Understanding the Bicycle Journey Direction Problem

This question asks us to determine the direction of the boy's starting point relative to his school after a specific bicycle journey. To solve this, we need to track the boy's movements and figure out his final position relative to his start, and then reverse that perspective to find the start's position relative to the final point (the school).

Tracing the Boy's Path and Displacement

Let's break down the boy's journey step-by-step:

1. He starts at a point (let's call it S).
2. He rides 4 km towards North. From S, he is now 4 km North.
3. He takes a right turn. If he was going North, a right turn means he is now going East.
4. He rides 6 km after the right turn. So, from his current position (4 km North of S), he moves 6 km East.
5. He takes a left turn. If he was going East, a left turn means he is now going North.
6. He rides 5 km after the left turn to reach his school (let's call it School). So, from his current position (4 km North and 6 km East of S), he moves another 5 km North.

Determining the School's Position Relative to the Starting Point

Now, let's calculate the total displacement from the starting point (S) to the school:

• Total North displacement: 4 km (initial North) + 5 km (final North) = 9 km North.
• Total East displacement: 6 km (after first turn) = 6 km East.

So, the school is located 9 km North and 6 km East from the starting point (S).

Finding the Starting Point's Direction Relative to the School

The question asks for the direction of the starting point (S) with respect to the location of the school. This means we need to look from the school back towards the starting point.

If the school is 9 km North and 6 km East of the starting point, then the starting point must be located in the opposite direction and distance from the school.

• Opposite of North is South. So, the starting point is 9 km South of the school's North-South line.
• Opposite of East is West. So, the starting point is 6 km West of the school's East-West line.

Therefore, from the school, you would need to go South and West to reach the starting point.

The combination of South and West directions is known as Southwest.

Visualizing the Path

We can think of this using coordinates, with the starting point at the origin (0,0):

• Start (S): (0,0)
• After 4 km North: (0, 4)
• After 6 km East (right turn): (6, 4)
• After 5 km North (left turn): (6, 4 + 5) = (6, 9) - This is the School location.

So, the school is at (6, 9) relative to the start (0,0).

To find the direction of the start (0,0) from the school (6,9), we look at the difference:

• Change in X (East/West): \(0 - 6 = -6\) (which means 6 units West)
• Change in Y (North/South): \(0 - 9 = -9\) (which means 9 units South)

So, the starting point is 9 units South and 6 units West of the school, confirming the Southwest direction.

Summary of Directions

Total displacement of School from Start: 9 km North and 6 km East.

Direction of Start from School: South and West, which is Southwest.

Conclusion

Based on the movements described, the starting point is located to the Southwest with respect to the school's location.

Revision Table: Key Concepts for Direction Problems

Additional Information: Solving Direction and Distance Questions

Direction and distance problems are common in reasoning and aptitude tests. They often involve tracking a person's movement through a series of steps, including turns and distances. Here are some tips for solving them:

• Draw a diagram: Sketching the path helps visualize the movements and final position. Use arrows to indicate direction and label distances.
• Use a reference point: Usually, the starting point is taken as the origin or reference.
• Break down movements: Separate movements along North-South and East-West axes.
• Calculate net displacement: Sum up movements in the same direction and subtract movements in opposite directions for each axis (e.g., total North - total South = Net North/South).
• Determine final position relative to start: The net North/South and East/West displacements give the final position relative to the start.
• Determine start position relative to final: If the final position is X North and Y East of the start, the start is X South and Y West of the final position.
• Identify the final direction: Based on the final net displacement (or the reversed displacement), determine the direction (e.g., North-East, South-West).

Practice with different scenarios, including varying distances and multiple turns, to improve your understanding and speed.