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Direction Sense Test

Reasoning · Verbal Reasoning

8 Part3 worked examples18 practice questions

A person leaves home and walks 5 km towards North, then turns right and walks 3 km, then turns left and walks 2 km. How far is he from home now, and in which direction? That is the Direction Sense Test. There is no strange formula in it — only drawing the right picture and counting carefully.

In the exam it comes in three forms: (1) final direction — in which direction he is from the starting point, or which way he now faces; (2) distance — how far the straight line from the starting point is; (3) direction and distance together. Questions on the directions between villages, the shadow of the sun and a turned direction post come along with them.

We start from zero: first the eight directions and the arithmetic of turning, then the method for distance (which sometimes needs Pythagoras), and then each kind of question one by one. In every question the final point is fixed by two numbers — the distance East-West and the distance North-South. This chapter teaches you to handle exactly those two numbers.

How to use this chapter

  1. Read — each part first explains the idea, then the method.
  2. Study the solved example — at every step the position (East-West, North-South) is written, with a picture of the walk below.
  3. Try it yourself — draw the route on paper: the house in the middle, North up, an arrow for every leg. Then pick an option. The solution also shows the coordinates after each leg.
  4. Know the level — Basic, Standard, Advanced. Basic has plain turns and short routes; advanced has U-turns, diagonal directions and “cannot be determined” questions.
NoteThe questions you have solved are remembered in this browser. Look at the panel on the right (the “Contents” button on a phone) to see how many you have got right.
1

Foundation — directions, turning, distance

Every question rests on these three tools

The eight directions and their angles, turning left-right and in degrees, and Pythagoras for the distance.

The eight directions and their angles

Four main, four in between — all clockwise

North, East, South, West and the four directions between them; the angle of each direction and its opposite.

There are four main directions — North, East, South, West — and four intermediate directions between them: North-East, South-East, South-West, North-West. On a map North is up and East is to the right; so South is at the bottom and West on the left.

N0°NE45°E90°SE135°S180°SW225°W270°NW315°
Angles measured clockwise from North: North 0°, North-East 45°, East 90° … North-West 315°
  • Trick for the order: clockwise the four main directions come as N – E – S – W.
  • The name of an intermediate direction is made from the two main ones, with North / South written first: North-East (not East-North), South-West.
  • Between two neighbouring main directions the angle is 90°, and between a main and an intermediate direction it is 45°.
  • The opposite of every direction lies at 180°: North ↔ South, East ↔ West, North-East ↔ South-West, South-East ↔ North-West.
DirectionAngleOpposite
North0°South
North-East45°South-West
East90°West
South-East135°North-West
South180°North
South-West225°North-East
West270°East
North-West315°South-East
RememberThe sun rises in the East and sets in the West (roughly) everywhere. When you lose your way this is the surest landmark, and in the exam it is used directly in the shadow questions.
Example 1 · Basic
What is the opposite of North-West?
Solution, step by step
  1. 1North-West is at 315°.
  2. 2For the opposite add 180°: 315° + 180° = 495°.
  3. 3495° is above one full turn (360°), so subtract 360°: 135°. The direction at 135° is South-East.
Answer South-East
The opposite of North-West (North + West) reverses both parts: South + East.
BasicTry it yourself first
Kiran stands facing North. She takes a U-turn, then turns left, then turns left. In which direction is she facing now?
BasicTry it yourself first
Dinesh stands facing East. He turns right, then turns left, then turns right. In which direction is he facing now?
BasicTry it yourself first
Mahesh stands facing South. He turns 270° clockwise, then turns 45° clockwise, then turns 90° clockwise. In which direction is he facing now?

Turning — left, right and in degrees

Right = 90° clockwise, left = 90° anticlockwise

The new facing after every turn, and adding and subtracting clockwise / anticlockwise angles.

A turning question rests on two things: which way he faces now and which way and how much he turned. For a walking person “right” means his own right hand, that is a quarter turn clockwise (90°). “Left” is 90° anticlockwise; “about turn” (U-turn) is 180°.

Facing beforeAfter a right turnAfter a left turnAfter a U-turn
NorthEastWestSouth
EastSouthNorthWest
SouthWestEastNorth
WestNorthSouthEast
TrickThere is no need to memorise the table: North → East → South → West → North … is the clockwise cycle. Right turn = next direction in the cycle, left turn = previous one, U-turn = two ahead.

Turning in degrees

Many questions say “turned 135° clockwise”. Then replace every direction by its angle (North 0°, East 90°, South 180°, West 270°, the in-between directions 45°, 135°, 225°, 315°). Then add the clockwise angles and subtract the anticlockwise ones. If the result is 360° or more, or below 0°, subtract or add 360° (the circle starts again).

Clockwise turns +, anticlockwise turns −; then bring the result between 0° and 360°.
Example 2 · Standard
Dinesh stands facing West. He turns 135° clockwise, then 270° anticlockwise. In which direction is he facing now?
Solution, step by step
  1. 1West is at 270°.
  2. 2Clockwise 135°: 270° + 135° = 405°, and 405° − 360° = 45° (North-East).
  3. 3Anticlockwise 270°: 45° − 270° = −225°; adding 360° gives 135°.
  4. 4The direction at 135° is South-East.
Answer South-East
Do not forget to remove 360° when the circle completes; you may also remove it once at the end.
BasicTry it yourself first
Tarun stands facing East. He turns right, then turns left, then takes a U-turn, then turns right. In which direction is he facing now?
BasicTry it yourself first
Meena stands facing West. She turns left, then turns left, then turns right. In which direction is she facing now?
StandardTry it yourself first
Dinesh stands facing North. He takes a U-turn, then turns left, then turns left. In which direction is he facing now?
StandardTry it yourself first
Om stands facing North. He takes a U-turn, then turns left, then takes a U-turn. In which direction is he facing now?
StandardTry it yourself first
Vandana stands facing South. She turns 90° anticlockwise, then turns 270° clockwise, then turns 135° clockwise. In which direction is she facing now?
StandardTry it yourself first
Bindu stands facing North-East. She turns 270° clockwise, then turns 225° clockwise. In which direction is she facing now?
AdvancedTry it yourself first
Seema stands facing South-East. She turns 270° clockwise, then turns 135° anticlockwise, then turns 45° anticlockwise. In which direction is she facing now?
AdvancedTry it yourself first
Kiran stands facing North. She turns 90° clockwise, then turns 45° clockwise. In which direction is she facing now?

Distance — walked distance versus straight distance

Handle two numbers: East-West and North-South

The difference between the total distance walked and the straight distance from home, coordinate bookkeeping and Pythagorean triples.

Two different things are often mixed up in the exam: the distance walked (the sum of the lengths of all legs) and the straight distance from the starting point (called displacement). If someone walks 3 km East and then 4 km North, he has walked 7 km, but the straight distance from home is only 5 km.

Coordinate bookkeeping

After every leg write two numbers: how far East of the house (West counts negative) and how far North (South counts negative). For example the house (0, 0); 3 km East → (3, 0); then 4 km North → (3, 4). At the end everything comes from these two numbers: the sign tells the direction, the size tells the distance.

Straight distance: Pythagoras

If the end point is a away East-West and b away North-South, then the house, the corner and the end point make a right-angled triangle and the straight distance is its hypotenuse. (If a or b is zero the distance is simply the other number.)

The straight distance
total distance East-West
total distance North-South

In exam questions the numbers are usually such that the hypotenuse comes out whole. Learn to recognise these triples; their multiples (double, triple) are just as useful:

SideSideHypotenuse (straight distance)
345
6810
91215
121620
51213
81517
72425
202129
TrickWhen you get two numbers a and b, first see whether they are 3 : 4, 5 : 12, 8 : 15, 7 : 24 or a multiple of these. For example (15, 20) = 5 × (3, 4), so the hypotenuse is 5 × 5 = 25.
Example 3 · Basic
Om walks 3 km towards South from his house. After that he turns right and walks 4 km. How far (in a straight line) is he now from his house?
Solution, step by step
  1. 1Take East as + and West as −, North as + and South as −, call the house (0, 0), and write the position (East-West, North-South) after every leg.
  2. 21) towards South 3 km → (0, -3); 2) right turn, now towards West 4 km → (-4, -3)
  3. 3The final position is (-4, -3) — that is 4 km East-West and 3 km North-South.
  4. 4These are the two sides of a right angle, so the straight distance = √(4² + 3²) = √25 = 5 km.
Answer 5 km
The straight distance is the hypotenuse — do not add the sides, use Pythagoras.
34ABCN
A = the house (start), the coloured circle = the end; numbers are the distances of the legs; the arrow at the top right shows North (N)
BasicTry it yourself first
Geeta walks 6 m towards South from her house. After that she turns left and walks 8 m. How far (in a straight line) is she now from her house?
BasicTry it yourself first
Jaya walks 5 km towards North from her house. After that she turns left and walks 12 km. How far (in a straight line) is she now from her house?
BasicTry it yourself first
Geeta walks 9 km towards South from her house. After that she turns right and walks 9 km. Next she turns left and walks 3 km. What total distance did she walk?
BasicTry it yourself first
Zoya walks 7 km towards South from her house. After that she turns left and walks 9 km. Next she turns right and walks 8 km. Then she turns left and walks 4 km. What total distance did she walk?
StandardTry it yourself first
Meena walks 4 km towards East from her house. After that she turns right and walks 8 km. Next she turns right and walks 4 km. How far (in a straight line) is she now from her house?
StandardTry it yourself first
Neha walks 9 km towards East from her house. After that she turns right and walks 5 km. Next she turns right and walks 9 km. Then she turns right and walks 3 km. How far (in a straight line) is she now from her house?
StandardTry it yourself first
Mahesh walks 4 km towards North from his house. After that he turns left and walks 5 km. Next he turns right and walks 9 km. Then he turns right and walks 5 km. Then he turns left and walks 6 km. How far (in a straight line) is he now from his house?
2

Final direction (type 1)

In which direction is he now, and which way does he face

Final direction and facing from a route with turns, and questions on the directions between villages.

Walking and turning — final direction and facing

Take the house as (0, 0) and write the position at every leg

The picture of a route with turns, the direction of the end point, the final facing and the way back.

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Directions between villages

Only directions are given, no distances — what is certain then?

The direction of a pair from a chain of statements like “P is to the East of Q”, and “cannot be determined” questions.

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3

Distance (type 2)

How far is the straight line from the starting point

The straight distance as against the distance walked, equal and opposite diagonal legs cancelling, and two persons leaving the same place.

Straight distance — coordinates and Pythagoras

Two totals, then the hypotenuse

Cancelling the going and coming back in a route with turns and getting the straight distance from the two remaining numbers.

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Legs along diagonal directions

Equal and opposite diagonal legs cancel each other

The reckoning of distance when walking North-East, South-West and the like, when two diagonal legs are equal and opposite.

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Two persons leaving the same place

At a right angle, opposite or the same — only three possibilities

The distance between two persons and the direction of one from the other.

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4

Direction and distance together (type 3)

How far, in which direction — and the way back

Questions that ask for distance and direction in one answer, and the direction of the way home.

Distance and direction at once

Direction from the signs of the coordinates, distance from their sizes

Both distance and direction from the final coordinates, the usual slips and the direction of the way home.

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5

The sun, shadows and a direction post

When the direction must be caught from an outside sign

The facing and the hand direction of a person from the morning and evening shadow, and the real direction from a turned direction post.

Morning and evening shadows

From the direction of the sun to the direction of the shadow

In the morning the sun is in the East so the shadow falls West, in the evening the reverse; from this the facing and the direction of the hands.

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The turned direction post

All four arms turn together

Finding the amount of turn of the post from one arm and telling the real direction of any arm.

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6

Strategy guide

The order of the picture and the slips to avoid

What to do first in any direction question, what to do for each form of question, and where slips happen.

What to look at first

Spot the form of the question, then its method

A table of methods by form of question and the firm order for every question.

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Common traps and time-savers

The mistakes that keep happening

The commonest reasons for losing marks in direction questions and how to avoid them.

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7

Mixed practice

All forms mixed — first spot the form, then solve

Turning, final direction, distance, directions between villages, shadows and posts — in three levels.

Basic practice

Plain turns, short routes

Practice of turns, direction and distance on short routes, and easy shadow questions.

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Standard practice

U-turns, degrees, distance + direction, two persons

Routes of up to five legs, turns in degrees, distance and direction of two persons, and diagonal legs.

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Advanced practice

Five legs, U-turns, “cannot be determined” and mixed forms

Long routes, turns by named direction, uncertain relations, diagonal legs, posts and shadows.

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8

Revision — cards

Flip each card and test yourself

Think of the answer yourself first, then flip.

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