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Getting a sense of directions

Introduction

Somewhere in almost every competitive examination will come a question that will ask you to track movements and direction. You will be confronted by a character who has been given a licence to wander (the licence to kill had already been handed out to Mister. Bond!). He will go south, north, east and west, and turn left and right for no reason whatsoever. And at the end of it all, you will be asked to find his exact location and how far he has travelled.

Sounds familiar? Welcome to the world of direction-based exercises. Love them or hate them, you cannot afford to neglect them, as they are an integral part of most competitive examinations today. And believe us,they are not as difficult as they seem to be.

All one needs is some common sense allied at times with some very basic Mathematics

Contents

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Nature of exercises


As their very name indicates, direction-based questions revolve around the direction taken by a person or an object. We will term this person or object the 'subject' as he / she or it is at the centre of the problem.

Most direction-based exercises consist of a description of a route taken by the subject. As this is after all an exercise to test your ability, this route often involves three to four turns in different directions after travelling a certain distance. In most cases, you will be asked to find the following

  • How far is the subject from the place where it started its journey
  • In which direction is the subject from the place where it started its journey
  • In which direction did the subject start its journey
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What you need to solve them


As we said in the introduction, all that is needed to solve these questions is common sense along with some basic mathematics. And the mathematics is not always necessary!

More specifically, you need the following to tackle direction-based questions:

1. A sense of direction: You need to have a clear idea of north, south, east and west. Let us face it the subject of the question does not seem to have any sense of direction. If you do not have it either, you stand little chance of solving the question.

2. An ability to visualise: One can always trace the movements of the subject by drawing a diagram showing its approximate movement. However, this can be time consuming and even inconvenient for some candidates. A solution is to develop an ability to visualise the movement of the subject. One is still drawing a diagram, only this time it is in one's own mind. It does take a lot of practice to get used to this but believe us it is well worth the effort.

3. Some basic mathematics: Here comes the dreaded mathematical angle! Well, it is actually rather simple geometry. The subject more often than not ends up moving in a certain geometrical shape - it could be a rectangle, a square, a parallelogram or a triangle. In these cases, it does help to keep the following in mind:

 a. If the subject moves in a square shaped path: All the sides of a square are equal.
 b. If the subject moves in a rectangular path: The opposite side of a rectangle are equal
 c. If the subject moves in a path similar to a parallelogram: The opposite sides of a parallelogram are equal, just  like a rectangle.
 d. If the subject moves in a triangular path: As in most cases, the subject moves in a straight line, there is a fair chance that its path will take the form of a right triangle. A right triangle, incidentally, is a triangle in which one of the angles is 90 degrees in measure.
 e. If the subject's path [or some part of it] is indeed in the form of a right triangle, you can use Pythagoras' Theorem to find out the measurements of one of its sides, provided the measurements of the other two sides are given [which they usually are]. If you have forgotten the Pythagoras Theorem, relax, because we are explaining it once again below.

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Quite simply, this theorem states that in a right triangle, the square of the hypotenuse [the side opposite the right angle and also the longest side of the triangle] will be equal to the sum of squares of the other two sides.

So, if in a right triangle ABC,
BC= the hypotenuse
And AB and AC are the other two sides,

BC2= AB2 + AC2

Or, BC = v[AB2 + AC2]

The theorem also enables us to find the measurements of the other sides of a triangle, if one is given the measurement of the hypotenuse and any other side. To take the example of our triangle ABC once again

If BC2= AB2 + AC2
Then, AB2 = BC2 - AC2
Or AB = v [BC2 - AC2]

Similarly, AC2= BC2-AB2
Or AC= v [BC2-AB2]

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Some strategies


Your ability to visualise the movement of the subject will determine how successfully you solve direction-based questions. Mind you, visualising is not the easiest task at the best of times, especially if the subject moves in a wide range of directions. Now, unless you are born with a magnetic compass inside your head, tracking movements can be a bit of a pain. Hence the need for some strategies for visualisation. While every person has his or her own special way of visualising movements, here are a few strategies that can help in this regard:

1. The front-to-back strategy: Quite simply, the area in front of you is north, the area behind you is south, the area to the right is east and the area to the left is west. This strategy is the simplest and we will be using it to solve our sample questions.

2. The crosshair strategy: Consider a cross made by a vertical line and a horizontal line. Beginning with the top of the vertical line, move clockwise and label each end point of the lines as north, east, south and west. The place where the lines intersect is the place from which the subject starts travelling and the four segments show different directions.

3. The clock strategy: Quite simply, 12 o'clock is north, 3 o'clock is east, 6 o'clock is south and 9 'clock is west.

4. The face strategy: One can use one's own face to keep a rough track of directions. If your nose is the point of origin, your forehead is north, your right cheek is east, your chin is south and your left cheek is west.

5. Tackling left and right: Some exercises state that the subject moved to its right or to its left. One can tackle these by assuming that one is standing at the point where the turn is needed and taking a turn towards the side stated.

Remember, these strategies only give you a sense of direction. There is no real strategy for visualising how far the subject has moved. That one is up to you - the strategies can only point out the direction of the subject's movement.

Feel free to adopt any of them. Whichever it is, do practice it thoroughly as nothing aids effective visualisation more than practice. And if you do have a strategy of your own which you would like to share, mail it to us at Feedback@enablingdimensions.comWe will post it on the site, and give you due credit for it.

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Points to keep in mind:


1. Sometimes questions contain references to directions such as north-east, south-east, north-west and south west. These directions refer to the regions that cannot be strictly labelled as east west, north or south but fall in between them. For instance, north-west would refer to a direction between north and west! Generally, these directions are meant to indicate the fact that the subject is not travelling in a straight line but is moving diagonally.

2. Unless otherwise specified, it must be remembered that the subject always begins its journey from the very centre.

3. Geographical directions like north, south, east and west are always taken with reference to the centre. However, a turn to the right or the left is made with reference to the subject's current position.

4. When asked how far the subject has to travel to return to the point from which its journey started, assume that the subject will be taking the shortest possible route [a straight line] from his current location.

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Sample questions with solutions


[All these questions have been solved using the front-to-back strategy]

A person walks 6 kilometers to the north, then turns towards the east and walks 4 kilometers. He then turns to the south and walks a further 6 kilometers. How far will he have to travel to reach the place where he started his journey and in which direction?

a. 4 kilometers south b. 5 kilometers west c. 4 kilometers west d. 16 kilometers north e. None of the above

Solution:

1. The objective is to find the direction as well as distance.

2. The person walks 6 kilometers north, that is, in front.

3. He then turns towards east and walks 4 kilometers. The east is on the right hand side.
4. He then turns to the south and walks a further 6 kilometers. South is the area behind you, so the subject actually is walking back in the direction where he started out from [remember, he first went towards the north - in front of you].

5. Now that you have mapped the movement, take a good look at it. The subject goes 6 kilometers, turns right to travel a further 4 kilometers and then walks south for 6 kilometers.

6. The subject's path is taking the shape of an incomplete rectangle. Two opposite sides [the north path and the south path] are equal, both being 6 kilometers.

7. The distance between the subject's current location and his point of origin can be considered to be the fourth side of the rectangle. As the opposite sides of a rectangle are equal, this will be equal to the distance travelled east. Thus, the subject will have to travel the same distance he had travelled to the east to reach his point of origin. Only this time he will have to go in the opposite direction.

8. The subject will therefore have to travel 4 kilometers to the west to reach the place from where he started his journey. The answer is c.

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A person walks 20 meters to the north. He then turns to the right and walks another 30 meters. Once again, he turns to the right and walks 35 meters. He then turns left and walks 15 meters. Finally he turns left yet again and walks another 15 meters. In which direction is he from the starting point?

a. East b. West c. North d. South e. None of these

Solution:

1. The objective is to judge the direction and not the distance.

2. The subject walks 20 meters to the north, that is, in front of him.

3. He then turns to the right, that is, the east, and walks 30 meters.

4. He turns to the right again and walks 35 meters. This is the tricky part because he is moving to his right and not your right hand side. Turn yourself to the right and then turn again to the right. You will find that you are facing the area that was behind you when you started the turns. This means the subject has moved 35 meters to the south.

5. The subject then turns to the left and walks 15 meters. Remember the he is now facing south, so his left will be towards east [when he faced north, his right was towards the east!]

6. Finally, the subject who is now facing the east, turns left yet again and walks a further 15 meters. Turn yourself to the east [or your right] and you will find that your left hand is towards the north. So when the subject moves to his left, he heads 15 meters to the north.

7. Now take a final look at the subject's movements in geographical terms - remember we have to discover where he is in geographical terms from the place where he started. He has gone 20 meters to the north, then 30 meters to the east, followed by 35 meters to the south, 15 meters to the east and finally 15 meters to the north.

8. Visualise the directions and you will see that when the subject in his third move went south [35 meters], he actually went south of the place from where he started his journey. Remember that he had advanced only 20 meters to the north and then headed 30 meters to the east. So he actually went [35 minus 20] 15 meters south of the point from where he started out.

9. After heading south, the subject moved 15 meters to the east and then 15 meters to the north. By moving 15 meters to the north, he actually moved back into line with the place from where he had started the journey [remember, he had gone 15 meters south of it]. But in which direction? Well, he had gone 15 meters east before that. So he is actually to the east of the point from where he started out. The answer is A.

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Kevin drives his car 10 kilometers to the east. He then takes a right turn and travels 2 kilometers. Once again he turns to the right and goes a further 1 kilometer. In which direction is he from the starting point?

a. North b. East c. South-east d North-east e. None of the above

Solution:

1. Once again, the objective here is to find the direction rather than the distance.

2. To begin with, Kevin drives the car 10 kilometers to the east [that is, his right].

3. He then turns to the right and travels 2 kilometers. The right when he is facing east will be towards the south, so Kevin travels 2 kilometers to the south.

4. Finally he turns right again and drives for 1 kilometer. If he is facing the south, his right hand will be towards the west, so he drives 1 kilometer to the west.

5. Now take a final look at Kevin's movements in geographical terms - remember we have to discover where he is in geographical terms from the place where he started. He has gone 10 kilometers to the east, then gone 2 kilometers to the south and finally one kilometer to the west.

6. Visualise the directions and you will see that Kevin goes to the east, then to the south and then to the west. But he has travelled further east [10 kilometers] than he has in the west [1 kilometer]. So his direction is a mixture of south and east, that is, the south-east. The answer is c.

Send us a question


Is there a direction-based reasoning question that you cannot solve? Or a question that you would like to share with other candidates? If so, then do mail them to us, along with details of the examination paper / sample paper in which you found them. We will try to solve them and send the solution to you. Any questions that you send will also be uploaded to the Sample Questions section of the web site and you will be given credit for contributing them.

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