Time and Distance (நேரம் மற்றும் தூரம்)

Welcome to the foundational guide for Time, Speed, and Distance. This topic is crucial for almost every competitive exam, and mastering it is easier than you think.

🧠 Foundation: Understanding the Core Concept (அடிப்படை கருத்துகள்)

What is Speed? (வேகம் என்றால் என்ன?)

Speed is simply a measure of how fast an object moves. It tells you the distance covered in a specific unit of time.

Real-world examples:

• 🚶 Walking: Approx. 5 km in 1 hour → Speed = 5 km/hr
• 🚲 Cycling: Approx. 15 km in 1 hour → Speed = 15 km/hr
• 🚗 Car: Approx. 60 km in 1 hour → Speed = 60 km/hr

The Master Relationship

The entire topic revolves around one simple, powerful formula.

The Core Formula

Distance = Speed × Time
(தூரம் = வேகம் × நேரம்)

Why does this work? Imagine you are driving at a speed of 60 km/hr for 3 hours.

• In the 1st hour, you cover 60 km.
• In the 2nd hour, you cover another 60 km.
• In the 3rd hour, you cover yet another 60 km.
• Total Distance: 60 km/hr × 3 hr = 180 km.

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Level 0: Formula Playground

Let's get comfortable manipulating the core formula.

Formula Variations (சூத்திரத்தின் வகைகள்)

To Find	Formula	Tamil (தமிழ்)
Distance (தூரம்)	Speed × Time	வேகம் × நேரம்
Speed (வேகம்)	Distance ÷ Time	தூரம் ÷ நேரம்
Time (நேரம்)	Distance ÷ Speed	தூரம் ÷ வேகம்

Practice Drills

1. Speed = 60 km/hr, Time = 3 hours. Find Distance.

   • Answer: Distance = 60 × 3 = 180 km

2. Distance = 240 km, Time = 4 hours. Find Speed.

   • Answer: Speed = 240 ÷ 4 = 60 km/hr

3. Distance = 150 km, Speed = 30 km/hr. Find Time.

   • Answer: Time = 150 ÷ 30 = 5 hours

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⚙️ Unit Conversion Mastery (அலகு மாற்றம்)

A Critical Skill

90% of errors in this topic come from incorrect unit conversions. Master this section before moving forward.

Key Conversion Factors

• km/hr → m/s: Multiply by 5/18
• m/s → km/hr: Multiply by 18/5

The Logic Behind 5/18

Why this specific fraction? Let's break it down:

• 1 kilometer = 1000 meters
• 1 hour = 3600 seconds

So, 1 km/hr = 1000 meters / 3600 seconds. Simplifying this fraction gives us 10/36, which further simplifies to 5/18.

1 km 1000 m 5 m ------ = -------- = ---- 1 hr 3600 s 18 s

Conversion Practice

6. Convert 72 km/hr to m/s.

   • Answer: 72 × (5/18) = 4 × 5 = 20 m/s

7. Convert 25 m/s to km/hr.

   • Answer: 25 × (18/5) = 5 × 18 = 90 km/hr

8. Convert 108 km/hr to m/s.

   • Answer: 108 × (5/18) = 6 × 5 = 30 m/s

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Level 1: Problem Reading Strategy (கேள்வியை அணுகும் முறை)

The G-F-V Method

Before jumping into calculations, always pause and identify these three things:

G-F-V Framework

• Given: What information is provided? (கொடுக்கப்பட்டவை)
• Find: What is the question asking for? (கண்டுபிடிக்க வேண்டியது)
• Verify: Do the units match? (அலகுகள் சரியாக உள்ளதா?)

Simple Word Problems (எளிய வார்த்தைக் கணக்குகள்)

11. A car travels 200 km in 4 hours. What is its speed?

    • Given: Distance = 200 km, Time = 4 hours
    • Find: Speed
    • Verify: Units are km and hours, so the answer will be in km/hr. No conversion needed.
    • Solution: Speed = 200 ÷ 4 = 50 km/hr

12. A train runs at 80 km/hr for 3 hours. How much distance does it cover?

    • Given: Speed = 80 km/hr, Time = 3 hours
    • Find: Distance
    • Solution: Distance = 80 × 3 = 240 km

13. A bus covers 450 km at a speed of 90 km/hr. How much time does it take?

    • Given: Distance = 450 km, Speed = 90 km/hr
    • Find: Time
    • Solution: Time = 450 ÷ 90 = 5 hours

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Level 2: Pattern Recognition (கணக்குகளின் வகையை அறிதல்)

Now that the basics are solid, we move to the most important skill: identifying the problem pattern.

Key Problem Patterns

Pattern Type	English Keywords	Core Concept	Tamil (தமிழ்)
Meeting/Crossing	meet, approach, opposite	Add Speeds	சந்திப்பு / எதிர் திசை
Overtaking	catch up, overtake, same direction	Subtract Speeds	முந்துதல் / பின்தொடர்தல்
Boats & Streams	downstream, upstream, current	Add/Subtract Stream Speed	நீரோட்டம்
Average Speed	multiple stages, different speeds	Total Distance / Total Time	சராசரி வேகம்

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Pattern 1: Meeting/Crossing Problems (சந்திப்பு கணக்குகள்)

Keywords: meet, approach, towards each other, opposite directions

Core Insight: When two objects move towards each other, their speeds combine. We find their Relative Speed by adding their individual speeds.

Visual Diagram:

 <--- Distance Apart --->
A -----> <----- B
(Speed A) (Speed B)

Relative Speed = Speed A + Speed B

16. Two trains of lengths 100m and 150m are running in opposite directions at speeds of 45 km/hr and 55 km/hr respectively. In how much time will they completely cross each other?
    • Total Distance to Cover: When trains cross, the distance is the sum of their lengths.
      • Distance = 100m + 150m = 250m
    • Relative Speed: They move in opposite directions, so we add their speeds.
      • Speed = 45 + 55 = 100 km/hr
    • Unit Conversion: The distance is in meters, so convert speed to m/s.
      • 100 km/hr × (5/18) = 250/9 m/s
    • Time: Time = Distance / Speed
      • Time = 250 / (250/9) = 250 × (9/250) = 9 seconds

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Pattern 2: Chasing/Overtaking Problems (பின்தொடர்தல் கணக்குகள்)

Keywords: catch up, overtake, same direction, chase

Core Insight: When one object chases another in the same direction, the speed at which the gap closes is the difference between their speeds.

Visual Diagram: Initial Gap: [ Distance ] Thief -----> (Slower Speed)

Policeman --------> (Faster Speed)

Relative Speed = Faster Speed - Slower Speed

20. A thief is spotted by a policeman from a distance of 200m. If their speeds are 8 km/hr and 10 km/hr respectively, how much time will the policeman take to catch the thief?
    • Initial Gap (Distance): 200m
    • Relative Speed: They move in the same direction, so we subtract speeds.
      • Speed = 10 km/hr - 8 km/hr = 2 km/hr
    • Unit Conversion: Convert speed to m/s to match the distance unit.
      • 2 km/hr × (5/18) = 5/9 m/s
    • Time: Time = Distance / Speed
      • Time = 200 / (5/9) = 200 × (9/5) = 40 × 9 = 360 seconds
    • Final Answer: 360 seconds = 6 minutes

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Pattern 3: Boats & Streams Problems (படகு மற்றும் நீரோட்ட கணக்குகள்)

Keywords: downstream, upstream, with current, against current

Core Insight: The stream's current either helps (downstream) or hinders (upstream) the boat's actual speed.

• Downstream Speed (நீரோட்டத்தின் திசையில்): Boat Speed + Stream Speed
• Upstream Speed (நீரோட்டத்திற்கு எதிராக): Boat Speed - Stream Speed

Visual Diagram:

Downstream: Boat -----> + Stream -----> = Effective Speed (Faster)
Upstream: Boat -----> - <----- Stream = Effective Speed (Slower)

24. A boat travels 30 km downstream in 2 hours and 18 km upstream in 3 hours. Find the speed of the boat in still water and the speed of the stream.
    • Downstream Speed (D): 30 km / 2 hr = 15 km/hr
    • Upstream Speed (U): 18 km / 3 hr = 6 km/hr
    • Formulas:
      • Boat Speed = (D + U) / 2
      • Stream Speed = (D - U) / 2
    • Boat Speed: (15 + 6) / 2 = 21 / 2 = 10.5 km/hr
    • Stream Speed: (15 - 6) / 2 = 9 / 2 = 4.5 km/hr

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Level 3: Advanced Scenarios & Average Speed (சிக்கலான கணக்குகள் மற்றும் சராசரி வேகம்)

Keywords: average speed, multiple stages, stops, return journey

Common Trap!

Average speed is NOT the simple average of the speeds. It is always Total Distance / Total Time.

28. A person travels first 120 km at 60 km/hr, next 120 km at 40 km/hr, and the last 120 km at 30 km/hr. Find his average speed.
    • Total Distance: 120 + 120 + 120 = 360 km
    • Calculate Time for each part:
      • Time 1 = 120 km / 60 km/hr = 2 hours
      • Time 2 = 120 km / 40 km/hr = 3 hours
      • Time 3 = 120 km / 30 km/hr = 4 hours
    • Total Time: 2 + 3 + 4 = 9 hours
    • Average Speed: Total Distance / Total Time = 360 km / 9 hours = 40 km/hr

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⚠️ Common Beginner Mistakes (பொதுவான தவறுகள்)

1. Unit Mixing: Using km/hr with meters or seconds without conversion.

   • Fix: Always convert to a consistent set of units (either km/hr or m/s) before calculating.

2. Average Speed Error: Calculating (Speed1 + Speed2) / 2.

   • ❌ Wrong: For speeds 60 and 40, average is (60+40)/2 = 50.
   • ✅ Correct: You must find Total Distance / Total Time.

3. Relative Speed Confusion: Adding speeds when you should subtract.

   • Fix:
     • Opposite Direction → Speeds Add up.
     • Same Direction → Speeds Subtract.

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🚀 Key Takeaways (முக்கிய குறிப்புகள்)

1. Foundation First: Master D = S × T and unit conversions (5/18).
2. Identify the Pattern: Before you solve, ask: Is this a meeting, chasing, or boat problem?
3. Check Units: Always verify that all units are consistent.
4. Logical Check: Does the answer make sense? (e.g., time to catch up can't be negative).

Final Tip: Success in this topic is a formula itself: Mastery = (Strong Basics + Pattern Recognition) × Consistent Practice

Happy learning!

📝 Problems to Practice (பயிற்சிக் கேள்விகள்)

Test your understanding with these carefully selected problems. Each problem is designed to reinforce a specific pattern.

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Pattern 1: Meeting/Crossing Problems Practice

Problem 1

Two cars start from cities A and B simultaneously and travel towards each other. Car from A travels at 50 km/hr and car from B travels at 70 km/hr. If the distance between the cities is 360 km, after how much time will they meet?

Solution: Pattern Identification: This is a Meeting/Crossing Problem because two objects are moving "towards each other" from different starting points.

Step-by-step Solution:

• Given: Speed of Car A = 50 km/hr, Speed of Car B = 70 km/hr, Distance between cities = 360 km
• Pattern Rule: When objects move towards each other → Add their speeds
• Relative Speed: 50 + 70 = 120 km/hr
• Time to Meet: Distance / Relative Speed = 360 km / 120 km/hr = 3 hours

Problem 2

Two trains 150m and 200m long are running towards each other at speeds of 40 km/hr and 50 km/hr respectively. In how many seconds will they cross each other completely?

Solution: Pattern Identification: This is a Meeting/Crossing Problem because two trains are moving "towards each other" and we need to find crossing time.

Step-by-step Solution:

• Given: Train 1 length = 150m, Train 2 length = 200m, Speed 1 = 40 km/hr, Speed 2 = 50 km/hr
• Total Distance to Cover: Sum of lengths = 150m + 200m = 350m
• Pattern Rule: Trains moving towards each other → Add speeds
• Combined Speed: 40 + 50 = 90 km/hr
• Unit Conversion: 90 km/hr × (5/18) = 25 m/s
• Time to Cross: 350m / 25 m/s = 14 seconds

Problem 3

A and B start walking from two points 15 km apart towards each other at 9 AM. A walks at 4 km/hr and B walks at 5 km/hr. At what time will they meet?

Solution: Pattern Identification: This is a Meeting/Crossing Problem because A and B are walking "towards each other" from different points.

Step-by-step Solution:

• Given: Initial distance = 15 km, Speed of A = 4 km/hr, Speed of B = 5 km/hr, Start time = 9 AM
• Pattern Rule: Moving towards each other → Add speeds
• Combined Speed: 4 + 5 = 9 km/hr
• Time to Meet: 15 km / 9 km/hr = 5/3 hours = 1 hour 40 minutes
• Meeting Time: 9:00 AM + 1:40 = 10:40 AM

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Pattern 2: Chasing/Overtaking Problems Practice

Problem 4

A thief running at 8 km/hr is chased by a policeman running at 10 km/hr. If the policeman is 100m behind initially, how long will it take for him to catch the thief?

Solution: Pattern Identification: This is a Chasing/Overtaking Problem because the policeman is chasing the thief, both moving in the same direction.

Step-by-step Solution:

• Given: Thief speed = 8 km/hr, Policeman speed = 10 km/hr, Initial gap = 100m
• Pattern Rule: Same direction movement → Subtract speeds
• Relative Speed: 10 - 8 = 2 km/hr
• Unit Conversion: 2 km/hr × (5/18) = 5/9 m/s
• Time to Catch: 100m / (5/9) m/s = 100 × (9/5) = 180 seconds = 3 minutes

Problem 5

Two trains are traveling in the same direction. Train A is 300m long and travels at 72 km/hr. Train B is 200m long and travels at 54 km/hr. How much time will Train A take to completely overtake Train B?

Solution: Pattern Identification: This is a Chasing/Overtaking Problem because both trains move in the same direction and the faster one overtakes the slower one.

Step-by-step Solution:

• Given: Train A length = 300m, Speed A = 72 km/hr, Train B length = 200m, Speed B = 54 km/hr
• Total Distance: For complete overtaking = 300m + 200m = 500m
• Pattern Rule: Same direction → Subtract speeds
• Relative Speed: 72 - 54 = 18 km/hr
• Unit Conversion: 18 km/hr × (5/18) = 5 m/s
• Time to Overtake: 500m / 5 m/s = 100 seconds

Problem 6

Runner A starts at 8 AM at 6 km/hr. Runner B starts at 9 AM from the same point in the same direction at 8 km/hr. When will B catch up with A?

Solution: Pattern Identification: This is a Chasing/Overtaking Problem because B is chasing A, both running in the same direction, with A having a head start.

Step-by-step Solution:

• Given: A starts at 8 AM at 6 km/hr, B starts at 9 AM at 8 km/hr
• A's Head Start: A runs for 1 hour before B starts
• Distance covered by A: 6 km/hr × 1 hr = 6 km (initial gap)
• Pattern Rule: Same direction → Subtract speeds
• Relative Speed: 8 - 6 = 2 km/hr
• Time for B to catch up: 6 km / 2 km/hr = 3 hours after B starts
• Meeting Time: 9:00 AM + 3:00 = 12:00 PM (Noon)

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Pattern 3: Boats & Streams Problems Practice

Problem 7

A motorboat takes 2 hours to travel 20 km downstream and 4 hours to travel the same distance upstream. Find the speed of the boat in still water and the speed of the stream.

Solution: Pattern Identification: This is a Boats & Streams Problem because it involves downstream and upstream motion with current affecting the boat's effective speed.

Step-by-step Solution:

• Given: Downstream: 20 km in 2 hours, Upstream: 20 km in 4 hours
• Calculate Effective Speeds:
  • Downstream speed = 20 km / 2 hr = 10 km/hr
  • Upstream speed = 20 km / 4 hr = 5 km/hr
• Pattern Formulas:
  • Boat speed in still water = (Downstream + Upstream) / 2
  • Stream speed = (Downstream - Upstream) / 2
• Boat Speed: (10 + 5) / 2 = 7.5 km/hr
• Stream Speed: (10 - 5) / 2 = 2.5 km/hr

Problem 8

A swimmer can swim at 4 km/hr in still water. If the river current is 2 km/hr, how long will it take him to swim 9 km downstream and return to the starting point?

Solution: Pattern Identification: This is a Boats & Streams Problem involving a round trip with downstream and upstream segments.

Step-by-step Solution:

• Given: Swimming speed = 4 km/hr, Stream speed = 2 km/hr, Distance each way = 9 km
• Calculate Effective Speeds:
  • Downstream speed = 4 + 2 = 6 km/hr
  • Upstream speed = 4 - 2 = 2 km/hr
• Calculate Times:
  • Time downstream = 9 km / 6 km/hr = 1.5 hours
  • Time upstream = 9 km / 2 km/hr = 4.5 hours
• Total Time: 1.5 + 4.5 = 6 hours

Problem 9

An airplane flies 600 km with the wind in 2 hours and 400 km against the wind in 2 hours. Find the speed of the airplane in still air and the wind speed.

Solution: Pattern Identification: This is a Boats & Streams Problem (air version) because wind affects the airplane's effective speed, similar to how current affects boats.

Step-by-step Solution:

• Given: With wind: 600 km in 2 hours, Against wind: 400 km in 2 hours
• Calculate Effective Speeds:
  • Speed with wind = 600 km / 2 hr = 300 km/hr
  • Speed against wind = 400 km / 2 hr = 200 km/hr
• Pattern Formulas:
  • Airplane speed in still air = (With wind + Against wind) / 2
  • Wind speed = (With wind - Against wind) / 2
• Airplane Speed: (300 + 200) / 2 = 250 km/hr
• Wind Speed: (300 - 200) / 2 = 50 km/hr

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Pattern 4: Average Speed Problems Practice

Problem 10

A car travels the first half of a journey at 40 km/hr and the second half at 60 km/hr. Find the average speed for the entire journey.

Solution: Pattern Identification: This is an Average Speed Problem involving two different speeds for equal distances.

Step-by-step Solution:

• Given: First half at 40 km/hr, Second half at 60 km/hr
• Key Insight: For equal distances, we cannot simply average the speeds
• Assume Total Distance: Let total distance = 120 km (LCM of 40 and 60 for easy calculation)
• Calculate Times:
  • Time for first half (60 km) = 60 km / 40 km/hr = 1.5 hours
  • Time for second half (60 km) = 60 km / 60 km/hr = 1 hour
• Total Time: 1.5 + 1 = 2.5 hours
• Average Speed: Total Distance / Total Time = 120 km / 2.5 hr = 48 km/hr
• Quick Formula: For equal distances: Average = (2 × S1 × S2) / (S1 + S2) = (2 × 40 × 60) / (40 + 60) = 4800 / 100 = 48 km/hr

Problem 11

A person walks 3 km in 30 minutes, then runs 6 km in 20 minutes, then walks again 3 km in 40 minutes. Find his average speed for the entire journey.

Solution: Pattern Identification: This is an Average Speed Problem with multiple segments at different speeds and times.

Step-by-step Solution:

• Given: Walk 3 km in 30 min, Run 6 km in 20 min, Walk 3 km in 40 min
• Pattern Rule: Average Speed = Total Distance / Total Time
• Calculate Totals:
  • Total Distance = 3 + 6 + 3 = 12 km
  • Total Time = 30 + 20 + 40 = 90 minutes = 1.5 hours
• Average Speed: 12 km / 1.5 hr = 8 km/hr

Problem 12

A bus travels from city A to city B at 45 km/hr and returns from B to A at 55 km/hr. If the total journey time is 10 hours, find the distance between the cities.

Solution: Pattern Identification: This is an Average Speed Problem involving a round trip with different speeds for each direction.

Step-by-step Solution:

• Given: Speed A to B = 45 km/hr, Speed B to A = 55 km/hr, Total time = 10 hours
• Let distance between cities = d km
• Calculate Times:
  • Time A to B = d / 45 hours
  • Time B to A = d / 55 hours
• Set up Equation: Total time = Time A to B + Time B to A
  • 10 = d/45 + d/55
• Solve for d:
  • 10 = d(1/45 + 1/55) = d(55 + 45)/(45 × 55) = d(100)/2475
  • d = 10 × 2475 / 100 = 24750 / 100 = 247.5 km

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Practice Strategy (பயிற்சி உத்தி)

1. Pattern Recognition First: Always identify the pattern before calculating
2. Formula Selection: Use the appropriate formula based on the pattern
3. Unit Consistency: Check that all units match before final calculation
4. Logical Verification: Does your answer make practical sense?