Races and Circular Tracks: Head Start, Beats By, Dead Heat and Meeting Point Formulas for SSC Maths

Race and circular track problems are an application of the speed, distance and time concept, and they are asked in SSC CGL and CHSL. This guide on races and circular tracks for SSC explains the terms used in races, the speed ratio method and the formulas for meeting times on a circular track, with solved examples, tricks and practice questions.

Terms Used in Races

Basic Ideas

• Distance covered in the same time is proportional to speed.
• If A beats B by x metres in a race of D metres, then when A covers D, B covers D − x. The ratio of speeds A : B = D : (D − x).
• If A beats B by t seconds in a race, then B takes t seconds more than A for the same distance.
• To make a dead heat, the faster runner gives the slower runner a head start equal to the difference in distance covered.

Circular Track Formulas

Let the length of the track be L and the speeds of two runners be a and b (with a > b).

For three runners, the time to meet again at the start is the LCM of the three lap times. To meet at any point (same direction), take the LCM of the individual meeting times.

Solved Examples

Example 1. A runs 100 m in 20 seconds and B runs 100 m in 25 seconds. By how many metres and by how many seconds does A beat B?
In 20 s B covers 100 × 20/25 = 80 m. A beats B by 20 m. By time: 25 − 20 = 5 seconds.

Example 2. In a 150 m race A beats B by 30 m, and in a 100 m race B beats C by 20 m. By how much does A beat C in 150 m?
A : B = 150 : 120 = 5 : 4. B : C = 100 : 80 = 5 : 4. A : C = 25 : 16. When A runs 150, C runs 150 × 16/25 = 96. A beats C by 54 m.

Example 3. The track is 400 m long and two runners run at 5 m/s and 3 m/s in the same direction from the same point. When do they meet?
400/(5 − 3) = 200 seconds.

Example 4. In the same case, if they run in opposite directions, when will they meet?
400/(5 + 3) = 50 seconds.

Example 5. Two runners run on a 600 m track at 4 m/s and 6 m/s. When will they meet at the starting point?
Lap times are 150 s and 100 s. LCM = 300 seconds.

Example 6. A runs twice as fast as B. In a 1,200 m race, what head start must A give B for a dead heat?
When A runs 1,200 m, B runs 600 m. So A must give a start of 600 m.

Example 7. In a 1 km race, A beats B by 100 m. If B takes 5 minutes to finish 900 m, find the time A takes for 1 km.
When A runs 1,000 m, B runs 900 m, taking 5 minutes. So A takes 5 minutes for 1 km.

Tricks

• Convert every statement into a speed ratio first.
• To compare three runners, chain the ratios A : B and B : C into A : C.
• On a circular track, use relative speed: subtract for the same direction and add for opposite.
• Use LCM for meeting again at the starting point.
• Convert km/h to m/s by multiplying by 5/18.

Quick Tips

• Convert beats-by statements into speed ratios.
• Use relative speed on circular tracks.
• Take the LCM for the meeting at the starting point.
• Convert km/h into m/s with 5/18.

Quick Revision Points

• If A beats B by x in D, speed ratio A : B = D : (D − x).
• Same direction: time = L/(a − b).
• Opposite direction: time = L/(a + b).
• Meeting at the start: LCM of lap times.
• Dead heat means both reach the end at the same time.

Practice Questions With Answers

1. A runs 200 m in 40 s and B in 50 s. By how many metres does A beat B?
   Answer: 40 m
2. In a 100 m race A beats B by 10 m. Find the ratio of their speeds.
   Answer: 10 : 9
3. On a 300 m track, two runners at 4 m/s and 6 m/s run in the same direction. When do they meet?
   Answer: 150 s
4. In the same case, in opposite directions, when do they meet?
   Answer: 30 s
5. A can run 1 km in 4 min 50 s and B in 5 min. By how much time does A beat B?
   Answer: 10 seconds
6. A and B run on a 360 m track, with lap times of 60 s and 90 s. When do they meet at the start?
   Answer: 180 s
7. A is twice as fast as B. In a 500 m race, what head start should be given to B for a dead heat?
   Answer: 250 m
8. If A : B = 5 : 4, how far will B run when A runs 250 m?
   Answer: 200 m
9. Two persons run at 8 km/h and 12 km/h on a 2 km circular track in opposite directions. When do they meet?
   Answer: 6 minutes
10. In a 200 m race A beats B by 20 m. By how much does A beat B in a 500 m race?
    Answer: 50 m

How to Practise This Chapter

Practise 10 questions every day and write the steps on paper. Then attempt timed sets from our practice papers and the previous year papers page. See all topics on the Quantitative Aptitude notes page, read the previous topic Set Theory and Venn Diagram and the next topic Remainders and Unit Digit.

Frequently Asked Questions

What does A beats B by 20 metres mean?

It means that when A reaches the finish line, B is 20 m behind.

What is a dead heat?

A race in which both runners finish at the same time.

What is a head start?

The distance by which a runner starts ahead of the other.

How do I find the speed ratio from the beats-by data?

If A beats B by x in a race of D, A : B = D : (D − x).

What is the time to meet on a circular track in the same direction?

L / (a − b), where L is the track length.

What is the time to meet on a circular track in opposite directions?

L / (a + b).

How do I find when runners meet again at the start?

Take the LCM of the times that each takes to complete one lap.

How do I convert km/h to m/s?

Multiply by 5/18.

Are race questions common in SSC?

Yes. They appear in SSC CGL and CHSL as speed-distance-time applications.

What is the difference between beats by distance and by time?

Beats by distance is the gap in metres when the winner finishes; beats by time is the gap in seconds between finishing times.

Is relative speed used in races?

Yes. It is used mainly on circular tracks.

What are common mistakes in race problems?

Mixing up the distance and the time versions of beats-by, and adding speeds instead of subtracting for the same direction.