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Arithmetic - Tough Question on Races

Today let's look at a tough question on races. We discussed how to guess intelligently on it in this post. Today we will look at the complete solution. 

 

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

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.

 

Answer (B)

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