Time, Speed and Distance Explained with Formulas and Examples - Aptitude Questions & Answers

Category: Quantitative Aptitude Views: 14

Time, Speed and Distance Explained with Formulas and Examples is one of the most important topics in Quantitative Aptitude. In this lesson, you will learn concepts, formulas, shortcuts, solved examples, and aptitude questions with answers. This topic is useful for exams like SSC, Bank, CAT, TCS, and other competitive exams.

Time, Speed and Distance: Complete Guide with Formulas & Examples

What are Time, Speed and Distance?

Time, Speed and Distance are fundamental concepts in quantitative aptitude that deal with motion. These concepts are used to calculate how fast objects move, how long they take to travel, and the distances they cover. Understanding these relationships is crucial for solving problems in competitive exams and real-world scenarios.

Basic Relationship: Distance = Speed × Time

Basic Concepts and Formulas

Distance

The total length of path traveled by an object. Distance is a scalar quantity (magnitude only).

Distance = Speed × Time
Unit: meters (m), kilometers (km), miles (mi)

Example: If a car travels at 60 km/h for 3 hours, distance = 60 × 3 = 180 km

Time

The duration over which an object moves to cover a certain distance.

Time = Distance / Speed
Unit: seconds (s), minutes (min), hours (hr)

Speed

Rate at which an object covers distance. Speed is a scalar quantity.

Speed = Distance / Time
Unit: m/s, km/h, mph

Unit Conversion Formulas

km/h to m/s Conversion

1 km/h = 1000 m / 3600 s = 5/18 m/s
To convert km/h to m/s: Multiply by 5/18

Example: Convert 72 km/h to m/s

72 × (5/18) = 20 m/s

m/s to km/h Conversion

1 m/s = (1/1000) km / (1/3600) h = 18/5 km/h
To convert m/s to km/h: Multiply by 18/5

Example: Convert 15 m/s to km/h

15 × (18/5) = 54 km/h

Average Speed Formulas

General Formula

Average Speed = Total Distance / Total Time

Example Problem: A truck covers 900 km in 30 hours and next 500 km in 40 hours. Find average speed.

Total distance = 900 + 500 = 1400 km
Total time = 30 + 40 = 70 hours
Average speed = 1400/70 = 20 km/h

Special Case 1: Equal Distances

When distances are equal at speeds x and y:
Average Speed = 2xy / (x + y)

Example: Raman goes Delhi to Pune at 40 km/h and returns at 60 km/h

Average speed = (2 × 40 × 60) / (40 + 60) = 4800/100 = 48 km/h

Special Case 2: Equal Time Intervals

When time intervals are equal at speeds x and y:
Average Speed = (x + y) / 2

Example: Car travels at 48 km/h for 1 hour, then 36 km/h for 1 hour

Average speed = (48 + 36) / 2 = 42 km/h

Relative Speed Concepts

Objects Moving in Opposite Directions

Relative Speed = Speed₁ + Speed₂

Example Problem: Two cars 140 km apart move towards each other at 45 km/h and 25 km/h. When will they meet?

Relative speed = 45 + 25 = 70 km/h
Time = Distance / Relative speed = 140/70 = 2 hours

Objects Moving in Same Direction

Relative Speed = |Speed₁ - Speed₂|

Example Problem: Two bikes 50 km apart move in same direction at 40 km/h and 30 km/h. When will they meet?

Relative speed = 40 - 30 = 10 km/h
Time = 50/10 = 5 hours

Boat and Stream Problems

Upstream Speed

Moving against the direction of stream/current.

Upstream Speed = Boat Speed in Still Water - Stream Speed

Example: Boat speed in still water = 16 km/h, Stream speed = 3 km/h

Upstream speed = 16 - 3 = 13 km/h

Downstream Speed

Moving with the direction of stream/current.

Downstream Speed = Boat Speed in Still Water + Stream Speed

Example: Boat speed in still water = 9 km/h, Stream speed = 4 km/h

Downstream speed = 9 + 4 = 13 km/h

Finding Boat and Stream Speeds

Boat Speed = (Downstream Speed + Upstream Speed) / 2
Stream Speed = (Downstream Speed - Upstream Speed) / 2

Example: Downstream = 20 km/h, Upstream = 10 km/h

Boat speed = (20 + 10)/2 = 15 km/h

Stream speed = (20 - 10)/2 = 5 km/h

Train Problems

Train Crossing a Stationary Object

Distance = Length of Train
Time = Length of Train / Speed of Train

Example: Train 264 m long passes pole in 24 seconds

Speed = 264/24 = 11 m/s

Train Crossing a Platform/Bridge

Total Distance = Length of Train + Length of Platform
Time = (Train Length + Platform Length) / Train Speed

Example Problem: Train 260 m long crosses bridge in 30 seconds at 45 km/h

Speed = 45 × (5/18) = 12.5 m/s
Let bridge length = x meters
Total distance = 260 + x
30 = (260 + x)/12.5
260 + x = 375
x = 115 meters

Two Trains Moving in Same Direction

Relative Speed = Difference of Speeds
Time to cross = (Length₁ + Length₂) / Relative Speed

Example: Trains 700 m and 500 m long at 12 m/s and 13 m/s in same direction

Relative speed = 13 - 12 = 1 m/s

Time = (700+500)/1 = 1200 seconds

Two Trains Moving in Opposite Directions

Relative Speed = Sum of Speeds
Time to cross = (Length₁ + Length₂) / Relative Speed

Example: Trains 1100 m and 900 m long at 12 m/s and 13 m/s in opposite directions

Relative speed = 12 + 13 = 25 m/s

Time = (1100+900)/25 = 80 seconds

Speed-Time Inverse Relationship

When distances are equal:
Speed ∝ 1/Time
S₁/S₂ = √(T₂/T₁)

Example Problem: Two trains meet at a point. After meeting, Train A takes 4 hours to destination, Train B takes 16 hours. Speed ratio?

S₁/S₂ = √(16/4) = √4 = 2:1

Train A is twice as fast as Train B

Race and Circular Track Problems

Linear Race Concepts

A Beats B

A finishes before B. Difference in time or distance can be calculated.

Example: In 100m race, A beats B by 10m means when A finishes, B has covered 90m

Dead Heat

Both finish at exactly the same time.

Head Start

One participant starts before others or from ahead.

Linear Race Calculations

Example Problem: In 450m race, X beats Y by 60m or 20 seconds

When X finishes, Y has covered 450-60 = 390m
Y\'s speed = 60m/20s = 3 m/s
Y\'s time for full race = 450/3 = 150 seconds

Example Problem: In 200m race, X gives Y 40m headstart and beats by 10s. With 80m headstart, dead heat. Find speeds.

Difference in headstarts = 80-40 = 40m
Difference in time = 10 seconds
Y\'s speed = 40/10 = 4 m/s
With 80m headstart: Y runs 120m, X runs 200m
Time same (dead heat): 120/4 = 200/X_speed
X_speed = (200×4)/120 = 20/3 ≈ 6.67 m/s

Circular Track Problems

Meeting at Starting Point

Time when all meet at start = LCM of individual lap times

Example: Track = 2400m, Speeds = 9, 18, 27 km/h

Convert to m/s: 2.5, 5, 7.5 m/s

Lap times: 960s, 480s, 320s

LCM(960,480,320) = 960s = 16 minutes

Meeting Anywhere on Track

Time to meet = Track Length / Relative Speed
For same direction: Use difference of speeds
For opposite direction: Use sum of speeds

Time, Speed and Distance: Frequently Asked Questions

What\'s the difference between speed and velocity?

Speed is scalar (magnitude only), velocity is vector (magnitude and direction). In most aptitude problems, we use speed and ignore direction unless specified.

How to convert km/h to m/s quickly?

Multiply by 5/18. For mental calculation: divide by 2, then by 3.6. Example: 72 km/h → 72/2=36 → 36/3.6=10 m/s (actual: 20 m/s, so this is approximate).

When to use 2xy/(x+y) formula?

Use when equal distances are covered at two different speeds. Example: Going and returning same distance at different speeds.

How to solve train problems with platforms?

Remember: Total distance = Train length + Platform length. Time = (Train length + Platform length) / Speed.

What\'s the key to circular track problems?

For meeting at starting point, find LCM of lap times. For meeting anywhere, use relative speed concepts similar to linear tracks.

Practice Problems

Problem 1

A man travels 600 km by train at 80 km/h, 800 km by ship at 40 km/h, and 100 km by car at 50 km/h. Find average speed for entire journey.

Solution: Total distance = 1500 km. Train time = 600/80=7.5h, Ship time=800/40=20h, Car time=100/50=2h. Total time=29.5h. Average speed=1500/29.5≈50.85 km/h.

Problem 2

Two trains 150m and 120m long run at 45 km/h and 54 km/h respectively. Find time taken to pass each other when running in opposite directions.

Solution: Relative speed = 45+54=99 km/h = 27.5 m/s. Total distance=150+120=270m. Time=270/27.5=9.82 seconds.

Problem 3

A boat goes 30 km upstream and 44 km downstream in 10 hours. It goes 40 km upstream and 55 km downstream in 13 hours. Find boat and stream speeds.

Solution: Let u=upstream speed, d=downstream speed. 30/u+44/d=10, 40/u+55/d=13. Solve: u=5 km/h, d=11 km/h. Boat speed=(11+5)/2=8 km/h, Stream speed=(11-5)/2=3 km/h.

Problem 4

In a 100m race, A beats B by 10m and B beats C by 10m. By what distance does A beat C?

Solution: When A runs 100m, B runs 90m. When B runs 100m, C runs 90m. So when A runs 100m, C runs 90% of 90m=81m. A beats C by 19m.

Problem 5

Two persons start from opposite ends of 90km road. One cycles at twice the speed of other. If they meet in 2 hours, find speeds.

Solution: Let speed of slower=x km/h, faster=2x km/h. Relative speed=3x km/h. Distance=90km, Time=2h. So 3x=90/2=45 → x=15 km/h. Speeds: 15 km/h and 30 km/h.

Important Formulas Summary

Basic Formulas:
• Speed = Distance/Time
• Distance = Speed × Time
• Time = Distance/Speed

Unit Conversion:
• km/h to m/s: × 5/18
• m/s to km/h: × 18/5

Average Speed:
• General: Total Distance/Total Time
• Equal distances: 2xy/(x+y)
• Equal times: (x+y)/2

Relative Speed:
• Same direction: |x-y|
• Opposite direction: x+y

Boats & Streams:
• Upstream = b - s
• Downstream = b + s
• Boat speed = (d+u)/2
• Stream speed = (d-u)/2

Trains:
• Crossing object: Time = Length/Speed
• Crossing platform: Time = (L_train + L_platform)/Speed

Frequently Asked Questions

What is Time, Speed and Distance Explained with Formulas and Examples?

Time, Speed and Distance Explained with Formulas and Examples is an important aptitude topic used in competitive exams that tests your logical reasoning and problem-solving abilities.

Is Time, Speed and Distance Explained with Formulas and Examples important for competitive exams?

Yes, Time, Speed and Distance Explained with Formulas and Examples is frequently asked in SSC, Bank, CAT, TCS, and other placement exams. It's essential to master this topic for better scores.

How to prepare Time, Speed and Distance Explained with Formulas and Examples easily?

Practice solved examples, learn formulas and shortcuts, and attempt practice questions regularly to master Time, Speed and Distance Explained with Formulas and Examples.

What are the important formulas in Time, Speed and Distance Explained with Formulas and Examples?

Key formulas vary by topic, but generally include basic concepts, shortcuts, and standard problem-solving approaches specific to Time, Speed and Distance Explained with Formulas and Examples.

How many questions come from Time, Speed and Distance Explained with Formulas and Examples?

Typically 5-10 questions come from Time, Speed and Distance Explained with Formulas and Examples in most competitive exams, making it a high-scoring section.

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