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Arithmetic II

Time & Speed

ALL MODULES

CAT 2025 Lesson : Time & Speed - Circular Track

8. Distance Covered on a circular track

8.1 Running in the same direction from the same starting point

If two people A and B are running in the same direction around a circular track of

l

metres at speed of

s_1

and

s_2

respectively (where

s_1

s_2

), their relative speed is

s_1- s_2

. The first time they meet after starting is when A overlaps B. So, the distance covered is the length of the track.

Time taken to meet for the first time

= \dfrac{l}{s_1 - s_2}

Time taken to meet for the second time

= \dfrac{2l}{s_1 - s_2}

Time taken to meet for the

n^{\text{th}}

time =

\dfrac{nl}{s_1 - s_2}

8.2 Running in the opposite direction from the same starting point

If two people A and B are running in the opposite directions around a circular track of

l

metres at speed of

s_1

and

s_2

respectively, their relative speed is

s_1 + s_2

. The first time they meet after starting is when a complete lap is covered by the two. So, the distance covered is the length of the track.

Time taken to meet for the first time

= \dfrac{l}{s_1 + s_2}

Time taken to meet for the second time

= \dfrac{2l}{s_1 + s_2}

Time taken to meet for the

n^{\text{th}}

time

= \dfrac{nl}{s_1 + s_2}

Note that the ratio of the distance covered by them individually will be the ratio of their speeds.

8.3 Time taken to meet for the first time at the Starting Point

When two people are running in the same or opposite direction, the time taken to meet for the first time at the starting point

=

LCM of the individual time taken to complete a lap

=

LCM of {

\dfrac{l}{s_1}, \dfrac{l}{s_2}

}

Example 26

A and B start running, in the same direction, around a circular track of length 400 metres at speeds of 25 m/s and 15 m/s respectively. When A and B meet for the

20^{th}

time, what is the distance covered by A (in km)?

Solution

As they are running in the same direction, relative speed

= 25 - 15 = 10

m/s

Relative distance covered when they meet for the

20^{\text{th}}

time is the

20

overlaps of A. So, the distance is

20 \times 400 = 8000

metres.

So, time taken

= \dfrac{8000}{10} = 800

seconds.

Distance covered by A

= 25 \times 800 = 20000

metres =

20

km

Answer: 20

Example 27

Parul and Ramnique take 3 minutes and 5 minutes to go around a circular track. How many minutes does it take for them to meet for the fourth time if they started running from the same point and in the same direction?

Solution

Given the length of the track being constant for both, ratio of speeds is the ratio of reciprocal of the times taken.

Ratio of times taken by Parul and Ramnique

= 3 : 5

Ratio of speeds of Parul and Ramnique

= 5 : 3

Let the length of track be

15

metres and the speeds of Parul and Ramnique be

5

metres/minute and

3

metres/minute respectively.

Time taken to meet for the first time

= \dfrac{15}{5 - 3} = 7.5

minutes

Time taken to meet for the fourth time

= 4 \times 7.5 = 30

minutes

Answer: 30 mins

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Back

Quant

Arithmetic II

Time & Speed

ALL MODULES

CAT 2025 Lesson : Time & Speed - Circular Track

8. Distance Covered on a circular track

8.1 Running in the same direction from the same starting point

If two people A and B are running in the same direction around a circular track of

l

metres at speed of

s_1

and

s_2

respectively (where

s_1

s_2

), their relative speed is

s_1- s_2

. The first time they meet after starting is when A overlaps B. So, the distance covered is the length of the track.

Time taken to meet for the first time

= \dfrac{l}{s_1 - s_2}

Time taken to meet for the second time

= \dfrac{2l}{s_1 - s_2}

Time taken to meet for the

n^{\text{th}}

time =

\dfrac{nl}{s_1 - s_2}

8.2 Running in the opposite direction from the same starting point

If two people A and B are running in the opposite directions around a circular track of

l

metres at speed of

s_1

and

s_2

respectively, their relative speed is

s_1 + s_2

. The first time they meet after starting is when a complete lap is covered by the two. So, the distance covered is the length of the track.

Time taken to meet for the first time

= \dfrac{l}{s_1 + s_2}

Time taken to meet for the second time

= \dfrac{2l}{s_1 + s_2}

Time taken to meet for the

n^{\text{th}}

time

= \dfrac{nl}{s_1 + s_2}

Note that the ratio of the distance covered by them individually will be the ratio of their speeds.

8.3 Time taken to meet for the first time at the Starting Point

When two people are running in the same or opposite direction, the time taken to meet for the first time at the starting point

=

LCM of the individual time taken to complete a lap

=

LCM of {

\dfrac{l}{s_1}, \dfrac{l}{s_2}

}

Example 26

A and B start running, in the same direction, around a circular track of length 400 metres at speeds of 25 m/s and 15 m/s respectively. When A and B meet for the

20^{th}

time, what is the distance covered by A (in km)?

Solution

As they are running in the same direction, relative speed

= 25 - 15 = 10

m/s

Relative distance covered when they meet for the

20^{\text{th}}

time is the

20

overlaps of A. So, the distance is

20 \times 400 = 8000

metres.

So, time taken

= \dfrac{8000}{10} = 800

seconds.

Distance covered by A

= 25 \times 800 = 20000

metres =

20

km

Answer: 20

Example 27

Solution

Given the length of the track being constant for both, ratio of speeds is the ratio of reciprocal of the times taken.

Ratio of times taken by Parul and Ramnique

= 3 : 5

Ratio of speeds of Parul and Ramnique

= 5 : 3

Let the length of track be

15

metres and the speeds of Parul and Ramnique be

5

metres/minute and

3

metres/minute respectively.

Time taken to meet for the first time

= \dfrac{15}{5 - 3} = 7.5

minutes

Time taken to meet for the fourth time

= 4 \times 7.5 = 30

minutes

Answer: 30 mins

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