Let’s discuss races today. It is a very simple concept but questions on it tend to be tricky. But if you understand how to handle them, most questions can be done easily.

A few points to remember in races:

1. Make a diagram. Draw a straight line to show the track and assume all racers are at start at 12:00. Then according to headstart, place the participants.

2. There are two types of head starts: Time and distance

Say there is a 1000 feet race between A and B which starts at 12:00.

Time – A gives B a headstart of 1 min means B starts running at 12:00 and A waits at the start point. Then A starts running from the start point at 12:01.

Distance – A gives B a headstart of 10 feet means A starts from the start point but B starts from the point 10 feet ahead (and hence runs only 990 feet to complete the race)

3. A dead heat is a race in which both the participants finish exactly at the same time. Most races in race questions end in a dead heat!

4. There are two ways in which a participant can beat another: Time and distance

Say A beats B in the 1000 feet race in which both start from the start point at 12:00.

Distance – If A beats B by 20 feet,  it means A finishes the race (full 1000 feet) and at that time, B is 20 feet away from the finish line.

Time – If A beats B by 2 mins, it means that if A finished at 12:10, B is still 2 mins away from the finish line i.e. at his/her speed, B takes 2 mins to reach the finish line.

That is all! Now let’s look at some questions:

Question 1: A’s speed is 20/17 times that of B. If A and B run a race, what part of the length of the race should A give B as a head start, so that the race ends in a dead heat?

(A)   1/17

(B)   3/17

(C)   1/10

(D)   3/20

(E)    3/10

Solution: We have the ratio of A’s speed and B’s speed. This means, we know how much distance A covers compared with B in the same time.

This is what the beginning of the race will look like:

(Start) A_________B______________________________

If A covers 20 meters, B covers 17 meters in that time. So if the race is 20 meters long, when A reaches the finish line, B would be 3 meters behind him. If we want the race to end in a dead heat, we want B to be at the finish line too at the same time. This means B should get a head start of 3 meters so that he doesn’t need to cover that. In that case, the time required by A (to cover 20 meters) would be the same as the time required by B (to cover 17 meters) to reach the finish line.

So B should get a head start of 3/20th of the race.

This question was relatively very straight forward and we gave it only to help you apply the concepts discussed above. Let’s make it slightly tricky now.

Question 2: A’s speed is 20/17 times that of B. If A and B run a race, what part of the length of the race should A give B as a head start, so that B beats A by 20% of the length of the race?

(A)   15%

(B)   20%

(C)   28%

(D)   32%

(E)    35%

SolutionAgain, we have ratio of A’s speed and B’s speed given as 20:17. If A covers 20 meters, B covers 17 meters in that time. This time, let’s assume that the length of the race is 25 meters.

At the beginning, this is what the 25 meter track will look like with a head start to B:

(Start ) A_________B_______________________________

Since A will give B a head start so A must start from the start line while B will start from ahead.

Since A should cover only 80% of the length of the race, when B reaches the finish line, A should still have 20% of the track leftover.

20% of the track will be (20/100)*25 = 5 meters. So A should be at 20 meters when B is at the finish line.

So this is what the finish of the race will look like:

____________20_________________A____5_____B (Finish)

A will cover a total of 20 meters when B should be at the finish line. In this time, B will cover only 17 meters. But the total track is of 25 meters. So the rest of the 25-17 = 8 meters, B should get as a head start.
Head start will be 8/25 *100 = 32% of the race.

If you found it tricky, we would suggest you to practice some more races questions. It is usually easy to “figure out” the answer logically and the calculations required are minimum.

Now try this official question. We will solve it for you next week.

Question 3: A and B run a race of 2000 m. First, A gives B a start of 200m and beats him by 30 seconds. Next, A gives B a start of 3mins and is beaten by 1000m. Find the time in minutes in which A and B can run the race separately.

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!

### 3 Responses

1. Sriram says:

A slightly different approach to problem 2. We could figure out the time taken by A to cover 80% of the distance and then find out the corresponding distance covered by B in the same time. This difference from the total distance would give us the head start B needs.

2. manu says:

Found this quite difficult. Solved normal math and eventually reached there in 5 min.

Key is when time is constant , velocity is proportional to distance .
There are two cases and in each case ratio of speeds is given.

From case1 : A beats B by 30 secs.–>
In the same time, B covered 1800 minus velocity of B times 30 sec, whereas A covered 2000m fully.

[1800-(Vel.B*30)]/2000 = Vel.B/Vel.A

From case second one:- In 180 secs . Vel.B*30 is distance covered by B.
Hence , 2000-(vel.B*180)/ 1000 = vel.B/Vel.A

Equating both got Vel.B 20/3. And vel.A =25/3 . Thus times are 300s and 240s
respectively by B and A

• Karishma says:

The point is that the more variables you take, the more time you will consume.Try to refer to the diagram to understand what is going on. Imagine what the race looks like at the beginning, at important junctures and at the end. Logic beats Math in GMAT.