5 rounds.

Let B complete 1 lap in x seconds

x/70 = 4.5/3.5 = 9/7
x = 9*70/7 = 90 seconds

IMO A­
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kkannan2
Thanks KarishmaB for the explanation! The point here is that with circular motion, the faster person will meet the slower person if and only if the faster person has "lapped" the slower person, right? Meaning that the faster person will be exactly 1 lap ahead. Visualizing it that way (as if two people were running on a track) and thinking about when they would meet helped me understand your explanation clearer.
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­Yes, exactly!­
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My approach to the solution:

They meet for the first time when A was in the middle of his 5th round => 4.5 rounds.
Time taken by A to complete 4.5 rounds = 70*4.5 seconds = 315 seconds.

Now, since they are moving in the same direction, they meet for the first time when there is a lap difference of 1 between A and B and so on (second meet when lap difference is 2, .....).

This implies in the same interval, B must have travelled 1 less round than A => 4.5 -1 = 3.5 rounds.

Therefore, time required for B to complete 1 round = 315/3.5 seconds = 90 seconds!
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"A" and "B" run around a circular track starting from the same point s 05 Nov 2025, 09:47
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Hi KarishmaB , I have the same doubt still and the below explanation is not able to clear that. If they start at the same time, A should have met B on his 2nd lap only since it's a circle, how is A meeting B on his 5th lap coz no matter how I try to imagine it, as A travels he will pass B on his 2nd lap, otherwise where will B disappear while A is making the 2nd lap? Please clear this confusion

KarishmaB

Divyadisha

A covers 4.5 laps in 315 seconds. This is when they meet. So B has also been running for 315 seconds. How much distance will he cover in this time?
Think about it: If B was still on his first lap, A would have met him in his second lap. Why? Because B would be somewhere on the circular path when A takes his second lap. So he would have met B then. But A met him for the first time after 4.5 laps.
Since they are both running in the same direction, till the time they cover equal number of laps (perhaps A is very slightly faster than B), they don't meet. B lags behind A. But when A covers a distance of 1 lap more than B, he has to cross B again (since they are running on the same circular path).

Try to draw a diagram to imagine it. They meet for the first time when A covers exactly 1 lap more than B.
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"A" and "B" run around a circular track starting from the same point s 05 Nov 2025, 22:15
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Natansha, imagine the case in which their speeds are almost the same but B is very slightly slower than A. So B will be shadowing A for a long time and A will take many laps to finally cover one whole circle more than B and come up behind him.

Natansha
Hi KarishmaB , I have the same doubt still and the below explanation is not able to clear that. If they start at the same time, A should have met B on his 2nd lap only since it's a circle, how is A meeting B on his 5th lap coz no matter how I try to imagine it, as A travels he will pass B on his 2nd lap, otherwise where will B disappear while A is making the 2nd lap? Please clear this confusion


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"A" and "B" run around a circular track starting from the same point s 21 Mar 2026, 03:06
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Great solution! I was however trying to find out a way with the distinct "first time meet" exactly at the half way stage. Can we also thus say that the number of distinct meeting points is 2? As the meeting points are always equally distributed across the length of the track no matter the length of the track. How can we solve the problem looking at it from that perspective? I am getting stuck somewhere, kindly help me out.
KarishmaB
"A" and "B" run around a circular track starting from the same point simultaneously in the same direction. "A" meets "B" for the first time when "A" is exactly in the middle of his 5th round. If "A" is faster than "B" and take 70 seconds to complete 1 lap, how long will B take to complete 1 lap?

A) 90 seconds
B) 54.44 seconds
C) 110 seconds
D) 63 seconds
E) 77 seconds

Show SpoilerSOLUTION
A is faster than B. Every second he increases distance between A and B. They will meet for the first time again when he increases distance between them by one full circle. (Say the lap is of 100m. He keeps increasing distance between them. When he is 90 m ahead of him, it is the same as 10 m behind him because they are moving in a circle. Finally when he is 100 m ahead of him, he is exactly 1 lap ahead and hence both are at the same point).

A takes 70 secs for one full lap. So he covers 4.5 laps in exactly 70*4.5 = 315 seconds.

In 315 secs, B completes 3.5 laps. So for each lap, he takes 315/3.5 = 90 secs

Answer (A)
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"A" and "B" run around a circular track starting from the same point s 21 Mar 2026, 05:04
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Since A is faster than B, we can automatically rule out options B and D. This leaves 90, 77, and 110 seconds as the possible lap times for B.

A -> 70 Seconds for 1 lap -> 315 Seconds for 4.5 laps.

Because they meet halfway across the track at the 315-second mark, B must have run a distance ending in a half-lap (e.g., 0.5, 1.5, 2.5, or 3.5 laps) in that exact same amount of time (315 Sec).

Using trial & error with the remaining time options (77, 90, and 110 seconds) to see which fits this 315 second window

So, answer is A. 90 Seconds


KarishmaB
"A" and "B" run around a circular track starting from the same point simultaneously in the same direction. "A" meets "B" for the first time when "A" is exactly in the middle of his 5th round. If "A" is faster than "B" and take 70 seconds to complete 1 lap, how long will B take to complete 1 lap?

A) 90 seconds
B) 54.44 seconds
C) 110 seconds
D) 63 seconds
E) 77 seconds

Show SpoilerSOLUTION
A is faster than B. Every second he increases distance between A and B. They will meet for the first time again when he increases distance between them by one full circle. (Say the lap is of 100m. He keeps increasing distance between them. When he is 90 m ahead of him, it is the same as 10 m behind him because they are moving in a circle. Finally when he is 100 m ahead of him, he is exactly 1 lap ahead and hence both are at the same point).

A takes 70 secs for one full lap. So he covers 4.5 laps in exactly 70*4.5 = 315 seconds.

In 315 secs, B completes 3.5 laps. So for each lap, he takes 315/3.5 = 90 secs

Answer (A)
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"A" and "B" run around a circular track starting from the same point s 22 Mar 2026, 08:41
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Check out the key points discussion on circular motion here: https://youtu.be/Jz-mSB1xNFY


Adit_
Great solution! I was however trying to find out a way with the distinct "first time meet" exactly at the half way stage. Can we also thus say that the number of distinct meeting points is 2? As the meeting points are always equally distributed across the length of the track no matter the length of the track. How can we solve the problem looking at it from that perspective? I am getting stuck somewhere, kindly help me out.

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