Races - Solutions

Have given below solutions to the races questions that can be found here.

1. Two friends A and B simultaneously start running around a circular track . They run in the same direction. A travels at 6m/s and B runs at b m/s. If they cross each other at exactly two points on the circular track and b is a natural number less than 30, how many values can b take?

Let track length be equal to T. Time taken to meet for the first time = T / relative speed
= T/(6-b) or T/(b-6)

Time taken for a lap for A = T/6
Time taken for a lap for A = T/b

So, time taken to meet for the first time at the starting point = LCM (T/6, T/b) = T / HCF (6,b)

Number of meeting points on the track = Time taken to meet at starting point/Time taken for first meeting = Relative speed / HCF (6,b). For a more detailed discussion on this have a look at the last few slides in this presentation.

So, in essence we have to find values for b such that 6-b/ HCF(6,b) = 2 or b-6/ HCF(6,b) = 2

b = 2, 10, 18 satisfy this equation. So, there are three different values that b can take.

2. Three friends A, B and C decide to run around a circular track. They start at the same time and run in the same direction. A is the quickest and when A finishes a lap, it is seen that C is as much behind B as B is behind A. When A completes 3 laps, C is the exact same position on the circular track as B was when A finished 1 lap. Find the ratio of the speeds of A,B and C?

Let track length be equal to T. When a completes a lap, let us assume B has run a distance of (t-d). At this time, C should have run a distance of (t-2d)

After three laps C would have traveled a distance of 3 * (t-2d) = 3t - 6d.

After 3 laps C is in the same position as B was at the end one lap. So, the position after 3t-6d should be the same as t-d. Or, C should be at a distance of d from the end of the lap. C will have completed less than 3 laps (as he is slower than A), so he could have traveled a distance of either t-d or 2t-d.

=> 3t-6d = t-d => 2t = 5d => d = 0.4t => The distances covered by A,B and C when A completes a lap will be t, 0.6t and 0.2t respectively. Or, the ratio of their speeds is 5:3:1

In the second scenario, 3t-6d = 2t-d => t = 5d=> d = 0.2t => The distances covered by A,B and C when A completes a lap will be t, 0.8t and 0.6t respectively. Or, the ratio of their speeds is 5:4:3

The ratio of the speeds of A, B and C is either 5:3:1 or 5:4:3

Races - Two Questions

Have given below two questions on races. For a recap of the basics on Races, see questions here and the solutions here

1. Two friends A and B simultaneously start running around a circular track . They run in the same direction. A travels at 6m/s and B runs at b m/s. If they cross each other at exactly two points on the circular track and b is a natural number less than 30, how many values can b take?

2. Three friends A, B and C decide to run around a circular track. They start at the same time and run in the same direction. A is the quickest and when A finishes a lap, it is seen that C is as much behind B as B is behind A. When A completes 3 laps, C is the exact same position on the circular track as B was when A finished 1 lap. Find the ratio of the speeds of A,B and C?

Races Basic Questions Solutions

Have given below the solutions to the basic questions on Races.

Races

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Races - Basic Questions

Have given below a few basic questions in Races

1. In a 100m race, A can give B a start of 12 meters or 3 seconds. What is the speed of B in m/sec and how long will A take to run 100m?

2. A can give B a start of 30 in a km race and A can give C a start of 50m in a km race. How much start will B be able to give C in a km race?

3. A runs a 400m race at the speed of 8 m/sec. He gives B a start of 40m and still beats him by 10 seconds. What is B's speed?

4. If A gives B a start of 20m in a race, A beats him by 10 seconds. On the contrary, if A gives B a start of 40m, they finish the race in a dead heat. If the length of the race is 400m, how long will A take to run the race?

5. A and B start around a circular track of length 600m at speeds of 6m/s and 9 m/sec from the same point, simultaneously and in the same direction. When will they meet for the first time? How far from the starting point will they meet, when they meet for the first time?

6. A and B run around a circular track at speeds of 8m/sec and 5m/sec starting from the same point, simultaneously and in the same direction. How many points with respect to the starting point on the circular track will they meet, if the run at the above mentioned constant speeds endlessly?