This is an offshoot of time, speed and distance based questions wherein, you will have races being run by 2 or more people and there would be certain scenarios given which would allow you to find the distance or time by which one person wins races against another.

Points to remember in these questions:

(a) The distance is constant in any race. If it is a 100 m race, it is expected that one of the sprinters finishes the race while the other is a bit behind in terms of time/distance

(b) A head start/handicap is an advantage that is given to a participant in terms of time/distance. This is the extra distance that the other racer will have to cover in order to win the race

(c) As was the case with escalators, the racers are expected to run at constant speeds unless it’s explicitly mentioned. The distance between the faster sprinter and the slower sprinter increases gradually from start to finish

(d) Dead heat, deadlock, standoff, draw, tie essentially mean that the racers have reached the end of the race together

(e) If A beats B by 5 seconds, it means that B would have covered the distance if B had continued for 5 more seconds

The basic concepts remain the same:

Speed = Distance/Time

If distance is constant, speed is inversely proportional to time and if time is constant, distance will be directly proportional to the speeds. As the topic involves races, speeds will always be different except in special cases.

Let’s see a few examples to understand the concept in a better manner.

In a 100 m race, A gives B a start of 20 m but still wins the race by 5 seconds. But if A gives a head start of 40 m to B, the race ends in a dead heat. How long does A take to run 200m?

(a) 10 sec (b) 20 sec (c) 30 sec (d) 40 sec

A gives B a start of 20 m which effectively means that while B has to cover 80 m to reach the finishing point, A has to travel 100 m. But A still wins the race by 5 seconds. So, as mentioned above, B would have reached the end point if he had traveled for another 5 seconds. So, if B covers 80 m in x seconds, he would cover (400/x) m in 5 seconds and so, that is the margin of defeat. So, the time in which A covers 100 m, B covers (80-400/x) m. As time is the same, the ratio of their speeds are

In the second case, A travels for 100 m in the same time that B travels for 60 m. So, we get

x = 20 seconds. So, we get S_{B} = 4 m/s and so, S_{A} = 200/30 m/s. So, A will take 30 seconds to run 200 m.

In a kilometer race, Ram beats Shyam by one minute and Shyam beats Tarun by 30 seconds. If Ram beats Tarun by 250m in the same race. Find the time taken by Ram to run the race (in seconds).

Let Ram cover 1000 km in x minutes. Shyam will cover 1000 km in (x+1) minutes. Also, Tarun will cover 1000 km in (x+1.5) minutes.

So, we get

Equating the two, we get x = 4.5 minutes = 270 seconds which is the time taken by Ram to cover the race.

A few examples from previous year CAT papers. Try them out on your own:

Karan and Arjun run a 100-metre race, where Karan beats Arjun by 10 metres. To do a favour to Arjun, Karan starts 10 metres behind the starting line in a second 100-metre race. They both run at their earlier speeds. Which of the following is true in connection with the second race? (CAT 2004)

(a) Karan and Arjun reach the finishing line simultaneously.

(b) Arjun beats Karan by 1 metre.

(c) Arjun beats Karan by 11 metres.

(d) Karan beats Arjun by 1 metre.

Ram and Shyam run a race between points A and B, 5 km apart. Ram starts at 9 a.m. from A at a speed of 5 km/hr, reaches B, and returns to A at the same speed. Shyam starts at 9:45 a.m. from A at a speed of 10 km/hr, reaches B and comes back to A at the same speed. (CAT 2005)

At what time do Ram and Shyam first meet each other?

(a) 10 a.m. (b) 10:10 a.m. (c) 10:20 a.m. (d) 10:30 a.m.

At what time does Shyam overtake Ram?

(a) 10:20 a.m. (b) 10:30 a.m. (c) 10:40 a.m. (d) 10:50 a.m.

A question from Data Sufficiency

If A and B run a race, then A wins by 60 seconds. If B and C run the same race, then B wins by 30 seconds. Assuming that C maintains a uniform speed what is the time taken by C to finish the race? (CAT 2003 Leaked)

- A and C run the same race and A wins by 375 metres.
- The length of the race is 1 km.

Three runners A, B and C run a race, with runner A finishing 12 m ahead of runner B and 18 m ahead of runner C, while runner B finishes 8 m ahead of runner C. Each runner travels the entire distance at a constant speed. What was the length of the race? (CAT 2001)

(a) 36 m (b) 48 m (c) 60 m (d) 72 m

Do let us know if you have any novel approaches while solving these questions.

We are starting with the CAT 2015 booster workshops from the coming weekend. Do register using the link in the article. Also, you can read the entire 75 days to CAT 2015 series to cover the topics that you might have missed.

All the best!