# Race Questions: Simple Time and Distance Diagrams Demystify Head Starts and Dead Heats YouTube Lecture Handouts

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## Races

These are a class of quantitative aptitude problems so this is the basic formula this is the basic concept of speed is equal to distance into time which is used by all the problems in all the race problems.

Speed

So, this is how this class fits in our overall strategy we will be taking several classes which cover the race and speed problems and today we would we discussing problems which involve Head Start and Linear Track which means the track which goes straight later on we will be covering Circular Tracks and will understand how clocks fit into this overall scheme of things.

There are two concepts involving all race problems and you should understand these two concepts:

One is the concept of a dead heat; a dead heat means that all the participants in the race reach the finish line at the same time that is what is dead heat and then the second concept is Head Start that means that a participant is given more advantage over other participants. So that, they are two participant A and B. A is allowed to start first and that means A is given the heat start so, this is an example:

## Head Start in Time ⇾ Distance Covered is Same

Here, these two participants are stuck on the start line whereas these two participants started early so they were given advantage of Head Start. Now the Head Start can be it՚s so clearly distinguish these two things that I′m telling you now the head start can be it terms of time or it can be in terms of distance. They are two distance things this particular head start shown in this diagram this figure is in terms of time which means these two participants started early and these two participants will start late they would cover the same length or they would cover the same distance another kind of head start we can imagine is distance which means that two participants start from different locations on the track.

Let us say A start from here and B starts further down the track and hence B is being given the head start. B has been given advantage over A we can say it has A gives a Head Start to B and this head start is in terms of distance when they when the starting pistol is fired these two participants would start running at the same time but they would be running from different positions and they would end up at the finish line from both of them. So if the head start is in distance time remains same that is they start at the same time if it՚s in terms of time then they cover the same distance.

## Winning by Distance ⇾ Time Remains Same (No Head Start)

Similarly, at the finish line they can be two different things either we can say that a participant won by certain margin of distance. Let՚s say this is much distance may be let՚s say 10 meters or I can keep time I can say that all this participant cross the finish line let us say at 10 o′clock and this participant took two more seconds so I can either give the participant who won either I can specify that in terms of margin of time and I can specify in terms of margin of distance when I will specify in terms of margin of distance let՚s say 10 meter it means that these two participants were at these positions at the same instance of time they were just at different locations. When I am giving the margin or the winning margin in terms of time that means these two participants cause the same distance that these two participants move the same distance but at different times.

## Time & Distance Diagrams- Most Important

Now on this problem we will apply our knowledge of this distinction between time and distance we՚ll apply that to draw two kinds of diagram:

- Time diagram and
- Distance diagrams.

Just using these two diagrams we will be able to solve each and every problem they are all on races.

## Speed as Rate

- Speed = Distance/Time
- Mind the units
- Prerequisites:
- Manipulate speed, distance and time: Watch the class on speed questions.
- Linear equations in 2 variables: Watch the class on linear equations.

## One Slide for All Problems

## (1) in a 2000 M Race between a and B. A Gives B a Start of a Minute but Still Beats Him by 200 M. When He Increases the Head Start to 80 Seconds, the Race Ends in Dead Heat. Find the Speed of A

In this, it is a 2000 meter raise so, let՚s draw distance diagram and I would be drawing a time diagram

So, the entire track is 2000 meter it gives the A head start of a minute but it still beats in by 200 meter so we ignore all the times we drawing the distance diagram we will only concern ourselves with the distances. So what this says that when A reach is here A and B they start from the same point. There is no head start in terms of distance and the heat start in terms of time. So they start from the same point but A covers this to 2000 meters and which is A′ and when he reaches A′. B is beats impact 200 meters so B is still 200 meters behind so this is 200 meters so obviously this is 1800 meters. So, now this is the distance diagram:

Let՚s see how this things looks in terms of time so what is the first thing that happens A gives B a start of a minute so who goes first, B goes first B is given advantage here and after some time which is 60 seconds A starts and then they both are running running and finally they reach certain configuration after A has finished the race and B is 200 meters away from the finish line so this is the time at which A finishes the race, so this is time diagram.

So what is aA′ what is the time it takes a to cover this entire distance from A to A′ what՚s the time let us assume that the speed of A is small a and speed of B is small b then I can write this A′ this time is 2000 meters that is the distance that A is covered with the speed of a and what the time B took to run 1800 meters. He took 18000 divided by b seconds so, this is the time diagram. This is the time B took, this is the time A took and here is the advantage the B was given so now with this time diagram I can write a equation

What does the equation say 1800 divided by B this much – 2000 divided by a is equal to 60 seconds.

So, equation

I can write this equation it but they are two distinct concept distance and time. Let՚s see what happens in the second case we know that the waves ends in a dead heat which means they are both covering the same amount of distance is just that one of them started first so again be B started first so I put B and then after A while 80 seconds they started and so I will put A and then they finished at the same time so this is 80 seconds now again how much distance a is covering 2000 with the speed of a so this is the time a will take to cover this 2000 meters and of course B is slower B would cover this 2000 meter in this and he՚ll negate the advantage of 80 seconds to meet A at the finish line so this is seconds time diagram and from here I can derive other equation

2000 divided by b – 2000 divided by a is equal to 80 second.

So, equation

Now I want to tell you what happens if we subtract these two equations then a vanishes and I am left with this term with a vanishes only b so I can simply solve for b here and b would come out to be 10 m/sec and finally I can solve for a also which comes out to be 16.67 m/sec so this is final result two equations to know I can solve it.

If I go by these first principles I draw all my diagrams I can easily solve this question in under a minute easily but if you notice one thing here there is a tick here and we can use a kind of shortcut here.

Let՚s look at this in little more detail there is a head start of a minute here 60 seconds and this was increased to a head start of 80 seconds in this additional 20 seconds enable be to cover this 200 meters .

Therefore speed of B is equal to 200 divided by 20 m/sec

So, I can directly arrive at the speed of b. but this shortcut does not work all the time whereas our first principle method of a diagram or our equations they work all the time so I should use the diagrams and equations

## (2) a and B Run a Race of 2000 M. First, a Gives B a Head Start of 200 M and Beats Him by 30 S. 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?

Let՚s start 2000 meters so I am going to draw distance diagram. A gives B a head start of 200 meter this time the head start is in terms of distance so A starts from here B starts 200 meters further down the track so B starts from here and then finally they arrive here and beats 30 seconds that A and B both reach the finish line A goes first after 30 seconds B crosses the finish line but they both cross the finish line so they have covered the same distance that they have reached the same point in the end so, this B′ is 1800 meter.

Now time diagram, A and B they start from at the same time they starting from different locations but from the same starting pistol when the short is fired from the pistol they start at the same time so they are starting from the time line and what happens in the end first A reaches and I am standing there at the finish line what will I say after 30 seconds B reaches so this is how time would look can I put more numbers here this distance what is the time it tool to reach the finish line if the speed of A is small a then he took and what is the time B took so, now with this I can write my first equation

So, equation

Let՚s see what happens in the second case.

In this case the head started in terms of time 3 minutes which is 180 seconds. So let՚s draw the distance diagram first beaten by 1000 meters I am not worrying about this yet because this is distance diagram.

A covers 1000 meters. B again this time the Head Start is in terms of time so they start from the same location they start from the same line but at different times so B covers the entire distance this is 2000 meters and these is 1000 meters.

In time diagram who starts first B is been given the advantage here so B would start first 180 seconds later A would start and when B crosses the finish line this is the time so what is the amount of time B took to cross the finish line I know he is covering time 2000 meter at the speed of B. what is the time between A is starting and B crossing the finish line l know that A is only covered 1000 meters at the speed of small a so this is my second equation.

Let be write that down,

Again I have two equations and two variables I can solve them and finally what will I get B is equal to that is B the speed b would come out to be 20 by 3 speed a would come out to be 25 by 3 and the time this is asking the time how many minutes they raised separately the times would respective times would come out to be 5 minutes and 4 minutes so you can solve these equations yourself and get here with this diagrams we were able to arrive at these equations very quickly.

Moving on this is relatively simple question right I mean we have done two very difficult problems already so we are all set actually with these basic concepts we can solve these other questions that I am going to go through very quickly.

## (3) a Runs 50 % Faster Than B. If a Gives a Start of 7 M Race Ends in Dead Heat. What is the Length of the Race?

A runs 50 % faster than B so, A is running faster than B

So if B runs at the rate of X: ,

A is running at the rate of 1.5 X:

It is what this thing is telling me A gives a head start of how much 7 meters. In the days anything dead so head start is in terms of distance. Head that it means that they reach the same end point in the same instant of time

So, let be draw that distance diagram.

So, this is A this is B. B is here 7 meters and this is the head start and then this is the finish line they reach the common to them there is this finish line at the same time and if this entire length of track let us assume it is then this becomes this is the distance B has covered.

Now what happens what about my time diagram mu time diagram would look very simple when the short was fired the both is started and they both finish again at the same time they might have started from different positions but they started at the same time and they finish at the same time so my time should be equal what does that tell me it tells me that this distance which b covered divided by X is same as the distance cover the time that a took divided by 1.5X now X and X cancel off I can solve if for and will come out to be 21 meters

Now this is not very complicated now you can easily use this method and do this question in under a minute. But if you really want to read more into this question if you really want to do careful analysis of this question there is a shortcut kind of thing here also and whats the shortcut.

Lets see if we assume that the length of track is 1.5 meter it is the shortcut can come from here actually we can derive the shortcut from just by looking at this diagram.

So A could cover this 1.5 meter and B would only cover 1 meter plus B is running at X speed so B would cover only 1 meter so how much Head Start should we give to B so then the race sensing dead heat we need to give 0.5 meter to B and if we do that then they both reach the end point at the same time so now with this what do I know if I give a head start of 0.5 meter my track length could be 1.5 m this is the head start I gave 0.5 meters similarly if I give a head start of 1 my track length could be and if I gave a head start of 7 my track length could be

So, this is the slightly shorter method of doing this same problem but if don՚t waste time arriving here with enough practice maybe naturally you will come here and that is okay but until and unless you come here you should use this first principle method which is applicable to any and all problems you should master this.

## (4) as Speed is 20 ⁄ 19 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? What if the a Wants to Beat B by 5 % ? What if the B Wants to Beat a by 10 % ?

Again we are given two people A and B. Speed of A is 20 by 19 times so let us say that the speed of A is then the speed of B becomes

So that the race ends in a dead heat

So let us try to draw a diagram here, if the length of the track is and I am required to give A head start to B. head start into a is the fraction of the length of the track which I am giving to B so the length of the head start becomes this is the head start I need to give to B so that they reach the end point the finish line at this same and this is what the question is asking and this becomes .

The time diagram they start at the same time I am not giving any head start in terms of time and then they finish also at the same time so, the time same.

I can say is the distance covered by B divided by the speed of B which is should be equal to a full length which is covered by a full length divided by the speed . and will be cancle and also cancel and finally a would come out to be 1 by 20.

This same time now the second part of the question is asking me a slightly different thing what it is saying that I don՚t want them to reach at the same location what I want is for A to win so when A reaches the finish line B should be away from the finish line this is B′ so again this is the same instant of time so I can equate the time this is the distance covered by B. now B is covering smaller distance at the last time because I want A to win and then again on this side everything remains the same and finally here my A comes out to be 0. what does that mean it means that there should not be any head start let them play fair is the thing and finally the third thing is I am reversing it now I want B to win so now here B reaches the finish line A should be so A should be how far . This is entire length this is so now how much distance B has covered B has covered distance from here with this distance this entire distance B has covered and how much distance A is covered A is covered distance from here which is and again cancel and cancel you can do the match finally you՚re a would come out to be

So, here I wanted this question to be in three parts to illustrate how we are fuzzing the distances how we are simply changing the distances the time can always be equated because this was a dead heat they are starting from the same time they are ending at the same time.

There are the three final questions are slightly trick questions in the sense that you can write equations but may be they are not going to help you as much you have to realize a few things.

## (5) a and B Run a 100 M Race, Where a Beats B by 10 M. To Do a Favour to B, a Starts 10 M Behind the Starting Line in a Second 100 M Race. They Both Run at Their Earlier Speeds. What is the Outcome of Second Race?

100 meter race, where A beats B by 10 meters so when A covers 100 meters B covers 90 meters

Now in the scenario given in the question what is happening A is starting further back he is moved back from the start line by how much 10 meters behind and B is starting from the same location so now when B has covered 90 meters A would have covered how many meters 100 meters. But we know that B when A is 100 meter and B is 90 meters they happen at the same moment in time when A covers 100 meters they can cover 90 meters so if they start together they would reach this place this common place 90 meter away from the start line at the same time so they reached here at the same time now this becomes our new assume that this becomes our new it starts A new fair is start line and now let՚s see what happened this start line is 10 meters away from the final finish line we՚ll draw it using two bars in 10 meters away so A can cover 10 meters.

When A covers these 10 meters how much B would have gone only 9 meters so B would have reached somewhere here B double prime and A would have reached the final finish line and this difference would be how much

So A would again beat B by 1 meter that is the answer.

Moving on next question,

## (6) in a Kilometer Race, a Beats B by One Minute and B Beats C by 30 Seconds. If a Beats C by 250 M in the Same Race. Find the Time Taken by a to Run the Race

Here given three things A, B and C. three participants are there.

First A and B runs and A beats B by 1 minute then B beats C by 30 seconds. So, if A and C are running then by how much A would beat C by

So, let՚s draw time diagram A and C start from the same time and then first A would reach after a while C would reach how much after 90 seconds that is the first diagram.

Now we also know that A beats C by 250 m in the same track so let us see the distance diagram.

A reaches is the finish line C is where C is 250 meters behind and how long does it take C to cover these 250 meters 90 seconds so as soon as you have these two diagrams you can quickly say that this is the instant when A crosses the finish line C is 250 meters behind now I am waiting at the finish line C arrives 90 seconds later which means that C to 90 is against to cover this 250 meters so now I know the speed of C.

The speed of C

So, now I go back to my first principle write the equations in both the cases or in one of the cases and then finally find the speed of A and the answer I can do that but there is a little bit of insight here just think about it so it took C this is 750 meters we know that because it՚s a kilometer is this entire thing is how much 1 kilometer which means 1000 meters we know that when C had covered 750 meters A have already reached the finish line that means that if we find the time it took C to cover 750 meters we would found the answer.

C covers 250 meters in 90 seconds we know that in 750 meters C would cover in how many seconds:

But if you want to solve it using equations you can do that too it՚s not very complicated I leave it up to you.

## (7) 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. What Was the Length of the Race?

Now this is the final question that we will be discussing today and again it might seem very complicated but if we draw a distance diagram properly for this question we՚ll get the inside required solve this problem.

There are 3 people now again A, B and C they start from the same location there is no there is no hand start business. A is the winner, A finishes and at the moment A crosses the finish line where is B 12 meters ahead of being so B is 12 meters behind, B is write here this is 12 meters this is B′ and A′

18 meters ahead of runner C and C is write here, this is 18 meters.

Now let us only draw the diagram for what happened when B crosses the finish line what are we told B and C we are only comparing them what happens when B crosses the finish line, C is 8 meters behind so C is somewhere let us say it call double prime so, B has moves these 12 meters and reached the finish line and during the same time and how much C has moved this distance is 8 meters. This distance we know is 18 meters so how much is this distance 10 meters so in the same time write A, B and C have respectively moved 12 meter and 10 meters.

So what՚s the ratio of the speed of B and C

Now if the length of this entire track let us say is again know that the ratio of their speeds do I know to find out initially A is the finish line C has covered how much distance and B has covered how much distance .

So what is the ratio of speed from here it comes out to be

In the next class, we will talk about races on circular tracks which get more interesting because on circular in circular tracks there could be Labs. Labs means that the participants can go around and many times and they would start crossing each other so complicated problems could be formed and at the same time we would discussing military speed.

## Next Circular Tracks

### Suppose a and B Are Running a 3 Km Race in a Circular Track of Length 300M. Speeds of a and B Are in the Ratio 4: 3. How Often and Where Would the Winner Pass the Other?

✍ Mayank