The topics based on age, directions, clock, arrangement etc. are found in logical reasoning. Typical 11 Plus exam papers have not shown much presence of these topics. But these are being asked in 2nd stages of certain exams and also in UKMT maths challenges. Typically, in such type of questions, the question stem contains a number of instructions regarding the movement of a person. We are asked to find the final location of the person or distance travelled by him or distance between initial or final position.
Introduction
Before doing this topic, the following points should be kept in mind.
- While attempting direction test, one must first understand directions. The following figure shows the four main directions i.e. North, South, West and East. Also four cardinal directions (North-East (NE), North-West (NW) South-East (SE) and South-West (SW) are shown to help students to know the direction
- One should start solving the question taking the initial position as origin
- For solving direction reasoning questions, it is critical that we have a direction sense after movement. Keep in mind the change in direction when a person takes right or left.
- One should keep in mind that in finding the distance between initial final position shortest distances is to be taken and use of by Pythagoras theorem is there.
Following direction and distance reasoning questions will help you cement your concepts:
Example No. 1: Abhishek starts walking from his house and goes 200 metres towards east. Then he turns to his left and walks 100 metres. Thereafter, he again turns to his left and walks 200 metres. In which direction and at what distance is he from his house now?
Solution: In this case, we have to do as it has been indicated by the examiner. First of all, he asks us to move 200 meters towards east. Thereafter, He asks us to move to our left, i.e. the north. Again, he requires us to move to our left, i.e. towards the west. As illustrated in the figure alongside, a rectangle is formed consisting of two sides 200 meters and 100 meters. Obviously, the person is 100 meters away from his starting point and at that moment, he is in the northern direction as compared to the starting point.
Example No. 2 : Gabriella starts from home and walks 80 metres towards the south. She turns to her left and walks 50 metres. She again turns to her right and walks 40 metres. She takes a turn towards her left and walks 60 metres. She takes a final turn towards her left and walks 120 metres. Where and at what distance is Gabriella from the starting point?
Solution : As usual, our starting point will be the point where the N-S and E-W lines intersect, as has been illustrated alongside. We follow her movement as has been asked of us in the question and finally arrive at a point in the east direction. Now the problem before us is to find her distance from the starting point. We will have to calculate it indirectly because there is no direct way to do so. If you look carefully, the two distances i.e. 50 and 60 metres, if joined together give us the distance traveled by her on the eastern side, taken from the starting point. If you transpose this line connecting 50 and 60 metres, you will see that this is exactly the distance traveled by her on the eastern side. So the answer in this case is 110 metres east.
Example No. 3 :In the evening at 8 p.m, Bittu starts from his office to meet his friend Bunty. He walks 60 metres towards the east, then takes a left turn and walks 80 metres. Find his distance from the starting point and in which direction is he now?
Solution: Now, this is posing a bit difficult question. As you can see for yourself, Bittu is somewhere in the middle of the northern and eastern direction. This is where you can make the use of Pythagorean Theorem. Look carefully, these two distances, 80 and 60 metres, are forming a right-angle triangle, and by applying the Pythagoras theorem we can get the third side.
Just for review, the Pythagorean Theorem states that the square of the hypotenuse of a right-angled triangle is equal to the sum of the squares of the other sides of the right angle.
Let us assume that the third (unknown) side is designated X.
Applying the theorem, X2 = (60)2 + (80)2 Þ X2 = 3600 + 6400 = 10000
Therefore, X = Ö10000 = 10 metres. Now he is in the middle of the northern and the eastern sides. Therefore, combing these two sides, we say he is at present in the north-east 10 metres away from his starting point. Do remember that it is not necessary for the person to be exactly in the middle of the two sides for you to say that he is in the north-east direction. Even if he is just 10 degrees in the east and mostly in the north, even then it should be stated that he is in the north-east direction.
Example No. 4: One morning Sandra and John were talking to each other face to face at a crossing. If John's shadow was exactly to the left of Sandra, which direction was Sandra facing?
Solution:
One morning John and Sandra were talking to each other face to face at a crossing. If John's shadow was exactly to the left of Sandra, Sandra was facing in the North direction.
Example No. 5: Y is in the East of W which is in the North of A. If Q is in the South of W, then in which direction of Y, is Q?
Solution:
Y is in the East of W which is in the North of A. If Q is in the South of W, then Q is in South-West of Y.
Example No. 6:
One morning after sunrise, Ali and Kavya were talking to each other face to face at Lake Square. If Kavya's shadow was exactly to the right of Ali, which direction Kavya was facing?
Solution:
In morning, sun rises in the east and so any shadow falls towards the west. Now, Kavya’s shadow falls to the right of Ali. Hence, Ali is facing South and Kavya is facing North.
Example No. 7 :
One morning after sunrise, Andrew and Lawrence were standing in lawn with their backs towards each other. Andre's shadow fell exactly towards left hand side. Which direction was Shaan facing?
Solution:
Since Andrew's shadow fell towards left, therefore, Andrew is facing north. So, Lawrence standing with his back towards Vipin, will be facing South.