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Arithmetic II

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Time & Speed

Time And Speed

MODULES

Basics of Ratios
Average Speed
Product Constancy
Relative Speed
Trains
Walking To & Fro
Circular Track
Boats, Races & Clock
Escalators
Past Questions

CONCEPTS & CHEATSHEET

Concept Revision Video

SPEED CONCEPTS

Time & Speed 1
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Time & Speed 2
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Time & Speed 3
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Time & Speed 4
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PRACTICE

Time & Speed : Level 1
Time & Speed : Level 2
Time & Speed : Level 3
ALL MODULES

CAT 2025 Lesson : Time & Speed - Circular Track

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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 lll metres at speed of s1s_1s1​ and s2s_2s2​ respectively (where s1s_1s1​ > s2s_2 s2​), their relative speed is s1−s2s_1- s_2s1​−s2​. 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 =ls1−s2= \dfrac{l}{s_1 - s_2}=s1​−s2​l​

Time taken to meet for the second time =2ls1−s2= \dfrac{2l}{s_1 - s_2}=s1​−s2​2l​

Time taken to meet for the nthn^{\text{th}}nth time = nls1−s2 \dfrac{nl}{s_1 - s_2}s1​−s2​nl​

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 lll metres at speed of s1s_1s1​ and s2s_2s2​ respectively, their relative speed is s1+s2s_1 + s_2s1​+s2​. 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 =ls1+s2= \dfrac{l}{s_1 + s_2}=s1​+s2​l​

Time taken to meet for the second time =2ls1+s2= \dfrac{2l}{s_1 + s_2}=s1​+s2​2l​

Time taken to meet for the nthn^{\text{th}}nth time =nls1+s2= \dfrac{nl}{s_1 + s_2}=s1​+s2​nl​

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 {ls1,ls2 \dfrac{l}{s_1}, \dfrac{l}{s_2}s1​l​,s2​l​}

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 20th20^{th}20th time, what is the distance covered by A (in km)?

Solution

As they are running in the same direction, relative speed =25−15=10= 25 - 15 = 10=25−15=10 m/s

Relative distance covered when they meet for the 20th20^{\text{th}}20th time is the 202020 overlaps of A. So, the distance is 20×400=800020 \times 400 = 800020×400=8000 metres.

So, time taken =800010=800= \dfrac{8000}{10} = 800=108000​=800 seconds.

Distance covered by A =25×800=20000= 25 \times 800 = 20000=25×800=20000 metres = 202020 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= 3 : 5=3:5
Ratio of speeds of Parul and Ramnique =5:3= 5 : 3=5:3

Let the length of track be 151515 metres and the speeds of Parul and Ramnique be 555 metres/minute and 333 metres/minute respectively.

Time taken to meet for the first time =155−3=7.5= \dfrac{15}{5 - 3} = 7.5=5−315​=7.5 minutes

Time taken to meet for the fourth time =4×7.5=30= 4 \times 7.5 = 30=4×7.5=30 minutes

Answer: 30 mins

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