Reasoning: How to Solve Distance and Displacement - Minimum Distance, Pythagoras in Reasoning

Introduction This topic typically carries 10-15 marks in competitive exams, and mastering it can make a huge difference in your score. So, let's dive into the crystal-clear guide on how to solve distance and displacement questions.

WHAT YOU NEED TO KNOW FIRST To tackle distance and displacement questions, you need to have the following basic concepts on your fingertips:

1. Direction Chart: A simple chart to help you visualize the direction of movement.
2. BODMAS: A rule to follow when solving mathematical expressions (Brackets, Orders, Division, Multiplication, Addition, and Subtraction).
3. Sitting Arrangement Conventions: Rules to follow when dealing with seating arrangements, such as "East" being to the right and "North" being upwards.

CRYSTAL‑CLEAR METHOD (Step‑by‑Step) To solve distance and displacement questions, follow these steps:

1. Read the question carefully: Understand what the question is asking for (distance or displacement).
2. Identify the direction: Use your direction chart to determine the direction of movement.
3. Calculate the distance: Use the Pythagoras theorem to calculate the distance (a² + b² = c²).
4. Determine the displacement: If the question asks for displacement, calculate the shortest distance between the initial and final positions.
5. Check the answer: Verify your answer by plugging it back into the question.

DEMO USING A SIMPLE EXAMPLE Let's say we have a question: "A person moves 3 km east and then 4 km north. What is the distance between the initial and final positions?"

Using the steps above, we can calculate the distance as follows:

1. Read the question carefully: We need to find the distance between the initial and final positions.
2. Identify the direction: Using our direction chart, we can see that the person moves east and then north.
3. Calculate the distance: Using the Pythagoras theorem, we can calculate the distance as follows: √(3² + 4²) = √(9 + 16) = √25 = 5 km.
4. Determine the displacement: Since the question asks for distance, we don't need to calculate the displacement.
5. Check the answer: We can verify our answer by plugging it back into the question.

WORKED EXAMPLES

Example 1 – Easy A person moves 5 km east and then 3 km west. What is the distance between the initial and final positions?

1. Read the question carefully: We need to find the distance between the initial and final positions.
2. Identify the direction: Using our direction chart, we can see that the person moves east and then west.
3. Calculate the distance: Since the person moves in opposite directions, we can calculate the distance as follows: 5 km + 3 km = 8 km.
4. Determine the displacement: Since the question asks for distance, we don't need to calculate the displacement.
5. Check the answer: We can verify our answer by plugging it back into the question.

What we learned: When the person moves in opposite directions, we can simply add the distances to find the total distance.

Example 2 – Medium A person moves 2 km north, 3 km east, and then 4 km south. What is the distance between the initial and final positions?

1. Read the question carefully: We need to find the distance between the initial and final positions.
2. Identify the direction: Using our direction chart, we can see that the person moves north, east, and then south.
3. Calculate the distance: Using the Pythagoras theorem, we can calculate the distance as follows: √((2² + 3²) + 4²) = √(4 + 9 + 16) = √29.
4. Determine the displacement: Since the question asks for distance, we don't need to calculate the displacement.
5. Check the answer: We can verify our answer by plugging it back into the question.

What we learned: When the person moves in multiple directions, we can use the Pythagoras theorem to calculate the distance.

Example 3 – Exam‑Style A person moves 5 km east, 3 km north, and then 2 km west. What is the distance between the initial and final positions?

1. Read the question carefully: We need to find the distance between the initial and final positions.
2. Identify the direction: Using our direction chart, we can see that the person moves east, north, and then west.
3. Calculate the distance: Using the Pythagoras theorem, we can calculate the distance as follows: √((5² + 3²) + 2²) = √(25 + 9 + 4) = √38.
4. Determine the displacement: Since the question asks for distance, we don't need to calculate the displacement.
5. Check the answer: We can verify our answer by plugging it back into the question.

What we learned: When the person moves in multiple directions, we can use the Pythagoras theorem to calculate the distance.

Common Mistakes

MISTAKE → WHY IT HAPPENS → CORRECT APPROACH 1. Mistaking displacement for distance: Why it happens: We might get confused between the two concepts. Correct approach: Always read the question carefully and identify what is being asked for. 2. Not using the Pythagoras theorem: Why it happens: We might forget to use the theorem when calculating the distance. Correct approach: Always use the Pythagoras theorem when calculating the distance. 3. Not considering the direction: Why it happens: We might forget to consider the direction of movement. Correct approach: Always use your direction chart to determine the direction of movement. 4. Not checking the answer: Why it happens: We might get lazy and not verify our answer. Correct approach: Always check your answer by plugging it back into the question. 5. Not using the correct formula: Why it happens: We might use the wrong formula to calculate the distance. Correct approach: Always use the correct formula (Pythagoras theorem) to calculate the distance.

EXAM TRAPS

Trap → How to Spot it → How to Avoid it 1. Trap: Using the wrong formula: How to spot it: We might use a formula that is not relevant to the question. How to avoid it: Always read the question carefully and identify the correct formula to use. 2. Trap: Not considering the direction: How to spot it: We might forget to consider the direction of movement. How to avoid it: Always use your direction chart to determine the direction of movement. 3. Trap: Not checking the answer: How to spot it: We might get lazy and not verify our answer. How to avoid it: Always check your answer by plugging it back into the question.

TIME‑SAVING SHORTCUTS

1. Elimination trick: When the question asks for distance, we can eliminate options that are not possible given the direction of movement.
2. Diagram hack: We can use a diagram to visualize the movement and determine the distance.
3. Formula shortcut: We can use the formula √(a² + b²) to calculate the distance quickly.

1‑MINUTE RECAP Hey there, student! It's the night before the exam, and you're feeling confident about your preparation. Let's quickly recap the strategy for solving distance and displacement questions.

First, read the question carefully and identify what is being asked for (distance or displacement). Next, use your direction chart to determine the direction of movement. Then, calculate the distance using the Pythagoras theorem. Finally, check your answer by plugging it back into the question.

Remember to avoid common mistakes like mistaking displacement for distance, not using the Pythagoras theorem, and not considering the direction. Also, be aware of exam traps like using the wrong formula and not checking the answer.

To save time, use elimination tricks, diagram hacks, and formula shortcuts. With this strategy, you'll be able to solve distance and displacement questions quickly and confidently. Good luck on your exam tomorrow!