This week we will look at the question on races that we gave you last week.
Question 3: A and B run a race of 2000 m. First, A gives B a head start of 200 m and beats him by 30 seconds. Next, A gives B a head start of 3 mins and is beaten by 1000 m. Find the time in minutes in which A and B can run the race separately?
(A) 8, 10
(B) 4, 5
(C) 5, 9
(D) 6, 9
(E) 7, 10
Solution: Now this question is a little tougher than the previous ones we saw last week.
There are two scenarios given:
1 – A gives B a head start of 200 m and beats him by 30 seconds.
2 – A gives B a head start of 3 mins and is beaten by 1000m.
Let’s study both of them and see what we can derive from them.
Scenario 1: A gives B a start of 200m and beats him by 30 seconds.
As we suggested before, we will start by making a diagram.
A runs from the Start line till the finish line i.e. a total distance of 2000 m.
A gives B a head start of 200 m so B starts, not from the starting point, but from 200 m ahead. A still beats him by 30 sec which means that A completes the race while B takes another 30 sec to complete it. So obviously A is much faster than B.
In this race, A covers 2000m. In the same time, B covers the distance shown by the red line. Since B needs another 30 sec ( i.e. 1/2 min) to cover the distance, he has not covered the green line distance. The green line distance is given by (1/2)*s where s is the speed of B in meters per minute. The distance B has actually covered in the same time as A is the distance shown by the red line. This distance will be (1/2)*s less than 1800 i.e. it will be [1800 – (1/2)*s].
Scenario 2: A gives B a head start of 3mins and is beaten by 1000m.
A gives B a head start of 3 mins means B starts running first while A sits at the starting point. After 3 mins, B covers the distance shown by the red line which we do not know yet. Now, A starts running too. B beats A by 1000 m which means that B reaches the end point while A is still 1000 m away from the end i.e. at the mid point of the 2000 m track.
In this race, A covers a distance of 1000 m only. In that time, B covers the distance shown by the green line. The distance shown by the red line was covered by B in his first 3 mins i.e. this distance is 3*s. This distance shown by the green line is given by (2000 – 3s).
Now you see that in the first race, A covers 2000m while in the second race, he covers only 1000m. So in the second race, he must have run for only half the time. Therefore, in half the time, B would also have covered half the previous distance.
Distance covered by B in first race = 2*Distance covered by B in second race
1800 – (1/2)*s = 2*(2000 – 3s) (where s is the speed of B in meters/min)
s = 400 meters/min
Time taken by B to run a 2000 m race = Distance/Speed = 2000/400 = 5 min
Only one option has time taken by B as 5 mins and that must be the answer.
If required, you can easily calculate the time required by A too.
Distance covered by B in scenario 1 = 1800 – (1/2)*s = 1600 m
In the same time, A covers 2000 m which is a ratio of A:B = 5:4. Hence time taken by A:B will be 4:5.
Karishma, a Computer Engineer with a keen interest in alternative Mathematical approaches, has mentored students in the continents of Asia, Europe and North America. She teaches the GMAT for Veritas Prep and regularly participates in content development projects such as this blog!