Aryan runs at a speed on 40 metre/ minute. Rahul follows him after an interval of 5 minutes and runs at a speed of 50 metre/minute. Rahul’s dog runs at a speed of 60 metre/minute and starts along with Rahul. The dog reaches Aryan and then comes back to Rahul, and continues to do so till Rahul reaches Aryan. What is the total distance covered by the dog?

1. Fundamentals of Speed, Distance, and Time (basic)

Welcome to the first step of your journey into quantitative aptitude! To master motion-based problems, we must start with the most fundamental building block: Speed. Simply put, speed is the distance covered by an object in a unit of time—this unit could be a second, a minute, or an hour Science-Class VII, NCERT (Revised ed 2025), Chapter 8, p. 113. It tells us how fast or slow an object is moving by comparing the distance it travels within that specific timeframe.

The relationship between speed, distance, and time is governed by a single, powerful mathematical triad. If you know any two of these variables, you can always find the third:

• Speed = Total Distance ÷ Total Time
• Distance = Speed × Time
• Time = Total Distance ÷ Speed

For instance, if a bus moves at a constant speed of 50 km/h for 2 hours, it will cover a distance of 100 km Science-Class VII, NCERT (Revised ed 2025), Chapter 8, p. 115.

In your exams, you will encounter two types of motion. When an object covers equal distances in equal intervals of time, it is in Uniform Linear Motion. However, in daily life, most objects exhibit Non-Uniform Motion, where their speed changes—like a train that starts slowly, picks up speed, and then slows down to a halt Science-Class VII, NCERT (Revised ed 2025), Chapter 8, p. 116-117. In such cases, we often calculate the Average Speed, which is the total distance divided by the total time taken Science-Class VII, NCERT (Revised ed 2025), Chapter 8, p. 118.

Sources: Science-Class VII, NCERT (Revised ed 2025), Chapter 8: Measurement of Time and Motion, p.113; Science-Class VII, NCERT (Revised ed 2025), Chapter 8: Measurement of Time and Motion, p.115; Science-Class VII, NCERT (Revised ed 2025), Chapter 8: Measurement of Time and Motion, p.116-117; Science-Class VII, NCERT (Revised ed 2025), Chapter 8: Measurement of Time and Motion, p.118

2. Average Speed and Total Distance Concepts (basic)

At its most fundamental level, speed is the measure of how much distance an object covers in a specific unit of time — whether that unit is a second, a minute, or an hour Science-Class VII, Chapter 8, p.113. While we often talk about speed as a constant number, in the real world, motion is rarely perfectly steady. A car navigating city traffic slows down at signals and speeds up on empty stretches; this is known as non-uniform motion Science-Class VII, Chapter 8, p.118. To simplify these complex movements, we use the concept of Average Speed, which treats the entire journey as if the object moved at one consistent rate.

The beauty of this concept lies in its mathematical reliability. The formula for speed is the bedrock of quantitative aptitude:

Speed = Total Distance Covered / Total Time Taken

From this single relationship, we can derive two other vital equations to find any missing variable in a problem:

• Total Distance = Speed × Time
• Time Taken = Distance / Speed

When you are solving a problem, always ensure your units are consistent. For instance, if distance is in kilometers and time is in hours, your speed must be in km/h. If you need to compare a horse galloping at 18 m/s to a train moving at 72 km/h, you must convert them to the same unit first Science-Class VII, Chapter 8, p.118.

In competitive exams, a common pitfall is assuming that the average speed is simply the average of two different speeds. This is rarely true. To find the true average speed, you must always return to the first principle: calculate the entire distance traveled and divide it by the entire time the journey took Science-Class VII, Chapter 8, p.119. Whether an object is moving in a straight line or a zigzag, if you know the total duration and the total path length, the average speed provides a complete bird's-eye view of the motion.

Sources: Science-Class VII, Measurement of Time and Motion, p.113; Science-Class VII, Measurement of Time and Motion, p.115; Science-Class VII, Measurement of Time and Motion, p.118; Science-Class VII, Measurement of Time and Motion, p.119

3. Unit Consistency and Conversions (basic)

In quantitative aptitude, the Golden Rule of Unit Consistency is simple: you cannot perform mathematical operations on values unless they share the same unit family. If a distance is given in kilometres but the speed is in metres per second, any direct calculation will lead to a wrong answer. We must first convert them into a uniform system. As per the standard SI system, speed is fundamentally defined as Distance / Time, usually expressed in metres per second (m/s), though kilometres per hour (km/h) is more common in daily transport scenarios. Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.113

To master these conversions quickly, you should memorize the relationship between km/h and m/s. Since 1 km = 1000 metres and 1 hour = 3600 seconds (60 min × 60 s), then 1 km/h = 1000/3600 m/s, which simplifies to 5/18. This fraction is the most powerful shortcut in speed-distance problems. While most terrestrial problems use these units, specialized fields like maritime navigation use knots (nautical miles per hour), where 1 international knot equals approximately 1.852 km/h or 0.514 m/s. Physical Geography by PMF IAS, Tropical Cyclones, p.372

To Convert From	To	Operation
km/h	m/s	Multiply by 5/18
m/s	km/h	Multiply by 18/5
Minutes	Hours	Divide by 60

Sources: Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.113; Physical Geography by PMF IAS, Tropical Cyclones, p.372

4. Ratio and Proportion in Speed-Time Problems (intermediate)

At the heart of every speed-time problem lies a simple yet powerful relationship: Speed = Distance / Time. As we explore in Science-Class VII . NCERT(Revised ed 2025), Chapter 8, p.113, speed is essentially the distance covered by an object in a unit of time (such as a second, minute, or hour). However, for competitive exams like the UPSC, the real magic happens when we look at this formula through the lens of Ratio and Proportion. By understanding how these three variables interact when one of them is held constant, we can solve complex chase or meeting problems without getting bogged down in heavy algebra.

There are two critical proportional relationships you must master:

• Inverse Proportion (Distance is Constant): If the distance to be traveled is fixed, speed and time are inversely proportional (s ∝ 1/t). This means if you double your speed, you will take half the time to reach your destination.
• Direct Proportion (Time is Constant): If two objects travel for the same duration, the distance they cover is directly proportional to their speeds (d ∝ s). As noted in Science-Class VII . NCERT(Revised ed 2025), Chapter 8, p.115, the faster object will always cover a greater distance in that same unit of time.

Constant Factor	Relationship	Mathematical Ratio
Distance (d)	Speed is Inversely Proportional to Time	s₁/s₂ = t₂/t₁
Time (t)	Distance is Directly Proportional to Speed	d₁/d₂ = s₁/s₂

In "chase" scenarios, we often use Relative Speed to find the time taken to close a gap. If two people are moving in the same direction, their relative speed is the difference between their individual speeds. Once you calculate the time required for the faster person to catch the slower person (Time = Gap / Relative Speed), you can then treat that time as a "constant" to find out how far a third moving object (like a bird or a dog) has traveled during that exact same interval.

Sources: Science-Class VII . NCERT(Revised ed 2025), Chapter 8: Measurement of Time and Motion, p.113; Science-Class VII . NCERT(Revised ed 2025), Chapter 8: Measurement of Time and Motion, p.115

5. Connected Topic: Time and Work Efficiency (intermediate)

In quantitative aptitude, Efficiency is essentially the rate at which work is performed. Think of it as the "speed" of doing a task. Just as speed is distance covered per unit of time Science-Class VII . NCERT(Revised ed 2025) | Measurement of Time and Motion | p.113, efficiency is the amount of work completed per unit of time. The fundamental relationship to remember is: Work = Efficiency × Time. If the total work to be done remains constant, efficiency and time are inversely proportional. This means a person who is twice as efficient as another will take exactly half the time to complete the same task.

This concept is closely mirrors the idea of Productivity in economics. For instance, if 5 laborers produce 2 tonnes of grain, their average productivity (or efficiency) is 0.4 tonnes per laborer Indian Economy, Vivek Singh (7th ed. 2023-24) | Fundamentals of Macro Economy | p.20. In competitive exams, efficiency is often expressed as a percentage or a ratio. If we say "A is 200% as efficient as B," it implies that in the same timeframe, A does twice the work that B does. Consequently, if B takes 20 days to finish a project, A would finish it in just 10 days.

When multiple people work together, their individual efficiencies are additive, provided they don't hinder each other. For example, if Machine X produces 10 units/hour and Machine Y produces 15 units/hour, their combined efficiency is 25 units/hour. This logic is fundamental when calculating how long a team will take to complete a project. We also look at Marginal Productivity—the additional output generated by adding one more unit of input (like an extra worker)—to understand how efficiency changes as a team grows Microeconomics (NCERT class XII 2025 ed.) | Production and Costs | p.51.

Scenario	Efficiency Relationship	Time Relationship
A is 3 times as fast as B	Efficiency Ratio (A:B) = 3:1	Time Ratio (A:B) = 1:3
A is 25% more efficient than B	Efficiency Ratio (A:B) = 125:100 = 5:4	Time Ratio (A:B) = 4:5

Sources: Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.113; Indian Economy, Vivek Singh (7th ed. 2023-24), Fundamentals of Macro Economy, p.20; Microeconomics (NCERT class XII 2025 ed.), Production and Costs, p.51

6. Relative Speed: Objects Moving in the Same Direction (exam-level)

Every movement we observe in the universe is relative. As we learn in basic physics, motion depends entirely on the observer's frame of reference. For instance, if you are sitting on a merry-go-round turning anti-clockwise, a stationary tree outside will appear to be moving in the opposite direction, or clockwise, relative to you Science-Class VII, Earth, Moon, and the Sun, p.170. When we apply this to two objects moving in the same direction, the core concept is simple: the relative speed is the difference between their individual speeds. If you are driving at 60 km/h and a car overtakes you at 70 km/h, it doesn't feel like they are zooming past at a high velocity; rather, it feels like they are slowly pulling away from you at just 10 km/h (70 - 60). In competitive exams, this concept is most frequently tested through "chase" or "catch-up" scenarios. To solve these, you must identify two critical components: the Relative Speed and the Initial Gap (or head start). If Object A starts moving before Object B, you first calculate how much distance Object A covered during that solo time. This distance becomes the 'gap' that Object B needs to close. Because both are moving in the same direction, Object B closes this gap at a rate equal to the difference in their speeds Science-Class VII, Chapter 8, p.113.

The fundamental formula for these problems is:

Scenario	Formula
Relative Speed (Same Direction)	Speed₁ - Speed₂ (where S₁ > S₂)
Time to Catch Up	Initial Distance / Relative Speed

Sources: Science-Class VII . NCERT(Revised ed 2025), Earth, Moon, and the Sun, p.170; Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.113

7. Head Start and Catch-up Time Scenarios (exam-level)

In competitive exams, Head Start and Catch-up scenarios are classic applications of Relative Speed. A head start occurs when one object begins its journey earlier than another, creating an initial distance gap. To solve these, we don't look at the absolute distances immediately; instead, we focus on how quickly that gap is being closed.

According to the principles of motion, speed is defined as the distance covered in a unit of time Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.113. When two objects move in the same direction, the Relative Speed is the difference between their individual speeds. This relative speed is the actual rate at which the 'gap' between them shrinks. If the pursuer is not faster than the leader, the catch-up will never happen!

To master these problems, follow this logical sequence:

1. Find the Head Start Distance: Multiply the speed of the first person by the time lead they have.
2. Calculate Relative Speed: Subtract the slower speed from the faster speed (Speed₂ - Speed₁).
3. Determine Catch-up Time: Divide the Head Start Distance by the Relative Speed.
4. The 'Continuous Traveler' Trick: If a third entity (like a dog or a bird) travels back and forth during the entire pursuit, its total distance is simply its speed multiplied by the total catch-up time calculated in step 3. You don't need to track its individual turns!

Step	Formula/Logic	Purpose
1. Gap Analysis	Distance = Speed × Time Lead	Finds the 'buffer' to be closed.
2. Closing Rate	Relative Speed = V_fast - V_slow	Determines how much the gap shrinks per minute/hour.
3. Time to Meet	Time = Gap / Relative Speed	The duration of the entire pursuit.

Sources: Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.113

8. Solving the Original PYQ (exam-level)

This problem is a classic application of the Relative Speed and Head Start concepts you have just mastered. To solve it efficiently, you must first determine the "catch-up time." Since Aryan starts 5 minutes earlier, he creates a head start of 200 metres (40 m/min × 5 min). Once Rahul begins his pursuit, he closes this gap at a relative speed of 10 m/min (50 m/min - 40 m/min). By dividing the head start by the relative speed, we find that it takes Rahul exactly 20 minutes to reach Aryan. This duration is the crucial link because the dog remains in motion for this entire 20-minute period.

The dog’s back-and-forth movement is a common conceptual trap designed to trick students into calculating a complex infinite series of individual trips. However, the logic remains simple: the dog travels at a constant speed for a specific duration. By applying the formula Distance = Speed × Time, we multiply the dog’s speed (60 m/min) by the total time (20 minutes) to arrive at the correct answer of 1200 metres (Option D). This "big picture" approach is essential for the CSAT, as it helps you avoid unnecessary complexity by focusing on the total time of motion, a principle grounded in Science-Class VII . NCERT(Revised ed 2025).

The other options represent common pitfalls in competitive exams. Option (A) 600 metres might be reached if a student incorrectly uses a 10-minute duration, perhaps by forgetting the head start calculation. Options (B) and (C) are distractors that do not follow the Relative Speed logic. Always remember: in "dog and runner" problems, do not get distracted by the path; focus solely on the total time the animal is moving at its constant speed.